the effort to design and build a quantum computer is motivated by the discovery of a number of quantum algorithms that are more efficient than their best known classical equivalents .some practical benefit could arise from these algorithms with a quantum computer of only tens or hundreds of qubits , but a quantum computer with thousands of logical qubits will be required to outperform the most powerful classical computers . to mitigate the effects of decoherence and systematic imprecisionit is expected that fault tolerant error correction will be required at the expense of additional quantum resources . motivated by the fact that a limited amount of resources can be allocated to error correction , here we present a new method for error correction which uses message passing to achieve lower error rates than conventional methods that incur the same resource cost .  with the notable exceptions of topological quantum computing and the surface code ,most methods for error correction involve concatenation .our result is in this context and so we briefly review this approach .  first , information represented by the state of a single physical qubit is encoded in the states of several physical qubits , which together form the logical qubit of an error correction code . using error correction circuitry constructed from physical operations ( including state preparation , measurement , and quantum logic gates such as hadamard and controlled - not ) some number of errors affecting the data qubits that make up the logical qubit can be detected and corrected . typically this is done by interacting the data qubits with some ancillary qubits and then measuring the ancillary qubits to extract a syndrome from which errors are diagnosed .then , to form a concatenated quantum code , this process is applied recursively - that is , logical qubits are used to encode higher level logical qubits and logical operations are used to construct higher level error correction circuits .it is known that by concatenating a quantum code , an arbitrarily accurate and arbitrarily large quantum computation can be performed efficiently with faulty components provided that the failure rate of all physical operations is below some threshold . with each level of concatenation the failure rate of the logical operationsreduces double - exponentially , however both the number of physical qubits to encode each logical qubit and the time required to perform each logical operation increase exponentially .this increase in resources makes it impractical to use more than just a few levels of concatenation .conventional implementations of quantum error correction implicitly assume that errors are equally likely on all data qubits and so the correction is always to apply the fewest possible bit or phase flips required to return the state to the code space .error correction at each level of concatenation operates independently in this way .however , not only does error correction reduce the probability of a logical error , it also gives some indication of the likelihood that an error has occurred at a higher level of encoding - a logical error is more likely to have occurred if a correction was applied during error correction than if a correction was not applied .if this classical information is passed from lower levels of error correction to higher levels it is possible to correct a larger set of errors .message passing in quantum computation was first considered in the context of an error detection code , where errors at the level below would become located errors at the next level up and thus be corrected .the idea was also applied to an error correction code in the context of communication across a noisy channel .  herewe extend this idea by applying message passing to fault tolerant error correction under the nine - qubit bacon - shor subsystem code .in particular we present a method for error correction at the second level of concatenation which uses both the regular syndrome and messages from the first level of concatenation .we refer to this method as message passing error correction ( mpec ) . using simulations we find that mpec will correct all cases with four physical errors , whereas conventional error correction can only guarantee success with up to three physical errors .we also show that mpec retains fault tolerance and so for physical error rates well below the threshold the use of this method can lead to a significant improvement in the fidelity of a large scale computation relative to conventional error correction .note that while message passing error detection has been shown to increase the asymptotic threshold , our method is designed to improve the performance of error correction without requiring additional resources .we do not explicitly consider any effect that message passing may have on the asymptotic threshold .in this paper we focus entirely on improving the performance of the second level of concatenated error correction using the [ [ 9,1,3 ] ] bacon - shor code .the bacon - shor error correction circuits we use are shown in fig .[ fig : ec circuit ] .  in the [ [ 9,1,3 ] ] codea logical failure can occur if at least two of the data qubits have errors .when the error correction circuit measures a non - zero syndrome we know that at least one of the data qubits has an error and that any additional error may cause failure . to have a logical error and observe no syndrome requires at least three physical errors .therefore logical errors are of order more likely at logical circuit locations which measured non - zero syndrome . at these level-1 locations , which have detected an error at the physical level, we raise a _ flag _ to indicate an increased probability of level-1 error .these flags are simply classical information that can be tracked through the quantum circuit .bit ( ) and phase ( ) errors can be treated independently .there will be flags and flags each indicating increased probability of the respective errors .since the and error correction circuits are similar the same rules apply independently to both and correction .table [ tab : outcomes ] shows the four possible outcomes of a level-1 error correction block ..possible outcomes of a level-1 error correction block .the error correction block is _ flagged _ if a non - zero syndrome is measured , indicating that at least one error has occurred .error correction can only detect deviations from the code space , and so logical failures can not be detected without higher level error correction . in terms of the probability of a physical gate failure , ,the weights of the outcomes are and respectively . [ cols="^,^,^ " , ]using conventional concatenation each level of error correction works independently .the level of error correction can correct any single gate failure at the level below .all gates in the error correction circuit are assumed to be equally likely to fail , and so when a non - trivial error syndrome is measured the lowest possible number of errors is assumed and the corresponding correction is applied .information about the syndrome and the corrections applied is then thrown away . in our methodwe use this same information to estimate the relative probability of gate failures at a higher level .the key ideas are that more errors can be corrected if the locations of the errors are known and that logical failure is more likely when a non - trivial correction is applied .that is , there is a significant correlation between flags and logical failure .codeword states with two errors share the same set of syndromes as codeword states with single errors .however , with the additional flag information , is it possible to distinguish between a larger set of errors .we call the combined information of the flags and the measured syndrome the _ super syndrome_. at the second level of error correction , the syndrome of a single may be indistinguishable from that of two , and so without message passing , four physical errors ( if they result in two ) can cause a logical error at level 2 .when the full super syndrome is considered one is distinguishable from two and so we can attempt to correct two level-1 errors at once .cases with two can be accurately diagnosed and corrected . share the same super syndrome with two , but is weight in terms of the physical error rate .it turns out that all weight 4 cases can be corrected using mpec .when the error correction circuitry itself is made from noisy gates the problem is significantly more complicated .the error correction circuits are designed to be fault tolerant - a fault at any given physical location will not spread to multiple data qubits in the same logical qubit . however, this type of fault tolerance does not guarantee the protection of the properties that make message passing assisted error correction superior .for example , consider an encoded level-1 cnot that is part of a level-2 error correction circuit .a single physical error during this cnot could result in a flag being raised on one or both of the qubits depending on the location of the error .two physical errors could result in , , , , etc .[ fig : cnot errors ] ) . with three physical errors ,all combinations of flags and failures are possible .therefore not all two cases are weight 4 for example .this effect complicates the process of interpreting the super syndrome and ensuring fault tolerance .we found that not all fault tolerant circuits for conventional error correction can be readily adapted to mpec .  to provide the best possible error correction, we aim to apply a correction corresponding to the most likely cause of the super syndrome .we assume a low physical error rate , and so the most likely cause is the one with the fewest physical errors .here is an overview of how our scheme works : the error correction circuit used is shown in fig .[ fig : ec circuit ] . and error correction are treated separately .there are flags and flags , which represent potential and errors respectively .a flag is raised on a level-1 qubit whenever lower level error correction measures a non - zero syndrome .each flag has its own identity .the propagation of each flag is tracked through the circuit as if it were an actual error , spreading through cnot gates . flags copy downwards through cnot gates and flags copy upwards .when the second level syndrome is measured we look for combinations of one , two , or three flags on the ancilla which if corresponded to actual errors would produce the syndrome that was measured .such a combination is called a _flag match_. ( for examples of this see figures in appendix a. ) if a flag match is found , we apply corrections to the data qubits that share the same flags as used in the flag match .some super syndromes have more than one flag match , in these cases we preference the match which implies the fewest errors and _ results in the fewest corrections to the data qubits_. if no flag match is found , we fall back to the default error correction action ; ie .we correct based on the syndrome alone . after error correction ,all flags found on the ancillary qubits at the point of measurement are cleared from the data qubits - these flags have either been established to be ( and have been corrected ) or , so they are of no further use .  note that although has equal weight to , triple flag matches must still be preferenced over the default correction to avoid particular cases in which four physical errors would otherwise cause level-2 failure due to certain combinations of 2-qubit gate failures .( see appendix a. )  further levels of concatenated error correction will compound the benefits of mpec .however , making use of message passing at all levels is not so easy . finding a flag scheme that works at all levels is non - trivial , andso for this paper we have focused on just two levels of error correction .a simple method for concatenation beyond level 2 is to alternate between error correction that provides flags for the next level , and error correction that uses flags from the previous level but provides no flags . for this alternating sequencethe number of physical errors required for logical failure at increasing levels of concatenation is ; in contrast with the regular scaling , , for the same physical resources .the procedure described in the previous section ensures that _ no combination of four physical errors will result in a level-2 logical error_. this is an improvement over conventional concatenation which can only guarantee the correction of three physical errors at level-2 .however , to show that our scheme is effective for large circuits we must show that an arbitrary length chain of level-2 locations will not fail for any combination of four physical errors in each level-2 extended rectangle .an extended rectangle is defined to be an encoded gate with error correction before and after .  usually , fault tolerance of error correction circuits is considered from the point of view that all levels act independently .the basic rule is that a single error should not spread to multiple errors on the same data qubit . for mpecthe fault tolerant condition is more complicated .since corrections are based on the entire super syndrome we must take into account the error weight of each of the possibilities in table [ tab : outcomes ] rather than just counting the failures .  to ensure fault tolerance, an mpec box must satisfy the following condition : where is the error weight of the state entering the mpec box , is the weight of errors occurring inside the box , and is the error weight of the state leaving the box after any corrections have been applied .error weights are calculated according to table [ tab : outcomes ] . evidence that the mpec scheme satisfies the condition can be found in appendix a. a proof that this condition guarantees the performance of the error correction can be found in appendix b.to test our message passing error correction we simulate a level-2 cnot extended rectangle .we compare the failure rates with and without message passing .the failure rate of this circuit is meant to approximate the failure rate of the level-2 cnot which might be an algorithmic location .  to simulate the error correction circuit we need only simulate the propagation of errors that occur during the circuit .this avoids having to store the complete state of the computer which we assume is an arbitrary codeword state perturbed by the errors .the circuit is deemed to fail if a readout of the data qubits at the end of the circuit in either the or basis would not produce the correct output .equivalently , a circuit is defined to have succeeded if an errorless error correction cycle applied to its output state would produce the correct state .  for perform many simulations of the entire two level circuit with a fixed error rate and tally the failures . at each physical circuit locationwe apply an error with probability . for all single qubit locations ( preparation , measurement , memory )the error is a randomly selected single qubit pauli error , , , or . for two - qubit locations ( cnot )the error is a randomly selected two - qubit pauli error ( , , , etc . )errors at all locations are independent .  for error rates below number of trials required to generate sufficiently accurate statistics directly is too large ; so we use an alternative approach . instead of giving each circuit location some probability of error , we simulate the circuit with exactly errors placed randomly .this is repeated many times for $ ] to generate the probability that the circuit fails given that there were errors .these conditional probabilities can be combined to give the failure rate of the circuit as a function of , where is the number of locations in the entire circuit , 72657 .  to ensure randomness over the large number of trials required we use the simd - oriented mersenne twister pseudo random number generator .the results of the simulations are in fig .[ fig : results ] .note that eq .[ eq : expansion ] is truncated after after which is set to equal zero .this is why the curves drop to zero at higher error rates .a second set of lines connect the direct data points taken above .our simulations indicate that all combinations of four errors are corrected by mpec .this is in contrast to standard error correction which fails for some four error combinations . for values of close to the threshold , ,the failure rate is not significantly reduced , but for the failure rate is reduced by a factor of the superscripts indicate standard error correction and message passing error correction .in the region of interest - for physical error rates where two or three levels of error correction is sufficient for logical computation - mpec is significantly better than standard quantum error correction .we emphasize that this benefit is achieved only at the cost of additional classical processing . for more than two levels of error correctionthe benefit can be compounded , but the problem of finding an optimal and general flagging scheme ( one that can be applied at all levels ) is an open question and the subject of further work . also of interestis that the complexity of correctly interpreting the super syndrome appears to depend on the circuits that are used to extract the syndrome .can mpec work fault tolerantly for other codes such as the steane code and if so is it compatible with the most compact steane circuits ?we acknowledge the assistance of anthony agius , geoff attwater , magdalena carrasco , andrew greentree , charles hill , and lloyd hollenberg .we acknowledge the support of the australian research council ( arc ) , the us national security agency ( nsa ) , and the army research office ( aro ) under contract number w911nf-04 - 1 - 0290 .the primary purpose of this section is to demonstrate that the circuits and rules described in section [ sec : method ] satisfy the fault tolerance condition , eq .[ eq : ft rule ] . in doingso we also give examples of flag propagation and flag matching which could be useful for a general understanding of the mpec scheme .it is important to note that the [ [ 9,1,3 ] ] bacon - shor subsystem code has some symmetries which can be used to greatly simplify the fault tolerance analysis .the nine data qubits form a 3x3 grid for which the rows essentially form a repetition code that protects against errors and the columns a repetition code that protects against errors ( fig .[ fig : ec circuit ] ) .the code is defined to take the binary addition of the columns / rows for the syndrome , so that any even number of errors on a column is benign and even number of errors on a row is benign .the entire error correction circuit consists of three sets of the circuit shown in fig .[ fig : ec circuit ] run in parallel followed by three sets of the equivalent syndrome extraction . as and error correctionare similar and operate independently , and all errors and flags from circuits performed in parallel are added in binary at end of the syndrome extraction , only a single block ( the circuit in fig .[ fig : ec circuit ] ) need be considered .all errors and flags on parallel circuit blocks are equivalent to the same errors and flags on a single block .although these symmetries are used for our explanations and diagrams , our simulations always used the full circuits with both and errors .cnot gates copy errors from the control to the target , and copy errors from the target to the control . flags represent potential errors and so they are propagated in the same way that the errors are .each flag that is raised is given a unique identity , but they can by copied to multiple qubits by cnot gates in the same way that errors are . for the diagrams in this section the flags are distinguishable by their patterns .a set of flags is said to match the syndrome if identifying those flags as errors can exactly describe the measured syndrome .when testing for a flag match , each ancillary qubit has its flags added in binary , flags that are not part of the set to be tested are ignored , if the result of all the ancillary qubits is identical to the syndrome then the set of flags is a match .  the figures shown in this section depict the propagation of flags and errors on the second level of error correction .it is implicit that at every circuit location there is a lower level of error correction being performed which can raise flags .these flags are indicated by the flags drawn on the circuits .crosses indicate a level-1 error ( or correction at the end of the circuit ) . the propagation of the errors is shown using bold lines . at the end of the circuit ,the figures show the values of the syndrome measurements and the flags that have propagated onto the ancilla .the flags used in a flag match are highlighted and corrections are shown by crosses at the end of the circuit .the flags and crosses result from physical errors during the lower level error correction .the error weights of these locations can be inferred from table [ tab : outcomes ] .errors during the cnot gates can result in flags and errors on both qubits ( fig .[ fig : cnot errors ] ) . in these casesthe number of physical errors during the cnot limit the error weight of each logical qubit .  to demonstrate that eq .[ eq : ft rule ] is satisfied ( ie . that the error weight of the output from an ec box is no greater than the number of errors occurring inside the box )we reason on a case by case basis with cases grouped according to the weight of the incoming errors .[ fig : mpec fail1 ] shows how a single can cause failure by masking the flags of two . for this failure to occur, the two and the must each affect different qubit lines ( rows for errors , columns for errors ) .[ fig : mpec fail2 ] shows that a pair of can match the syndrome of a and thus cause the wrong correction to be applied .again , each of the and the must affect different qubit lines . if there is a flag from an that follows the same path as a , the combination of the and the is equivalent to an .a single incoming can not be masked by one or two .the two can be positioned to make a two - flag match , but mpec will always prefer to use the single flag match which is provided by the .this time , we have the freedom to place a single anywhere within the circuit . but this does nt affect any of arguments given for .an can not interfere with the correction of a single incoming or no matter where it is placed .one of the important properties of the ec circuit we have chosen is that no single can mask an error to prevent its correction , because each incoming error on the data will affect two of the ancillary qubits .this property is not a feature of all ( conventionally ) fault tolerant circuits .cnot errors can that result in pairs in addition to an incoming ( fig .[ fig : mpec3 ] ) , but not in any configuration that will cause problems .the and from cnot failure can be thought of as an effect on the same data qubit line . with this in mindit is clear that no situation like fig .[ fig : mpec fail1 ] can occur without extra errors .with there must be an incoming on one of the data qubits .three physical errors inside the ec box is still only enough to cause failure at one location , but this time there may be a .if flag from the matches the syndrome , this will result in successful correction .there can be a that produces a syndrome with no flag match .when there is no flag match mpec falls back to the conventional bacon - shor error correction rules for this circuit .this will either result in a successful correction of the failure ( if it occurs before or during the first set of cnots ) or a parity mismatch on the syndrome and hence no correction .with it is acceptable to let a or leave the ec box , eg .[ fig : mpec4 ] .   with there can either be two separate flagged failure locations , or one location and one location .as we have already seen , the cases are all work in the same way as they do for conventional error correction .the worst case is shown in fig .[ fig : mpec5 ] , where two pairs of physical errors case two pairs .when this happens there will be a two flag match which will not correct all of the errors but will not cause logical failure . since it is acceptable for the ec box to let through a single or even two .with there are still not enough errors to affect two data qubits with errors and have a flag on the third line ( fig .[ fig : mpec fail1 ] ) or to have a on one data qubit that matches misleading flags on the other two qubit lines ( fig .[ fig : mpec fail2 ] ) .having demonstrated in appendix [ sec : ap .a ] that eq .[ eq : ft rule ] is satisfied , in this section we aim to prove that improved failure rate scaling for mpec extended rectangles found in section [ sec : simulation ] carries over to arbitrary length logical computation .for simplicity we consider only identity gates at the logical level , but it is not hard to extend this proof to an arbitrary logical circuit given standard requirements of fault tolerant circuits , which are already satisfied by the bacon - shor code ( eg . transversal gates ) .   an error correction box is defined to have succeeded if an errorless error correction cycle applied to its output state would produce the correct state .ie . if an ec box succeeded then any errors remaining on the output can be corrected .let be the physical error weight of the input state to the ec box , and be the number of physical errors occurring inside the ec box .let denote the success of the ec box .then the success condition can be written as    no initial errors : all extended rectangles have at most four errors , that is , the sum of the errors in any two adjacent ec boxes is no more than 4 : mpec is fault tolerant , as described in section [ sec : fault tolerance ] and demonstrated in appendix [ sec : ap .a ] :    suppose , then by the fault tolerance rule , eq. [ eq : ft logic ] ( with added to both sides ) .therefore , using eq .[ eq : ex - rects ] we have with eq .[ eq : a0b0 ] and eq .[ eq : induction rule ] , is true by induction . with eq .[ eq : ft logic ] we then have for all .therefore which with eq .[ eq : success condition ] means success for all ec boxes .

Abstract: inspired by knill s scheme for message passing error detection , here we develop a scheme for message passing error correction for the nine - qubit bacon - shor code . we show that for two levels of concatenated error correction , where classical information obtained at the first level is used to help interpret the syndrome at the second level , our scheme will correct all cases with four physical errors . this results in a reduction of the logical failure rate relative to conventional error correction by a factor proportional to the reciprocal of the physical error rate .


there has been much work on the issue of unlabeled data in classification problems .some papers , such as , have taken the point of view that the data is missing , due to some deficiency in the data collection process , while others , such as , are aimed at situations in which some observations are deliberately left unlabeled .the latter approach is motivated by the fact that in many applications it is very difficult or expensive to determine class membership .thus proposed that in part of the training set , class membership be left undetermined .the unlabeled observations would still provide information on the marginal distribution of the features , and it was shown that this information can contribute to increased classification accuracy for future observations .  in the present work , it is again assumed that we have both labeled and unlabeled data , but the focus is on estimation of the class prior probabilities rather than estimation of the classification function .we wish to estimate those probabilities via a mixture of labeled and unlabeled data , in order to economize on the time and effort needed to acquire labeling information .for example , consider the geographic application in .our class variable here is forest cover type , representing one of seven classes .suppose we wish to estimate the population proportions of the various cover types .the authors note that a problem arises in that cover data is generally `` directly recorded by field personnel or estimated from remotely sensed data ... [both of which ] may be prohibitively time consuming and/or costly in some situations . ''however , various feature variables are easily recorded , such as elevation , horizontal distance to the nearest roadway and so on . previous work in the estimation of classification functions suggests that we may use this other data , without labeling , as a means of reducing the time and effort needed to determine class membership .  as another example , consider the patent analysis reported in . the overall goal was to estimate the proportion of patents whose research was publicly fundedso , we would have just two classes , indicating public funding or lack of it .the determination of which patents were publicly funded involved inspection of not only the patents themselves , but also the papers cited in the patents , a very time - consuming , human - intensive task . however , using the approach discussed here , we could define our features to consist of some key words which have some predictive power for public funding status , such as appearance of the word _university _ in the assignee section of the patent document , and then use the feature data to help achieve our goal of estimating the class probabilities .  specifically , one might develop a classification rule from the labeled observations , and then use the rule to classify the remaining observations in the training set . in other words , we would find predicted forest cover types for each of the unlabeled observations . finally , one would obtain estimates for the proportions of forest cover types in the full training set , using the predicted cover types for those on which labels had not been collected .actually , one can do even better by using estimated conditional class probabilities of the unlabeled and labeled observations , as will be proven here .let us set some notation .suppose there are c classes , and let be the class identification vector , so that is 1 or 0 , according to whether the item belongs to class j. take on the values . however , the statistics notation will be convenient here in various ways , e.g. that is simply .] let be the feature vector .for some observations in our training set , we will have data on both and , but others will be unlabeled , i.e. we will have data only on . in our context here , our goal is to estimate , the prior probability of class j.  the key point is that the unlabeled observations provide additional information on the marginal distribution of .since in the law of total expectation ,  ] , and the unconditional probability , still using u , is the expected value of that quantity .accordingly , we take our binary - v analog of to be   } { \min(ev,1-ev)}\ ] ]  applying jensen s inequality to the convex function , and using $ ] , we have that   \leq \min(ev,1-ev)\ ] ]  thus , as with .one could use as a means of assessing whether to make use of unlabeled data .another approach , motivated by ( [ improvement ] ) , would be to use    note that in the extreme case , is a constant and , so unlabeled data would be useless .note too that since , this measure is in some sense scale - free .accordingly , we might base our decision as to whether to use unlabeled data on ( [ varghat ] ) , making use of the unlabeled data if this quantity is sufficiently far from 0 .rather than merely presenting some examples of our methodology s use on real data , we aim here to assess the methodology s _ performance _ on real data . the bootstrap will be used for this purpose .( the reader should note that this is intended just a means of assessing the general value of the methodology , rather than part of the methodology itself . )suppose we wish to study the efficiency of estimating some population value using , an asymptotically unbiased function of a random sample of size s from the population .we are interested in studying the efficiency of for various sample sizes s , as measured by mean squared error  \ ] ]  now suppose we have an actual sample of size h from the population .we take this as the `` population , '' and then draw m subsamples of size s. in the i of these subsamples , we calculate , denoting its value by .the point is that the empirical c.d.f . of the , i = 1, ... ,m , is an approximation to the true c.d.f . of .moreover ,    is an estimate of the true mean squared estimation error , ( [ msenu ] ) .this bootstrap framework was used to study the the efficiency of our classification probability estimator . on each of three real data sets ,the ratio was computed of the estimated mse for to that resulting from merely using and the labeled data .figure [ pima ] below shows the estimated mse ratios for the pima indians diabetes data in the uci machine learning repository , .( the figures here have been smoothed using the r * lowess ( ) * function . ) here we are predicting whether a woman will develop diabetes , using glucose and bmi as features .the methodology developed here brings as much as a 22 percent improvement .as expected , the improvement is better for larger values of n - r . in other words , for each fixed number r of labeled observations , the larger the number n - r of unlabeled observations , the better we do . on the other hand , for each fixed number of unlabeled observations , the smaller the number of labeled observations , the better ( since n - r is a larger proportion of r ) .    figure [ abalone ] shows similar behavior for another famous uci data set , on abalones .we are predicting whether the animal has less than or equal to nine rings , based on length and diameter .    as pointed out in section [ assessing ] , our method here may produce little or no improvement for some data sets .here is an example , from the 2000 census data , .the data consist of records on engineers and programmers in silicon valley , and we are using age and income to predict whether the worker has a graduate degree . the results , seen in figure [ pumsfig ] , show that our methodology produces essentially no improvement in this case .now let us examine the two possible measures proposed in section [ assessing ] , in the context of the above three data sets .table [ worthit ] shows the results , which are rather striking . even though the misclassification rate , , was approximately equal across the three data sets , the value of was an order of magnitude smaller for pums than for the other two data sets .the latter pattern exactly reflects the pattern we saw in the mse values in the figures .  in other words , classification power by itself does not suffice to indicate whether our method yields an improvement .it is also necessary that the feature vector have a substantial amount of variation .the measure provides a direct assessment of the potential for improvement ..comparison of measures proposed in section [ assessing ] [ cols="<,<,<",options="header " , ]here the focus will be on quantities of the form    for various sets w of interest .it was mentioned earlier , for instance , that one may wish to compare the proportions of public funding for subcategories of patents , say computers and optics . herewe outline how the methodology developed earlier in this paper can be extended to this situation , focusing on the semiparametric case .write ( [ sbc ] ) as  }{p(x \epsilon w)}\ ] ]  where is the indicator function for membership in w. by analogy with ( [ qhatj ] ) , then , our estimator for is    the `` ordinary '' estimator , based only on labeled values , is    the asymptotic variance of is   +  \frac{1}{r }  c_j^t a_j^{-1 } c_j\ ] ]  since    we can again apply slutsky s theorem and derive the asymptotic distribution of . toward that end , define  \ ] ]  and concentrate for the moment on estimating , using    the analog of ( [ taylor ] ) is then    \\ & + & \left [  n^{-1 } \sum_{i=1}^n g'(x_i,\tilde{\theta}_j ) i_w(x_i ) \right ] ^t ~ n^{\frac{1}{2 } } ( \hat{\theta}_j - \theta_j ) \nonumber\end{aligned}\ ] ]  for some between and .proceeding as in ( [ qasy ] ) we have that is asympotically equivalent to  \ ] ]  where  \ ] ]  thus is asympotically equivalent to  \ ] ]  continuing as before , the desired result follows :   +  \frac{1}{r }  c_j^t a_j^{-1 } c_j  \right ] \ ] ]  the asympotic variance of the estimator based on both the labeled and unlabeled data , , is less than or equal to that of the estimator based only on labeled data , .the inequality as strict as long as the random variable is not constant .define .theorem [ thm1 ] , applied to h and rather than g and , shows that    has asympotic variance less than that of    the result then follows by applying slutsky s theorem to the denominators in ( [ qhatjw ] ) and ( [ ordinary ] ) , and noting ( [ zz ] ) .suppose is discrete , taking on b vector values .we assume here that our labeled data is extensive enough that for each k , there is some i with .then we can find direct estimates of the without resorting to using a parametric model or smoothing methods such as nearest - neighbor .we can write the prior probabilities as    where and .let be the indicator variable for , for i = 1, ... ,n and k = 1, ... ,b. then denote the counts of labeled and unlabeled observations taking the value k by    and    also define the count of class - j observations among the labeled data having feature value k :    we can estimate the quantities and by    and    our estimates of the class prior probabilities are then , in analogy to ( [ prioreqn ] ) ,    note that if we do not use unlabeled data , this estimate reduces to the standard estimate of , the class sample proportion , in ( [ ybarj ] ) .the quantities , and have an asympotically multivariate normal distribution .thus from the multivariate version of slutsky s theorem is asympotically equivalent to    this is a key point in addressing the question as to whether the unlabeled data provide our estimators with smaller asympotic variance .since the only randomness in ( [ gjhconst ] ) is in the , the larger the sample used to calculate those quantities , the smaller the variance of ( [ gjhconst ] ) will be .using the unlabeled data provides us with that larger sample , and thus ( [ gjhconst ] ) will be superior to ( [ ybarj ] ) .  stating this more precisely , the variance of ( [ gjhconst ] ) is    without the unlabeled data, this expression would be the same but with a factor of 1/r instead of 1/n .so , again , it is demonstrated that use of the unlabeled data is advantageous .  to form confidence intervals and hypothesis tests from ( [ ourest ] ), we obtain a standard error by substituting and in ( [ ourestav ] ) , and taking the square root .this paper has developed methodology for estimating class prior probabilities in situations in which class membership is difficult / expensive to determine .asymptotic distributions of the estimators were obtained , enabling users to form confidence intervals and perform hypothesis tests .it was proven that use of unlabeled data does bring an improvement in asymptotic variance , compared to using only the labeled data .  in the parametric cases , the usual considerations for model fitting hold .one should first assess the goodness of fit of the model though many formal test procedures exist , with the large samples often encountered in classification problems it may be preferable to use informal assessment .one can , for instance , estimate the function nonparametrically , using say kernel or nearest - neighbor methods , and then compare the estimates to those resulting from a parametric fit .the paper first built from the semiparametric framework of , and then also derived a fully nonparametric estimator for the case of purely discrete features .the latter methodology requires that the training set be large enough that there is at least one observation for every possible value of x. if there are many such values , or if the training set is not large enough , one might turn to smoothing methods , such as kernel - based approaches .another possibility in the purely discrete case is to use the log - linear model [ ] , which is fully parametric.  one might pursue possible further improvement by using the unlabeled data not just to attain a better estimate for the marginal distribution of , but also to enhance the accuracy of the estimate of the conditional distribution of given . as noted in section[ intro ] , much work has been done on this problem in the context of estimating classification functions .it would be of interest to investigate the asymptotic behavior in our present context of estimation of class prior probabilities .it would also be useful to investigate whether the methods of section [ assessing ] can be applied to the classification function problem .j. blackard and d. dean .comparative accuracies of artificial neural networks and discriminant analysis in predicting forest cover types from cartographic variables , _ computers and electronics in agriculture _ , 1999 , 24 , 131 - 151 .u. muller , a. schick and w. wefelmeyer .imputing responses that are not missing ._ probability , statistics and modelling in public health _ ( m. nikulin , d. commenges and c. huber , eds . ) , 2006 , 350 - 363 , springer .

Abstract: work in the classification literature has shown that in computing a classification function , one need not know the class membership of all observations in the training set ; the unlabeled observations still provide information on the marginal distribution of the feature set , and can thus contribute to increased classification accuracy for future observations . the present paper will show that this scheme can also be used for the estimation of class prior probabilities , which would be very useful in applications in which it is difficult or expensive to determine class membership . both parametric and nonparametric estimators are developed . asymptotic distributions of the estimators are derived , and it is proven that the use of the unlabeled observations does reduce asymptotic variance . this methodology is also extended to the estimation of subclass probabilities .


as an indispensable type of information filtering techniques , recommendation systems have attracted a lot of attention in the past decade and have been successfully deployed in many e - commerce websites , such as amazon and netflix ._ collaborative filtering _ ( cf ) and _ content - based _ techniques are two widely adopted approaches for recommendation systems  .collaborative filtering  recommends items for a given user by referencing item ratings from other similar users , while content - based techniques  make recommendations by matching a user s personal interests ( or profiles ) with item content ( e.g. , item description or tags ) .some research works have also discussed approaches that integrate both techniques for item recommendation  .however , no emphasis has been placed explicitly on users _ social influence _ in these works . in our real life, we usually turn to our friends for recommendations of books , movies or restaurants . as evident by the dramatic expansion of social media and social networking systems , social influence from friendspresents new opportunities for recommendation systems but also bring many great challenges . in this paper, we aim to take social influence among users , along with user profile , user preference and item content , into the design of recommendation systems .  to meet users social demands , show that social influence is beneficial for item recommendations .the idea behind is that a user s friends may share common interests with the user , and have influence on the user s decisions . to incorporate the social influence to the recommendation system , the random walk approach  to incorporate user s social network for item recommendation . on the other hand ,model - based systems were also been extended to include social influence  . assuming that trust intensities among a user and his friends are available , some prior works propose to integrate users social trust network into their models through a linear combination  or as a regularization term  .however , most of the proposed methods either apply ad hoc heuristics to include social influence to their methods or assume quantified prior knowledge of social trusts which is handily available .there is a need to define comprehensible social influence , beyond random walking over the social network , to explicitly model and unveil the social influence from data available to the recommendation system .owing to the success of collaborative filtering and content - based recommendation ideas , in this paper we propose to incorporate social influence with these ideas in a unified fashion to design new recommendation systems . through our design , we aim to demonstrate the importance and strength of social influence to recommendation services . to our best knowledge , ideas for unifying social influence with collaborativefiltering and content - based recommendation are unexplored and very challenging . in this paper, we adopt the _ probabilistic generative model _ as a methodology to reach our goal . the basic idea behind probabilistic generative modelsis to `` mimic '' user behaviors in a process of decision making , e.g. , deciding which restaurant to dine . while there exists a prior study  on integrating collaborative filtering and item content into a probabilistic generative model for item recommendation, we want to point out that incorporating social influence into the probabilistic generative model is nontrivial .notice that the data used for cf and content based techniques ( i.e. , the user - item accessing history , user profiles and item content ) contain _ explicit _ observations .thus , the notions of user preferences , user profiles and item content can be easily modeled . on the other hand , social influencecan not be observed directly from the data ( i.e. , we only know who access which item but never know if this decision is influenced by other people ) , we aim to introduce a latent variable and develop algorithms to capture the social influence between friends in addition of the latent variable for user s topics .  the proposed probabilistic generative model is a latent class statistical mixture model .the model discovers ( 1 ) users personal preference distribution over latent topics ; ( 2 ) an item generative distribution for each topic ; and ( 3 ) a social influence distribution from friends for each user .the generative model aims to capture the process of human behaviors and/or reasonings for decision making .for example , a user ( ) wants to choose a restaurant ( ) for dinner .he may choose one based on his own tastes or turn to one of his friends ( ) for help .in the case that wants to choose the restaurant without any influence from his friends ( with a certain probability ) , he chooses a topic according to his personal preference distribution .then the selected topic in turn `` generates ''an item following on the topic s item generative distribution . in the casethat social influence from a friend is effective , would generate an item following s preference distributions similarly .thus , this model simulates the process that how picks the item , including how a friend influences s decision .as mentioned , both users preferences and social influence among friends are latent variables .thus , there is a need to devise new learning algorithms to estimate the model parameters . in this paper, we address this issue by devising a new model learning algorithm based on the idea of _ expectation maximization _moreover , due to the large volume of social network datasets and the excessive computational cost incurred in learning the generative model parameters , we devise a parallel algorithm under the map - reduce framework in addition to a single - thread algorithm , to process the large - scale datasets we collect . finally , to demonstrate the flexibility and applicability of our ideas to other recommendation services that may utilize social influence, we adapt our probabilistic generative model and develop an algorithm to support _group recommendations_. the primary contributions made in our research are summarized as follows .* we argue that social influence is important for item recommendations and devise probabilistic generative models that explicitly quantify and incorporate social influence from friends to a user in the recommendation process . *we provide model learning methods ( based on em algorithms ) to learn the model parameters from common user - item pairs .we implement the algorithms on single machine and parallel processing platform ( based on the map - reduce framework  ) to efficiently process large - scale data . * in addition to support item recommendation for individual users ,we demonstrate that the quantified social influence parameter is essential for supporting group recommendations .owing to the advantages of social influence learned in our model , the proposed social - influence - based group recommendation algorithms significantly outperforms conventional aggregation - based allgorithms .* we conduct a comprehensive performance evaluation on two real datasets crawled from last.fm and whrrl.com .experimental results show that our proposal to incorporate social influence into generative models for item recommendation techniques are very effective .the experimental results for group recommendation also confirm that the good estimation of social influence in our generative model is beneficial for group recommendation .the remainder of this paper is organized as follows .section [ sec : preliminary ] summarizes the related work and provide some background on probabilistic generative models .section [ sec : social ] introduces the design of our generative model which combines collaborative filtering and social influence into recommendation process .section [ sec : mapreduce ] discusses how the em algorithm is implemented on a single machine and on the map - reduce framework .section [ sec : extend ] demonstrates how to incorporate social influence , in addition of collaborative filtering and item content , into the probabilistic generative model .section [ sec : group ] reviews some previous group recommendation methods and proposes a new group recommendation method using the social influence obtained from our model .section [ sec : perform ] shows the result of an empirical evaluation of our proposal using two real datasets .finally , section [ sec : conclusion ] concludes the paper .in this section , we introduce some related works , including recommendation systems , recommendation in social networks and group recommendation .then , we provide the background about how to utilize probabilistic generative model for item recommendations .* recommendation system*. item recommendation has been a crucial service for many e - commerce and web services ( e.g. netflix.com and amazon.com ) .the goal is to recommend an accurate list of items that the targeted user may be interested in .collaborative filtering and content - based techniques are two widely adopted approaches for recommendation systems  .both of them discover users personal interests and utilize these interests to find relevant items .collaborative filtering techniques  automatically predict relevant items for a given user by referencing item rating information from other similar users .content - based techniques  make recommendations by matching a user s personal interests ( or profiles ) to descriptive item information .recommendation systems using pure collaborative filtering approaches tend to fail when little knowledge about the user is known or when no one has similar interests with the user .for example , if a user has little item rating / selection history or his interests are rare compared to others , the item rating / selection history of other users can not help .although content - based methods is able to cope with the issue of lacking knowledge , it fails to account for community endorsement .for example , even though we know a user is interested in chinese restaurants , content - based methods may possibly recommend a bad chinese restaurant to him due to the lack of consideration in users group consensus . as a result , there has been a continuous research interests and effort in combining the advantages of both collaborative filtering and content - based methods  .our proposal in this work not only is able to naturally integrate the ideas behind collaborative filtering and content - based methods but also incorporate social influence into the recommendation process .* social recommendation*. under the context of social networks , social friendship is shown to be beneficial for recommendation  .however , prior works in this area are mostly based on ad hoc heuristics .how a user is influenced by friends in the item selection process remains vague .for example , linearly combines social influence with conventional collaborative filtering ; employ the random walk  approach to incorporate social network information into the process of item recommendation ; while explores social friendship via matrix factorization technique , where social influence is integrated by simple linear combination  or as a regularization term  .  in this paper, we propose to employ the probabilistic generative model as a methodology to integrate social influence with collaborative filtering and content - based methods for item recommendation .our work is uniquely different from these previous works because we do not assume social influence is explicitly available . by leaning a quantitative parameter for social influence, we are able to obtain a better understanding of the social influence and improve the performance of recommendation systems .moreover , the quantified social influence obtained in our model can support related applications such as group recommendation  and viral marketing  .* group recommendation*. to explore how to utilize social influence for group recommendation , we provides an in - depth study and comparison on group recommendation techniques .group recommendations have been designed for various domains such as web / news pages  , tourism  , music  , and tv programs and movies  . in summary ,two main approaches have been proposed for group recommendation  .the first one creates an aggregated profile for a group based on its group members and then makes recommendations based on the aggregated group profile  .the second approach _ aggregates _ the recommendation results from individual members into a single group recommendation list . in other words , recommendations ( i.e. , ranked item lists ) for individual membersare created independently and then aggregated into a joint group recommendation list  , where the aggregation functions could be based on average or least misery strategies  .different from these proposed methods , our approach regenerates the process of how group members would _express _ their preferences and _ influence _ other members to reach the final decision .evaluation from real datasets demonstrates a significant improvement over the proposed method using social influence over the traditional methods .the recommendation techniques we proposed in this paper are inspired by the probabilistic generative model developed for collaborative filtering in .let and be the user set and item set , respectively .a latent topic set is assumed to capture latent user interests and item profiles . in the context of item recommendation ,an event of a user accessing an item is considered to be associated with one of the latent topic variables .conceptually , as shown in figure [ fig : cf ] , user chooses a topic according to his interest distributions , and in turn the topic probabilistically `` generates '' an item according to the distribution of items associated with . under this model, users are assumed to be independent of items given the chosen topic .the joint probability distribution over user , topic and item can be written as an equivalent specification of the joint probability distribution that treats users and items symmetrically is since we are only interested in how likely a user chooses an item , the joint distribution over and item is   this model has a set of parameters , and for all , which for simplicity is represented as . in  , the user - item concurrence history , which contains all the observed user - item , is used to learn the model parameters .one way to learn is to maximize the log - likelihood of history data which is : where each can be found using model parameters as in equation ( [ eq : cf ] ) .after model parameters are inferred , items can be ranked for a given user according to , which refers to the probability that the user selects the item . can be computed as since most recommendation systems only focus on recommending _ new _ items ( items not presented in for a particular user ) , items with the higher and not accessed by are good recommendations .  the probabilistic generative model described above is based on the ideas of collaborative filtering .although has extended the model to integrate item contents as an additional component , social influence has not been considered yet . moreover , as to be shown later , incorporating social influence into the generative model is fundamentally more challenging than integrating item contents into the model because item contents are _ observable _ from the training data , while the social influence is a hidden factor not directly observable . in this paper , we demonstrate how to integrate social influence into this model and introduce our approach to infer model parameters .in this section , we introduce our approach to incorporate social influence in a new probabilistic generative model for item recommendation . while the ultimate goal of our study is to unify the ideas of social influence , collaborative filtering and content - based methods as a model for item recommendation . for simplicity , here we first discuss how to integrate collaborative filtering and social influence into a probabilistic generative model .we will introduce the complete model ( including collaborative filtering , item content and social influence ) later in section [ sec : extend ] .    herewe propose a new probabilistic generative model that describes the process of how a user selects an item by taking into account the user s own preferences and the social influence from her friends .let denote the friend list of a user .the social influence introduced in this model aims to capture the scenario that one of s friends ( ) has contributed his opinions in the item selection process . here ,for simplicity , we assume is a special friend of himself ( i.e. , ) .therefore , our model can be depicted as in figure [ fig : cf - social ] .as shown , a user would first picks a friend ( including himself ) to make the item selection .if the picked friend happens to be himself , is not influenced by someone else in this selection .nevertheless , if the picked friend is not , is influenced by at this time and thus the selected item follows s interests rather than s own tastes . in this model, we define a parameter _ social influence distribution _ as the probability for to be influenced by a friend .after is chosen based on , randomly chooses a topic according to his interests , and then the topic generates an item according to the topic s item distribution .the joint probability distribution over users , friends , topics and items is as below . where , , and .the key observations from this model are 1 ) and are independently conditioned on , and 2 ) and are independently conditioned on . because we intend to model the item selection probability in terms of social influence and topics ( two latent parameters in our model ) , we transform equation ( [ eq : joint ] ) into the following form : thus , the joint distribution over users and items is:  different from the generative model in  ( see equation ( [ eq : cf ] ) ) , the newly proposed model has two latent variables , namely the topic variable ( ) and the social influence variable ( ) .correspondingly , the model parameters now include .note that the size of parameters is increased by for social influence .notice that while the friend space could potentially be the entire user space , the averaged number of friends per user is limited .this is very important because it ensures that our latent variable space is small enough and not to over - complicate the model .moreover , the small latent parameter space yields high - quality parameter estimations even when the available history is not large .  in this study, we employ _ expected maximization ( em ) _ to learn model parameters from the user - item history .however , the conventional expected maximization ( em ) algorithm developed for single latent variable is not applicable for our model because we now have two latent variables , i.e. , social influence and topics . to address this challenging issue ,we have performed a detailed mathematical derivation to develop an new em algorithm in order to infer the model parameters .the derived em algorithm iterates over the following steps :  1 .e - step : computes posterior of the latent variables as using the model parameters of previous iteration .m - step : computes new model parameters by maximizing the expected log - likelihood .  in the e - step ,we only need to use the previous iteration s model parameters to find as : also note that we only need to compute the posteriors of those pairs presented in the history instead of all the possible user - item pairs , because the expectation to be maximized only weights on the observed user - item pairs .the m - step shall find new model parameters to maximize the expected log - likelihood found in the e - step . according to our derivation, the new model parameters should be updated as   equation ( [ eq : prmstep ] ) shows that for each parameter distributions , the new number should be chosen as normalized corresponding posterior sums .for example , is obtained by taking the sum of all the _ related _ latent variable posteriors for the and .then , because of , we need to normalize the posterior sums with regarding to different to update the correct model parameters of .  by repeating the e - step and m - step ,the em algorithm improves the model parameters iteratively until they converge to a local log - likelihood maximum .the learned model parameters are used for item recommendations by ranking items for a given user according to which can be calculated by equation ( [ eqn : zifu ] ) .in this section , we discuss how to implement the em algorithm efficiently to learn the model parameters . herewe first present our algorithm for single machine .a challenge encountered in our initial research effort is that the em algorithm , while fine tuned , is still slow due to excessive computation incurred in processing large - scale datasets . to overcome this challenge, we develop a parallel processing version of the em algorithm on the map - reduce framework . through this effort, we demonstrate that our design of the em algorithm can be elegantly decomposed for efficient parallel processing using map - reduce .we first show an implementation that efficiently realize our em algorithm on a single machine . for simplicity ,we only present one iteration of the e - step and the m - step , which aims is to approach the model parameters based on the current approximate value of parameters .normalize [socialem : norm ]  algorithm [ alg : socialem ] executes one em iteration to find the next model parameters .notice that we do not execute the e - step separately . because we only need to compute once for each user - item pair observed in , we embed the e - step computation in the m - step sothe posteriors are computed only as needed .therefore , for each observed , the e - step is executed once in line [ socialem : estep ] , and the m - step is executed from line [ socialem : mstep ] to accumulate latent variable posteriors into the corresponding posterior sums ( e.g. now takes the sum of all the posteriors with the same friend and topic ids ) .after all the observed user - item pairs are examined , m - step need to normalize posterior sums as next iteration s parameters in line [ socialem : norm ] .these accumulation and normalization steps realize the m - step in equation ( [ eq : prmstep ] ) .the running time for this em algorithm is , where is the total number of observed user - item pairs , is the latent topic size and is the average number of friends per user .   in this section ,we show how we decompose the algorithm [ alg : socialem ] for parallel processing .notice that there are three computation components in one em iterations : 1 ) e - step to compute posteriors ; 2 ) accumulate posteriors to posterior sums ; and 3 ) normalization step to obtain the model parameters for next iteration . among them ,step 1 and 3 could not be parallelized because step 1 requires the knowledge about all the related model parameters and step 3 requires the entire set of for parameter normalization . therefore ,based on the design principle of map - reduce algorithms , we execute step 3 of previous iteration along with step 1 . as such, the non - parallelizable components are combined to avoid overhead of another round of map - reduce to achieve the same task .normalize [mapper : norm ] [ socialem : endmstep ]  the algorithms for mapper side and reducer side are shown in algorithm [ alg : socialmapper ] and algorithm [ alg : socialreducer ] , respectively . at its start , each mapper normalize posterior sums from previous results to construct model parameters ( see line [ mapper : norm ] ) .then the user - item pairs are processed in parallel at different mappers ( line [ mapper : mstep ] ) because each mapper now has the same global knowledge of . and the accumulation step is done in reducer to find posterior sums of for next iteration .since each mapper only processes a portion of the user - item history , one mapper does not have the entire knowledge of , and thus can not accumulate correct posterior sums ( e.g. ) . to address this problem , we move the accumulation step to reducers ( algorithm [ alg : socialreducer ] ) .in particular , we park posteriors for a specific user , item or friend i d together and emit the packed value ( line [ mapper : park]-[mapper : parkend ] ) . to ensure these posteriors are correctly accumulated in reducer , a standard shuffle - and - sort is performed to all the emitted key - value pairs . in this way , the map - reduce framework ensures that _ all the emitted values for the same key _ are grouped and processed together in the same reducer .thanks to the shuffle - and - sort step , the reducer algorithm ( algorithm [ alg : socialreducer ] ) is very simple , which only takes a sum of all the grouped values and output the sums . let us take the computation as an example to understand how the reducer performs its task as desired .when a mapper run algorithm [ alg : socialmapper ] , only a part of pairs are processed and the posteriors with respect to a particular are emitted to the mapper outputs .although emitted key - value pairs with the same key ( ) may come from different mappers , but they are grouped by key ( ) after keys are shuffled and sorted .because now all the values ( packed with posteriors ) for the same key are grouped , the reducer can simply sum the posteriors to find the correct posterior sums .similar steps can be done for all the other parameter computations , i.e. , ( and ) .a reducer does not need to differentiate key types , because the accumulation steps are the same for user / item / friend ids .recall that these reducer outputs are not the yet .what left is the normalization for each posterior sums to do in the next map - reduce iteration .the above map - reduce em algorithms addresses the scalability issue in learning model parameters .we find the map - reduce framework is quite suitable for expediting our em algorithm because the posterior computation and accumulation ( which incur significant cost ) can be done in parallel .as mentioned earlier , we aim to developed a new generative model to unify the ideas of social influence , collaborative filtering and content - based methods for item recommendation . in this section, we present the unified generative model developed for our ultimate goal .the unified generative model depicting a more general process of item selection is shown in figure [ fig : cf - social - content ] . as shown , the early steps of the process is similar to the model introduced earlier ( see figure [ fig : cf - social ] ) .however , the selected topic now generates an item and a content description .therefore , a topic in this new model is not only associated with a distribution of items but also a distribution of item content ( e.g. , tag words ) .notice that here we assume items and contents are independently conditioned on the topics .as such , the similarity of item contents is taken into account in the recommendation process . as a result ,the joint probability distribution over all factors is : where and is the space of possible item contents .for example , could be a word of the content vocabulary or a tag of the tag space .similar to equation ( [ eqn : zifu ] ) , the joint probability distribution can be rewritten as :   now the remaining issue is to learn the set of all the model parameters .different from what we discussed earlier in section [ sec : social ] , the dataset used for learning now consists of three elements , including users , items and tags , i.e. , , where , , and ( i.e. , denotes the tag / word set associated with item ) . note that an item may contain multiple tags / words . for a history record of a user selecting an item where , we have , .notice that now includes an extra parameter in addition to the other model parameters discussed earlier in our preliminary generative model ( see section [ sec : social ] ) .the approach of learning model parameters is still to maximize the log - likelihood of .however , the details are different .  in e - step , instead of computing the expectation of the log - likelihood for individual latent variables ( e.g. , or individually ) , we propose to compute the expectation of the log - likelihood for the joint latent latent variables ( i.e. , and together ) .more specifically , we calculate   in m - step , model parameters are computed to maximize the expected log - likelihood found on the e - step as below .[eq : pr - mstep ] the parameters maximization method is similar to equation ( [ eq : prmstep ] ) .note that the summed latent variable posterior is different and that we have an additional set of parameters in .after the model is learned , items can be ranked for a given user based on , which can be approximated by equation ( [ eq : cf_rec ] ) , in which note that we are only interested in the tags / words associated with the given item when we calculate the user - item joint probability .item with high that user has not yet accessed is a good candidate for recommendation .given a group of people , group recommendation aims to find items that are welcomed by the whole group instead of individual group members .this recommendation service has a very large application base , e.g. , coordinating a group of people to find quality activities / venues / restaurants / movies , etc .although the generative models we proposed earlier are targeting on item recommendation for an individual user , the social influence parameter learned in our models is very useful for group recommendation . in this section ,we first introduce the state - of - the - art approaches for group recommendation , namely _ aggregation - based group recommendation _ and then discuss how to apply the quantified social influence obtained from our models to develop a new algorithm , called _ social influence based group recommendation _, for group recommendation .   for group recommendation ,one popular approach is the _ ranking aggregation _ method which finds a `` consensus '' ranking / score for each item for the whole group . given individual ranking / score for each member , some aggregation methods are employed to obtain a group ranking / score from individual ranking / scores . in this paper , we review two popular aggregation strategies : _ average _ and _ least misery _recent proposed in  .* average * - with the average aggregation strategy , an item s group score is defined as an _ average _ of the scores from individual group members . by using the item access probability estimation as the score, the group score for an item to group is calculated as accordingly , the recommendation ranking can be computed by sorting the group scores in descending order .* least misery * - with the least misery aggregation strategy , the group score for item to a group is equal to the _ smallest _ predicted rating for item in the group , specifically following this strategy , whether an item is acceptable to the group depends on the least satisfied member .basically , the item least disliked by each individual member shall has the highest group score for recommendation .these two aggregation - based group recommendation approaches captures a group consensus of item ranking by considering all the decisions made by users to be independent and equally important .however , in a group activity , people interact with each other and thus influence each other. we aim to address this observation in our social influence based group recommendation algorithm .note that we restrict the recommendation to a group where every group member has at least one friend in the group .within such a group , friends may influence each other so there may exist a group consensus . while our generative models aim to capture the process where a given user ( influenced by a friend ) selects an item , we can also see the process as a group activity , i.e. , is influenced by to jointly select item .intuitively , our models can be used directly to support group recommendation for `` two - member '' groups .let denote a two - member " group . to select an item for the group ,user could influence user and vice versa .therefore , we define the score for recommending an item to the group as where can be easily obtained from the model parameters of our generative models .    the ideas described above can be generalized for groups with more than two members .a group of more than two members can be decomposed into a set of two - member groups based on the friendship of members ( see the example in figure [ fig : group ] for illustration ) . to make a group recommendation , we assume the social influence only takes action between friends .intuitively , if most pairs of friends in the group prefer a particular item , it would be a good candidate for recommendation to the group .let denote a group with arbitrary cardinality .the score for recommending an item to is defined as the sum of score over all possible friend pairs in the group .formally ,    the ranking of items for group recommendation is based on the sorted group scores of items as defined above .thus , the decision of selecting an item for the group naturally incorporates the social influence among members of the group .we find superior performance of our social - influence strategy over the two aggregation strategies ( to be shown in our evaluation ) .in this section , we validate our proposed probabilistic generative models using two real datasets , one from _ last.fm _ and the other from _ whrrl.com_. we develop web crawlers to collect theses two datasets , which include user - item accessing history , users friendship network and tags associated with each item .besides , we collect group check - in history data from whrrl.com to validate our group recommendation approach . in our evaluation, we adopt the user - based collaborative filtering approach ( denoted as cf ) as a baseline and propose to study the effectiveness of different factors ( i.e. , social influence , collaborative filtering and item content ) included in our unified generative model .the different configurations of factors included on our evaluation are : 1 ) cf factor ( cf - pgm ) ( see figure [ fig : cf ] ) , 2 ) combination of cf and social influence factors ( cf+si - pgm ) ( see figure [ fig : cf - social ] ) , 3 ) combination of cf and item content factors ( cf+ic - pgm ) ( this has been discussed in ) , and 4 ) combination of cf , social influence and item content factors ( cf+si+ic - pgm ) ( i.e. , our unified model ; see figure [ fig : cf - social - content ] ) . in this evaluation , we conduct a comprehensive set of experiments for item recommendation ( to a single user ) and group recommendation .   herewe first provide information about the datasets , i.e. , _last.fm _ and _ whrrl.com_ , used in our experiments .last.fm is an on - line music radio web service and whrrl.com is a location - based social network web service .the last.fm dataset contains music access history of users over unique songs ; while whrrl.com dataset includes the check - in history of users to unique places .it is worth noting that the whrrl.com dataset includes 17,587 _ group _ check - in records which are very valuable for evaluating the group recommendation approaches .additionally , both datasets have their user social networks available . the basic statistics of these two datasets are summarized in table [ tbl : datasets ] . cumulative distributions with respect to the number of items accessed by users ( user items ) , the number of friends of users ( user friends ) , and the number of tags associated with items ( item tags ) are shown in figure [ fig : lastfm ] and figure [ fig : whrrl ] for last.fm and whrrl.com , respectively ..datasets statistics [ cols=">,^,^",options="header " , ]   after the datasets are prepared , we are able to apply em algorithms to infer model parameters . however , for all em methods , model parameters need to be initialized and the iteration termination condition needs to be specified . after experimenting with different model parameter initialization methods , we decided to use latent dirichlet allocation ( lda )  to initialize the model parameters .although lda has been mainly used for clustering documents , it has similar parameters as the cf - pgm in figure [ fig : cf ] . to obtain initial parameters from lda, we treat each user as a `` document '' in the lda model and transform items accessed by the user as `` words '' in the document .after the lda model converges , we discard the document clustering but keep and as the model initialization values for em algorithms . for the social influence parameters required in cf+si - pgm , we use the normalized jaccard similarity as the initial values ( i.e. , , where and denote the accessed items for and , respectively ) .this initialization ensures friends having more commonly accessed items to have a higher social influence initially .note that in our models , is treated as a special friend of himself .since , a user s self influence is always larger than any social influence from his friends at the beginning . to terminate the em algorithms , we use log - likelihood as model converge indicators and terminate the em algorithms when an additional em iteration can not improve the training data s log - likelihood by or when the maximum iteration threshold ( empirically set with 50 ) is reached .we implement both the single machine em - algorithm and its map - reduce version and confirm that both implementations produce the same results with small datasets . for those more complicated models ( i.e. , cf+si - pgm and + cf+si+ic - pgm ) , we apply our map - reduce implementation on a cluster of 10 machines to infer the model parameters .we use item recommendation as the primary test case to evaluate the performance of the probabilistic generative models under evaluation .we apply cross - validation method to find item recommendation s precisions and recalls . for both datasets , we mark off itemassess history of each user for testing .in other words , the rest user - item pairs are used as training data to infer model parameters .then after each model is learned , we use the model parameters to find for all users .the items _ not in presence _ in the training dataset are ranked based on their . in this way, we prevent our recommendation system from `` repeating '' a user s item access history. therefore , all the recommendations for a user must be `` fresh '' items that have not been accessed by him in the training dataset .the precisions and recalls for top recommendations are used as the evaluation metrics , where ( 5 is the default value ) .precision is calculated as the ratio of the number of recommendation hits to the recommendation size ; and recall is calculated as the ratio of the number of recommendation hits to the size of user s validation item set. then the average precisions and recalls of different users serve as the evaluation metrics .  in figure[ fig : lastfm_topic ] and figure [ fig : whrrl_topic ] , the precision and recall of top 5 item recommendations for last.fm and whrrl.com under different latent topic sizes are presented .we find social influence indeed improves the recommendation performance , for both cf+si - pgm against cf - pgm and cf+si+ic - pgm against cf+ic - pgm .the result shows that the best recommendation performance is reached when the topic size is chosen around .therefore , we set the default value of the latent topic size to 60 for the remaining part of performance evaluation .   & cf - pgm & 0.0399 & 0.0372 & 0.0342 & 0.0296 + & cf+si - pgm & 0.0427 & 0.0429 & 0.0383 & 0.0315 + & cf+ic - pgm & 0.0470 & 0.0542 & 0.0494 & 0.0388 + & cf+si+ic - pgm & 0.0492 & 0.0566 & 0.0506 & 0.0398 + & cf & 0.0046 & 0.0066 & 0.0080 & 0.0085 + & cf - pgm & 0.0048 & 0.0085 & 0.0157 & 0.0329 + & cf+si - pgm & 0.0050 & 0.0100 & 0.0187 & 0.0363 + & cf+ic - pgm & 0.0054 & 0.0102 & 0.0198 & 0.0385 + & cf+si+ic - pgm & 0.0057 & 0.0117 & 0.0213 & 0.0411 + & cf & 0.0010 & 0.0024 & 0.0050 & 0.0122 +   & cf - pgm & 0.0048 & 0.0038 & 0.0035 & 0.0028 + & cf+si - pgm & 0.0053 & 0.0041 & 0.0036 & 0.0028 + & cf+ic - pgm & 0.0059 & 0.0048 & 0.0040 & 0.0029 + & cf+si+ic - pgm & 0.0062 & 0.0049 & 0.0041 & 0.0030 + & cf & 0.0015 & 0.0016 & 0.0015 & 0.0011 + & cf - pgm & 0.0062 & 0.0090 & 0.0141 & 0.0251 + & cf+si - pgm & 0.0069 & 0.0100 & 0.0146 & 0.0254 + & cf+ic - pgm & 0.0076 & 0.0119 & 0.0157 & 0.0252 + & cf+si+ic - pgm & 0.0081 & 0.0115 & 0.0154 & 0.0275 + & cf & 0.0020 & 0.0033 & 0.0051 & 0.0071 +  in table [ tbl : lastfm_topn ] and table [ tbl : whrrl_topn ] , we compare the item recommendation performance of different algorithms .as shown in these two tables , all the probabilistic generative model approaches clearly outperform the conventional user - based collaborative filtering ( cf ) .again , we find social influence factor indeed improves the recommendation performance , ( for both cf+si - pgm against cf - pgm and cf+si+ic - pgm against cf+ic - pgm ) .most importantly , the unified model we propose in this paper ( which integrates collaborative filtering , social influence and item content ) shows the best performance .  by comparing results from whrrl.com and last.fm datasets, we find that social influence is more important ( in terms of item recommendation ) in whrrl.com than last.fm .one possible reason is that in our collected datasets , users in whrrl.com are more _ social _ than users in last.fm , i.e. , the average number of friends in whrrl.com is compared to last.fms .we also observe this phenomenon from the statistics shown in figure [ fig : lastfm](b ) and figure [ fig : whrrl](b ) .in other words , it is more likely for users in whrrl.com to be influenced by their on - line friends than users in last.fm .consequently , the recommendation performance benefit from social influence in last.fm is less significant than that in whrrl.com .   in this section, we aim to study the social influence between friends , where the social influence is learned through our proposed models .for simplicity , we focus on cf+si - pgm .instead of investigating how social influence improves the recommendation performance , here we are interested in how significant a particular user influence his friends . as different people have different personalities , we plot the distributions of social influence probabilities among friend pairs ( that we learned through cf+si - pgm ) in figure [ fig : last_fm_influence ] and figure [ fig : whrrl_influence ] .note that we also consider the circumstance of self - influence and use to denote the probability .figure [ fig : last_fm_influence ] shows the learned social influence presented in last.fm. in general , people s self - influence ( ) in this dataset is significantly higher than the influence from their friends ( ) when choosing a music piece .figure [ fig : last_fm_self_influence ] shows that more than users self - influences are higher than . also , since each user may have several friends , each friend s social influence is thus quite small , e.g. , friends influence is smaller than .this observation indicates that most music pieces consumed by the users in last.fm are selected in accordance with users own preferences and tastes .figure [ fig : whrrl_influence ] demonstrates very different findings . as shown in figure [ fig : whrrl_self_influence ] , the self - influence is still quite significant but much smaller than that in last.fm , i.e. , users self - influence is lower than .the implication from this finding is that while people visit places mainly based on their own preferences , they would sometimes take friends suggestions to visit places . while users in whrrl.com have more friends than users in last.fm , ( i.e. , average friends compared to ) , we find a lot of friends are not influential and that usually a small portion of friends takes the most part of social influence . in general , people s social influence in place check - in activities are much more significant than music consumption one explanation is that check - ins are inherently social activities and music consumption are usually for self - entertainment .   finally , we report our findings on evaluation of group recommendation algorithms , including the social - influence based ( sig ) algorithm introduced in section [ sec : group ] along with two aggregation - based group recommendation strategies .to compare their performance , we use the group check - in records in whrrl.com . in our experiment, we consider a group check - in record ( i.e. , the ground truth ) at a time and take the average of tested records .notice that a record indicates a group of people visits a place .an effective group recommendation algorithm should have this place ranked high among all the places returned .therefore , we propose a metric called _ relative ranking _ to evaluate the performance of these group recommendation algorithms .suppose that a given group recommendation algorithm returns a ranked list of items ( i.e. , places in this experiment ) .if the actual visited place is ranked in the -th position of the returned list , the relative ranking is calculated as .for example , if an actual visited place is ranked 10th among a total of 100 items returned by a group recommendation algorithm , the relative ranking is ) .  in figure [ fig : whrrl_group_result ] , we compare the performance of our sig group recommendation method with the other two aggregation - based strategies , i.e. , average and least misery .the values in y - axis represent the relative rankings of actual visited places ( the lower the better ) . in figure[ fig : whrrl_group_size ] , we find that sig outperforms the average and least misery strategies for most of the varied group sizes .however , the larger the group is , the smaller improvement is reached from sig .this finding implies that for smaller groups , the social influence among group members plays a major role in item selection for the group .however , for larger groups , the group consensus aggregated from individual preferences may dominate the group decision .this finding is consistent with our common experience that in activity planning for a smaller group , one or two influencing members may significantly determine the activity venue . on the other hand , for a large group , the social influence from individuals may be difficult to take effect on the entire group . as a result , the group s common interest dominates .next , we evaluate the three group recommendation strategies by varying topic size .the result shown in figure [ fig : whrrl_group_topic ] indicates that sig always outperforms the other two and reach its optimal point when the topic size is configured to around .this research attempts to explore social influence for item recommendation .we propose a probabilistic generative model , called _ unified model _ , which naturally unifies the ideas of social influences , collaborative filtering and content - based methods in the recommendation process . to address the issue of hidden social influence in the available datasets ,we devise new algorithms to learning the model parameters based on the idea of expectation maximization ( em ) .moreover , we provide a map - reduce implementation , in addition to a single - machine version , of our em algorithm to process large - scale datasets . furthermore , by exploring the social influence quantitatively captured in our models , we develop a social influence based group recommendation algorithm to demonstrate the strength of our proposed models on group recommendation . finally , we conduct a comprehensive experimental study to evaluate the performance of our proposal for item recommendation to individual users and to groups .experimental results show that the unified probabilistic generative model proposed in this paper accommodates different factors very well to achieve a superior recommendation performance over other alternatives .our experimental results also facilitate a better understanding of the social influence between friends in social networks .it is interesting to note that users in whrrl.com are more likely to be influenced by friends than those in last.fm .finally , our experimental result also confirms that our social influence based algorithm outperforms the state - 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Abstract: in this paper , we propose a probabilistic generative model , called _ unified model _ , which naturally unifies the ideas of social influence , collaborative filtering and content - based methods for item recommendation . to address the issue of hidden social influence , we devise new algorithms to learn the model parameters of our proposal based on expectation maximization ( em ) . in addition to a single - machine version of our em algorithm , we further devise a parallelized implementation on the map - reduce framework to process two large - scale datasets we collect . moreover , we show that the social influence obtained from our generative models can be used for group recommendation . finally , we conduct comprehensive experiments using the datasets crawled from last.fm and whrrl.com to validate our ideas . experimental results show that the generative models with social influence significantly outperform those without incorporating social influence . the unified generative model proposed in this paper obtains the best performance . moreover , our study on social influence finds that users in whrrl.com are more likely to get influenced by friends than those in last.fm . the experimental results also confirm that our social influence based group recommendation algorithm outperforms the state - of - the - art algorithms for group recommendation .