causal bayesian networks are widely used for causal reasoning [ e.g. , , ( ) , ( ) ] . in particular ,if the causal structure is known and represented by a directed acyclic graph ( dag ) , this framework allows one to deduce post - intervention distributions and causal effects from the pre - intervention ( or observational ) distribution .hence , if the causal dag is known , one can estimate causal effects from observational data .covariate adjustment is often used for this purpose .the _ back - door criterion _ [ ] is a graphical criterion that is sufficient for adjustment , in the sense that a set of variables can be used for covariate adjustment if it satisfies the back - door criterion for the given graph .  in practice , there are two important complications . first , the underlying dag may be unknown . in this caseone can try to estimate the dag , but in general one can not identify the underlying dag uniquely .instead , one can identify its markov equivalence class , which consists of all dags that encode the same conditional independence relationships as the underlying dag .such a markov equivalence class can be represented uniquely by a different type of graph , called a completed partially directed acyclic graph ( cpdag ) [ ] .second , it is often the case that some important variables were not measured , meaning that we do not have causal sufficiency . in this case , one can work with maximal ancestral graphs ( mags ) instead of dags [ ( ) ] .finally , the underlying mag may be unknown , so that it must be estimated from data .again , there is an identifiability problem here , as we can generally only identify the markov equivalence class of the underlying mag , which can be represented uniquely by a partial ancestral graph ( pag ) [ ] .  in this paper, we therefore consider generalizations of the back - door criterion to the following three scenarios :  we assume causal sufficiency , and we only know the cpdag , that is , the markov equivalence class of the underlying dag ;  we do _ not _ assume causal sufficiency , and we know the mag on the observed variables ;  we do _ not _ assume causal sufficiency , and we only know the pag , that is , the markov equivalence class of the underlying mag on the observed variables .  in scenarios 2 and 3 , we allow for arbitrarily many hidden ( or unmeasured ) variables . we do not , however , allow for selection variables , that is , for unmeasured variables that determine whether a unit is included in the sample .since the back - door criterion is a simple criterion that is widely used for dags , it seems useful to have similar criteria for cpdags , mags and pags .we also hope that our generalized back - door criterion will make working with mags and pags less daunting , and more accessible to people in practice .our generalized back - door criterion for dags , cpdags , mags and pags is given in section [ secgeneralizedbackdoorcriterion ] ; see especially definition [ defbackdoorgraphical ] and theorem [ thgeneralizedbackdoorsufficientforadjustment ] .corresponding r - code is available in the function ` backdoor ` in the r - package ` pcalg ` [ ] .our results are derived by first formulating invariance conditions that are sufficient for adjustment , and then using the graphical criteria for invariance derived by .we also show that the generalized back - door criterion is equivalent to pearl s back - door criterion for single interventions in dags , and is slightly more general for multiple interventions in dags ( lemma [ lemmaourbackdoorstrongerthanpearls ] and example [ exourbackdoorstrongerthanpearls ] ) . in section [ secfindingbackdoorset ] ,we give necessary and sufficient criteria for the existence of a set that satisfies the generalized back - door criterion relative to a pair of variables and a dag , mag , cpdag or pag .moreover , if a generalized back - door set exists , we provide an explicit such set .these results are summarized in theorem [ thbackdoorsetforgeneralgraph ] , using a general framework that covers dags , cpdags , mags and pags .corollaries [ corbackdoorsetdag][corbackdoorsetmag ] specialize the results for dags , cpdags and mags , respectively .we illustrate our results with several examples in section [ secexamples ] .all proofs are given in section [ secproofs ] .we close this introduction by discussing related work . for a given causal dag , identifiability of causal effects in general or via covariate adjustment has been studied by various authors .in particular , there are complete graphical criteria for the identification of causal effects if a causal dag with unmeasured variables is given [ e.g. , , ( ) , ] .shpitser , van der weele androbins ( ) studied effects that are identifiable via covariate adjustment , and provided necessary and sufficient graphical criteria for this purpose , again if the causal dag is given .their results can be viewed as an improvement on the back - door criterion , which is only sufficient for adjustment . studied covariate adjustment for a given dag from an algorithmic perspective . among other things, they showed that the back - door criterion and the adjustment criterion of are equivalent if one is interested in minimal adjustment sets for a certain subclass of graphs . extended these necessary and sufficient graphical criteria for covariate adjustment to mags .there are also existing approaches that do not make the assumption that the causal dag or mag is given .the prediction algorithm [ , chapter 7 ] roughly starts from a pag and uses invariance results . in this senseit is probably closest to our work .the main difference between this method and our results is that the prediction algorithm is more complex .in particular , it searches over all possible orderings of the variables , which quickly becomes infeasible for large graphs .the prediction algorithm may , however , be more informative , in the sense that certain distributions may be identifiable by the prediction algorithm but not by the generalized back - door criterion .studying the exact relationship between these two approaches would be an interesting topic for future work .  other work on data driven methods for selection of adjustment variables forthe estimation of causal effects does not assume that the causal structure is known , but does make some assumptions about causal relationships between the variables of interest and/or about the existence of a set of variables that can be used for covariate adjustment [ ] . in the current paper , we do not make any such assumptions . on the other hand ,we start from a given dag , cpdag , mag or pag .we do not see this as a genuine restriction of our approach , however , since there are algorithms to estimate cpdags and pags from data ( e.g. , the pc algorithm [ ] , greedy equivalence search [ ] and versions of the fci algorithm [ ] ) .these algorithms have been shown to be consistent , even in certain sparse high - dimensional settings [ ] . in practice, one could therefore first employ such an algorithm , and then apply the results in the current paper .throughout this paper , we denote sets in a bold font ( e.g. , ) and graphs in a calligraphic font ( e.g. , or ) .a graph consists of a set of vertices and a set of edges .the vertices represent random variables , and the edges describe conditional independence and causal ( ancestral ) relationships .there is at most one edge between every pair of vertices , and the edge set can contain ( a subset of ) the following four edge types : ( _ directed _ ) , ( _ bi - directed _ ) , ( _ nondirected _ ) and ( _ partially directed _ ) . a _ directed graph _contains only directed edges , a _ mixed graph _ can contain directed and bi - directed edges and a _ partial mixed graph _ can contain all four edge types .the endpoints of an edge are called _ marks _ , and they can be _ tails _ , _ arrowheads _ or _ circles_. we use the symbol `` '' to denote an arbitrary edge mark .if we are only interested in the presence or absence of edges , and not in the edge marks , then we refer to the _ skeleton _ of a graph .two vertices are _ adjacent _ if there is an edge between them .the adjacency set of a vertex in , denoted by , consists of all vertices adjacent to in .a _ path _ is a sequence of distinct adjacent vertices .the _ length of a path _ equals the corresponding number of edges , in this case .the path is said to be _ out of _ ( _ into _ ) if the edge between and has a tail ( arrowhead ) at . a sub - path of from to denoted by .we denote the concatenation of paths by , so that , for example , for .we use the convention that we remove any loops that may occur due to the concatenation , so that the result does not contain duplicate vertices and is again a path .the path is a _ directed path _ from to if for all , the edge occurs , and it is a _ possibly directed path _ if for all , the edge between and is not into .a _ cycle _ occurs if there is a path between and of length greater than one , and and are adjacent . a directed path from to forms a _ directed cycle _ together with the edge , and an _ almost directed cycle _ together with the edge .a _ directed acyclic graph _ ( dag ) is a directed graph without directed cycles .an _ ancestral _graph is a mixed graph without directed and almost directed cycles .if , we say that is a _ child _ of , and is a _ parent _ of .the corresponding sets of parents and children are denoted by and .if there is a ( possibly ) directed path from to or if , then is a ( _ possible _ ) _ ancestor _ of and a ( _ possible _ ) _ descendant _ of .the sets of ancestors , descendants , possible ancestors , and possible descendants of a vertex in are denoted by an , , , and , respectively .these definitions are applied disjunctively to a set , for example , .a path is an _ unshielded triple _ if and are not adjacent .a nonendpoint vertex on a path is a _ collider _ on the path if the path contains  .a nonendpoint vertex on a path which is not a collider is a _ noncollider _ on the path .a _ collider path _ is a path on which every nonendpoint vertex is a collider .a path of length one is a trivial collider path .a bayesian network for a set of variables is a pair , where is a dag , and is a joint probability density for ( with respect to some dominating measure ) that factorizes according to : . if the dag is interpreted causally , in the sense that means that has a ( potential ) direct causal effect on , then we talk about a _ causal dag _ and a _ causal bayesian network_.  one can easily derive post - intervention densities if the causal bayesian network is given and all variables are observed .in particular , we consider interventions for [ ] , which represent outside interventions that set the variables in to their respective values in .we assume that such interventions are effective , meaning that after the intervention .moreover , we assume that the interventions are local , meaning that the generating mechanisms of the other variables , and hence their conditional distributions given their parents , do not change .we then have this is known as the g - formula or the truncated factorization formula [ ] .  in a bayesian network , the dag encodes conditional independence relationships in the density via d - separation [ ; see also definition [ defm - connection ] ] .several dags can encode the same conditional independence relationships .such dags form a markov equivalence class which can be uniquely represented by a cpdag .a cpdag is a graph with the same skeleton as each dag in its equivalence class , and its edges are either directed ( ) or nondirected ( ) .an edge in such a cpdag means that is present in every dag in the markov equivalence class , while an edge represents uncertainty about the edge marks , in the sense that the markov equivalence class contains at least one dag with and at least one dag with .( note that many authors use instead of ; we use to ensure that the cpdag satisfies the syntactic properties of a pag ; see below . )if some of the variables in a dag are unobserved , one can transform the dag into a unique _ maximal ancestral graph _( mag ) on the observed variables ; see [ ( ) , page 981 ] for an algorithm. in particular , two vertices and are adjacent in a mag if and only if no subset of the remaining observed variables makes them conditionally independent .moreover , a tail mark in a mag means that is an ancestor of in all dags represented by , while an arrowhead means that is not an ancestor of in all dags represented by .thus an edge in means that there is a directed path from to in all dags represented by , but we emphasize that it does not represent a direct effect with respect to the observed variables , in the sense that there may be other observed variables on the directed path .several different dags can lead to the same mag , and a mag represents a class of ( infinitely many ) dags that have the same d - separation and ancestral relationships among the observed variables .the mag of a causal dag is called a _causal mag_. a mag encodes conditional independence relationships via the concept of m - separation ( definition [ defm - connection ] ) . again, several mags can encode the same conditional independence relationships .such mags are called markov equivalent , and can be uniquely represented by a _ partial ancestral graph _ ( pag ) .this is a partial mixed graph with the same skeleton as each mag in its markov equivalence class . a tail mark ( arrowhead ) at an edge ( in such a pag means that ( ) in every mag in the markov equivalence class , while a circle mark at an edge represents uncertainty about the edge mark , in the sense that the markov equivalence class contains at least one mag with , and at least one mag with .we say that a density is _ compatible with a dag _ if the pair forms a causal bayesian network .a density is _ compatible with a cpdag _ if it is compatible with a dag in the markov equivalence class described by .a density is _ compatible with a mag _ if there exists a causal bayesian network ( including hidden variables ) , such that is the mag of and is the corresponding marginal of .finally , is _ compatible with a pag _ if it is compatible with a mag in the markov equivalence class described by .we now present our generalized back - door criterion in definition [ defbackdoorgraphical ] and theorem [ thgeneralizedbackdoorsufficientforadjustment ] , where the name `` generalized back - door criterion '' is motivated by lemma [ lemmaourbackdoorstrongerthanpearls ] .we first introduce some more specialized definitions . introduced the concept of ( _ definitely _ ) _ visible _ edges in mags and pags .the reason for this is as follows .a directed edge in a dag , cpdag , mag or pag always means that is a cause ( or ancestor ) of , because of the tail mark at .however , if we allow for hidden variables ( i.e. , in mags and pags ) , there may be a hidden confounding variable between and .visible edges refer to situations where there can not be such a hidden confounder between and .invisible edges , on the other hand , are possibly confounded in the sense that there is a dag represented by the mag or pag with , where is not measured ( in addition to ) .[ defvisibleedge ] all directed edges in dags and cpdags are said to be _visible_. given a , a directed edge in is _ visible _ if there is a vertex not adjacent to , such that there is an edge between and that is into , or there is a collider path between and that is into and every nonendpoint vertex on the path is a parent of .otherwise is said to be invisible .  ; , figure 6 . instead of the tail mark at , one can also have an arrowhead or circle mark . ]figure [ figvisibileedges ] illustrates the different graphical configurations that can lead to a visible edge .we note that used slightly different terminology , referring to _ definitely visible _ edges in a pag , while we simply say _visible _ for both mags and pags . used the term _ pure - causal _ edges instead of _ visible _ edges in mags .we can now generalize the concept of a _ back - door path _ in definition [ defbackdoorpath ] .[ defbackdoorpath ] let be an ordered pair of vertices in , where is a dag , cpdag , mag or pag .we say that a path between and is a _ back - door path from to _ if it does not have a visible edge out of .  in a dag, this definition reduces to a path between and that starts with , which is the usual back - door path as defined by . in a cpdag ,a back - door path from to is a path between and that starts with or . in a mag ,it is a path between and that starts with , or an invisible edge . finally , in a pag , it is a path between and that starts with , or an invisible edge .we also need generalizations of the concept of d - separation in dags [ definition 1.2.3 of ] . in mags, one can use m - separation [ section 3.4 of ] . in cpdags and pags, there is the additional complication that it may be unclear whether a vertex is a collider or a noncollider on the path .we therefore need the following definitions :  [ defdefinitenoncollider ] a nonendpoint vertex on a path in a partial mixed graph is a _ definite noncollider _ on the path if ( i ) there is a tail mark at , that is , or , or ( ii ) is unshielded and has circle marks at , that is , and and are not adjacent in .the motivation for conditions ( i ) and ( ii ) is straightforward .a tail mark out of  on the path ensures that is a noncollider on the path in any graph obtained by orienting any possible circle marks .condition ( ii ) comes from the fact that the collider status of unshielded triples is known in cpdags and pags .hence , if the graph contains an unshielded triple that was not oriented as a collider , then it must be a noncollider in all underlying dags or mags .if is a dag or a mag , then only condition ( i ) applies and reduces to the usual definition of a noncollider .a nonendpoint vertex on a path in a partial mixed graph is said to be of a _ definite status _ if it is either a collider or a definite noncollider on . the path is said to be of a _ definite status _ if all nonendpoint vertices on the path are of a definite status .a path of length one is a trivial definite status path .moreover , in dags and mags , all paths are of a definite status .we now define m - connection for definite status paths .[ defm - connection ] a definite status path between vertices and in a partial mixed graph is _m - connecting _ given a ( possibly empty )set of variables  if the following two conditions hold :  every definite noncollider on the path is not in ;  every collider on the path is an ancestor of some member of .if a definite status path is not m - connecting given , then we say that blocks  .  if , we usually omit the phrase `` given the empty set .'' definition [ defm - connection ] reduces to m - connection for mags and d - connection for dags .we note that used the notions of _ possible m - connection _ and _ definite m - connection _ in pags , where his notion of definite m - connection is the same as our notion of m - connection for definite status paths .we now define an adjustment criterion for dags , cpdags , mags and pags . throughout , we think of and as nonempty sets .[ defadjustmentcriterion ] let , and be pairwise disjoint sets of vertices in , where represents a dag , cpdag , mag or pag .then we say that satisfies the adjustment criterion relative to and if for any density compatible with , we have   if and , we simply say that a set satisfies the criterion relative to [ rather than and the given graph .we now propose our generalized back - door criterion for dags , cpdags , mags and pags .we will show in theorem [ thgeneralizedbackdoorsufficientforadjustment ] that this criterion is sufficient for adjustment .[ defbackdoorgraphical ] let , and be pairwise disjoint sets of vertices in , where represents a dag , cpdag , mag or pag .then satisfies the _generalized back - door criterion _ relative to and if the following two conditions hold :   does not contain possible descendants of in ;  for every , the set blocks every definite status back - door path from to any member of , if any , in .a set that satisfies the generalized back - door criterion relative to and is called a _ generalized back - door set _ relative to and .  [ rempossibledescendantsdefinitestatuspath ] condition ( b - i ) in definition [ defbackdoorgraphical ] is equivalent to the following :   does not contain possible descendants of _ along a definite status path _ in .  condition ( b - i) may be easier to check computationally than ( b - i ) .the equivalence of ( b - i ) and ( b - i) is shown in the proof of theorem [ thgeneralizedbackdoorsufficientforadjustment ] , using lemma [ lemmapossibledescendantsdefinitestatuspath ] .[ thgeneralizedbackdoorsufficientforadjustment ] let , and be pairwise disjoint sets of vertices in , where represents a dag , mag , cpdag or pag . if satisfies the _generalized back - door criterion _ relative to and ( definition [ defbackdoorgraphical ] ) , then it satisfies the adjustment criterion relative to and ( definition [ defadjustmentcriterion ] ) .the proof of theorem [ thgeneralizedbackdoorsufficientforadjustment ] consists of two steps .first , we formulate invariance criteria that are sufficient for adjustment ( theorem [ thinvarianceimpliesadjustment ] ) .next , we translate the invariance criteria into the graphical criteria given in definition [ defbackdoorgraphical ] , using results of ( theorem [ thbackdoorequivalenttoinvariance ] ) .we refer to definition [ defbackdoorgraphical ] as generalized back - door criterion because its conditions are closely related to pearl s original back - door criterion [ ( ) ] .  [ defbackdoorpearl ] a set of variables satisfies the back - door criterion relative to an ordered pair of variables in a dag if the following two conditions hold :  no vertex in is a descendant of in ;   blocks every path between and in that is into .  similarly ,if and are two disjoint subsets of vertices in , then is said to satisfy the back - door criterion relative to in if it satisfies the criterion relative to any pair such that and .  in particular , the conditions in definition [ defbackdoorgraphical ] are equivalent to pearl s back - door criterion for a dag with a single intervention ( ) . for a dag with multiple interventions ,any set that satisfies pearl s back - door criterion also satisfies the generalized back - door criterion , but not necessarily the other way around . in this sense ,our criterion is slightly better ; see lemma [ lemmaourbackdoorstrongerthanpearls ] and example [ exourbackdoorstrongerthanpearls ] .[ lemmaourbackdoorstrongerthanpearls ] let , and be pairwise disjoint sets of vertices in a dag .if satisfies pearl s back - door criterion ( definition [ defbackdoorpearl ] ) relative to and  , then satisfies the generalized back - door criterion ( definition [ defbackdoorgraphical ] ) relative to and .an important reason for the popularity of pearl s back - door criterion is the following .consider two distinct vertices and in a dag .then satisfies the back - door criterion relative to and , unless . in the latter case, there is no set that satisfies the back - door criterion relative to and , but it is easy to see that for any density compatible with , since there can not be a directed path from to in .  in this section , we formulate similar results for the generalized back - door criterion . in particular , we consider the following problem . given two distinct vertices and in a dag , cpdag , mag or pag , can we easily determine if there exists a generalized back - door set relative to and the given graph ? moreover , if this question is answered positively , can we give an explicit set that satisfies the criterion ?theorem [ thbackdoorsetforgeneralgraph ] addresses these questions in general , while corollaries [ corbackdoorsetdag][corbackdoorsetmag ] give specific results for dags , cpdags and mags .we emphasize that throughout this section , we focus on the setting with a single intervention variable and a single variable of interest .the setting with multiple interventions ( i.e. , a set ) is considerably more difficult , even for dags [ ] .it therefore seems challenging to generalize the results in this section to sets .handling sets seems less difficult , and we plan to study this in future work .  in a dag , the following result is well known .if and are not adjacent in a dag and , then blocks all paths between and . in mags , we have a similar result , but we need to use instead of the parent set ; see definition [ defd - sep ] and lemma [ lemmad - sep ] .[ defd - sep ] let and be two distinct vertices in a mixed graph .we say that if , and there is a collider path between and in , such that every vertex on this path ( including ) is an ancestor of or in .[ lemmad - sep ] let and be two distinct vertices in an ancestral graph .then the following statements are equivalent : and are m - separated in by some subset of the remaining variables , , and and are m - separated in by . moreover ,if is a mag , a fourth equivalent statement is and are not adjacent in .we now introduce important definitions that are needed to formulate our generalized back - door criterion in theorem [ thbackdoorsetforgeneralgraph ] .[ defr ] let be a vertex in , where is a dag , cpdag , mag or pag .let be a class of dags or mags , defined as follows .if is a dag or a mag , we simply let . if is a cpdag / pag , we let be the subclass of dags / mags in the markov equivalence class described by that have the same number of edges into as .  for any , let be the graph obtained from by removing all directed edges out of that are visible in ; see definition [ defvisibleedge ] .  for any given and ,we say that a graph satisfies definition [ defr ] if there exists an such that .lemma [ lemmamagwithsameindegreeaspag ] shows that the class is always nonempty .the definition of is related to the -lower manipulated mags that were used by .it is important to note , however , that is obtained from by removing the edges out of that are visible in ( rather than ) .moreover , zhang replaced invisible edges by bi - directed edges , but that is not needed for our purposes ( although it would not hurt to do so ) .finally , we note that is ancestral , since any is ancestral.  we can now present the main result of this section .[ thbackdoorsetforgeneralgraph ] let and be two distinct vertices in , where is a dag , cpdag , mag or pag .let be any graph satisfying definition [ defr ] .then there exists a generalized back - door set relative to and if and only if and .moreover , if such a generalized back - door set exists , then is such a set .the definitions of and in definition [ defr ] are needed in theorem [ thbackdoorsetforgeneralgraph ] to ensure that implies that there does not exist a generalized back - door set relative to and ; see also example [ expag ] . for dags , cpdags and magswe can simplify theorem [ thbackdoorsetforgeneralgraph ] somewhat ; see corollaries [ corbackdoorsetdag][corbackdoorsetmag ] .corollary [ corbackdoorsetdag ] is the well - known result for dags that we discussed earlier .corollary [ corbackdoorsetmag ] is given without proof , since it follows straightforwardly from theorem [ thbackdoorsetforgeneralgraph ] .[ corbackdoorsetdag ] let and be two distinct vertices in a dag .then there exists a generalized back - door set relative to and if and only if .moreover , if such a generalized back - door set exists , then is such a set .[ corbackdoorsetcpdag ] let and be two distinct vertices in a cpdag .let be the graph obtained from by removing all directed edges out of .then there exists a generalized back - door set relative to and if and only if and .moreover , if such a generalized back - door set exists , then is such a set .[ corbackdoorsetmag ] let and be two distinct vertices in a mag .then there exists generalized backdoor set relative to and if and only if and .moreover , if such a generalized back - door set exists , then is such a set .we now give several examples to illustrate the theory for dags , cpdags , mags and pags .we start with an example that shows that the generalized back - door criterion for dags is weaker than pearl s back - door criterion for dags , in the sense that it can happen that there is no set that satisfies pearl s back - door criterion , while there is a set that satisfies the generalized back - door criterion .  [ cols="^,^ " , ]   [ figpag->mag ]  [ expag ] consider the pag in figure [ figpag](a ) , and suppose we are interested in .note that the mag as given in figure [ figpag->mag](b ) is in  ; see definition [ defr ]. we will apply theorem [ thbackdoorsetforgeneralgraph ] using the corresponding graph , which is as but without the edge .we then have and .hence theorem [ thbackdoorsetforgeneralgraph ] implies that is a generalized back - door set relative to and  .one can easily verify that all subsets of are also generalized back - door sets relative to and , since all back - door paths from to are blocked by the collider on these paths .this shows that is not minimal .this example also shows the importance of definition [ defr ] . to see this ,let be as , but with the edge instead of , so that there is an additional edge into .then , and we get .this shows that applying theorem [ thbackdoorsetforgeneralgraph ] with instead of leads to incorrect results .in this paper , we generalize pearl s back - door criterion [ ] to a generalized back - door criterion for dags , cpdags , mags and pags .we also provide easily checkable necessary and sufficient criteria for the existence of a generalized back - door set , when considering a single intervention variable and a single outcome variable . moreover ,if such a set exists , we provide an explicit set that satisfies the generalized back - door criterion .this set is not necessarily minimal , so if one is interested in a minimal set , one could consider all subsets .although effects that can be computed via the generalized back - door criterion are only a subset of all identifiable causal effects , we hope that the generalized back - door criterion will be useful in practice , and will make it easier to work with cpdags , mags and pags .moreover , combining our results for cpdags and pags with fast causal structure learning algorithms such as the pc algorithm [ ] or the fci algorithm [ ] yields a computationally efficient way to obtain information on causal effects when assuming that the observational distribution is faithful to the true unknown causal dag with or without hidden variables . to our knowledge ,the prediction algorithm of is the only alternative approach under the same assumptions , but the prediction algorithm is computationally much more complex .the ida algorithm [ ] has been designed to obtain bounds on causal effects when assuming that the observational distribution is faithful to the true underlying causal dag without hidden variables .ida roughly combines the pc algorithm with pearl s back - door criterion .we could now apply a similar approach in the setting with hidden variables , by combining the fci algorithm with the generalized back - door criterion for mags .possible directions for future work include studying the exact relationship between the prediction algorithm and our generalized back - door criterion , generalizing the results in section [ secfindingbackdoorset ] to allow for sets and and extending the recent results of to cpdags and pags .in order to prove theorem [ thgeneralizedbackdoorsufficientforadjustment ] , we formulate so - called invariance conditions that will turn out to be sufficient for adjustment ; see definition [ definvariancecriterion ] and theorem [ thinvarianceimpliesadjustment ] below .first , we briefly define what is meant by invariance .we refer to for full details .let , and be three subsets of vertices in a causal dag , where .then a density is said to be entailed to be invariant under interventions on given if for all causal bayesian networks , where the subscript denotes .( this notation is used since and are allowed to overlap . )the density is said to be entailed to be invariant under interventions on given a cpdag , a mag or a pag if it is entailed to be invariant under interventions on given all dags represented by , or , respectively .[ definvariancecriterion ] let , and be pairwise disjoint sets of vertices in , where is a dag , cpdag , mag or pag .then satisfies the _ invariance criterion _ relative to and if the following two conditions hold for any density compatible with :   ;   .  in other words , conditions( i - i ) and ( i - ii ) state that and are entailed to be invariant under interventions on given .the conditions are also closely related to the conditions in equation ( 9 ) of .we note that condition ( i - i ) is trivially satisfied if .[ thinvarianceimpliesadjustment ] let , and be pairwise disjoint sets of vertices in , where is a dag , cpdag , mag or pag .if satisfies the invariance criterion relative to and , then it satisfies the adjustment criterion relative to and .if , condition ( i - ii ) immediately gives .otherwise , we have under conditions ( i - i ) and ( i - ii ) , the right - hand side of ( [ eq1 ] ) simplifies to . ( ) , formulated invariance results for dags , mags and pags .we derive a similar result for cpdags and then summarize the results for all these types of graphs in theorem [ thinvariancegeneral ] .  [ thinvariancegeneral ]let be three subsets of observed vertices in , where represents a dag , cpdag , mag or pag .moreover , let .then is entailed to be invariant under interventions on given if and only if :  for every , every m - connecting definite status path , if any , between and any member of given is out of with a visible edge ;  for every , there is no m - connecting definite status path between and any member of given ;  for every , every m - connecting definite status path , if any , between and any member of given is into .one can easily check that the conditions reduce to the appropriate conditions for dags , mags and pags [ , proposition 18 , theorem 24 and theorem 30 ] . the result for cpdagscan be proved analogously .note that , and form a partition of .hence , only one of the conditions in theorem [ thinvariancegeneral ] is relevant for a given .we also need the following basic property of pags and cpdags :  [ lemmapropertycp1forcpdagsandpags ] for any three vertices , and in a cpdag or pag , the following holds : if , then there is an edge between and with an arrowhead at , namely .furthermore , if the edge between and is , then the edge between and is either or ( i.e. , not ) .we now show that the invariance conditions in definition [ definvariancecriterion ] are equivalent to the graphical conditions of definition [ defbackdoorgraphical ] .[ thbackdoorequivalenttoinvariance ] the generalized back - door criterion ( definition [ defbackdoorgraphical ] ) is equivalent to the invariance criterion ( definition [ definvariancecriterion ] ) .we first show that condition ( b - ii ) of definition [ defbackdoorgraphical ] is equivalent to condition ( i - ii ) of definition [ definvariancecriterion ] .we use theorem [ thinvariancegeneral ] with , where , and .then , and clause ( 1 ) of the theorem yields that ( i - ii ) is equivalent to the following : for every , every m - connecting definite status path , if any , between and any member of given is out of with a visible edge .this is equivalent to condition ( b - ii ) by our definition of a back - door path ; see definition [ defbackdoorpath ] .  by lemma [ lemmapossibledescendantsdefinitestatuspath ] ( below ) ,condition ( b - i ) of definition [ defbackdoorgraphical ] is equivalent to condition ( b - i) of remark [ rempossibledescendantsdefinitestatuspath ] .  we now show that condition ( b - i) is equivalent to condition ( i - i ) in definition [ definvariancecriterion ] .we use theorem [ thinvariancegeneral ] with , where , and .then and clause ( 3 ) of the theorem yields that ( i - i ) is equivalent to the following condition ( i - i) : for every , every m - connecting definite status path , if any , between and any member of is into . we now show that ( i - i) is equivalent to ( b - i) .first suppose that violates ( b - i) .then there are and such that there is a possibly directed definite status path from to . since is possibly directed , it is not into and it can not contain colliders . hence , it is an m - connecting definite status path between and that is not into .this violates ( i - i) .now suppose that violates ( i - i) . then there are and such that there is an m - connecting definite status path between and that is not into .let be such a path .then every nonendpoint vertex on must be a definite noncollider .suppose that is not a possibly directed path from to , meaning that there exists an such that the edge between and is into .if , this means that the path is into , which is a contradiction . if , then the edge between and must be out of , since is a definite noncolliderbut this means that the edge must be into , since edges of the form or are not allowed .continuing this argument , we find that for all , the edge between and is into .but this means that the path is into , which is a contradiction .hence , is a possibly directed path from to .together with the fact that is of a definite status , this violates ( b - i) .[ lemmapossibledescendantsdefinitestatuspath ] let and be two distinct vertices in , where is a dag , cpdag , mag or pag .if , then there is a possibly directed definite status path from to .moreover , if for some , then for all .if is a dag or a mag , the lemma is trivially true .so let be a cpdag or a pag , and assume that .this implies that there is a possibly directed path from to in .let be a shortest such path .if is of length one , then the lemma is trivially true .so assume that the length of is at least two , that is , .we first show that is a definite status path .note that can contain the following edges , and ( ) .we now consider a sub - path of , for some .this sub - path can not be of the form or .to see this , suppose that the sub - path takes such a form .then lemma [ lemmapropertycp1forcpdagsandpags ] implies the edge .suppose that this edge is into ; that is , it is .then lemma [ lemmapropertycp1forcpdagsandpags ] applied to implies the edge , which is a contradiction .if the edge is not into , then is not a shortest possibly directed path .similarly , the sub - path can not be of the form or . to see this ,suppose that the sub - path takes such a form .then lemma [ lemmapropertycp1forcpdagsandpags ] implies the edge or . in either case , is not a shortest possibly directed path .  moreover ,if the sub - path is of the form , or , then it must be unshielded . to see this ,suppose that the sub - path takes such a form and is not unshielded .if the edge between and is into , then lemma [ lemmapropertycp1forcpdagsandpags ] applied to implies the edge , which is a contradiction .if the edge between and is not into , then is not a shortest possibly directed path .hence , can only contain triples of the form or , or of the form , or where and are not adjacent .in all these cases , the middle vertex is a definite noncollider , so that is a definite status path .finally , if for some , it follows that for all .proof of theorem [ thgeneralizedbackdoorsufficientforadjustment ] this follows directly from theorems [ thinvarianceimpliesadjustment ] and [ thbackdoorequivalenttoinvariance ] .proof of lemma [ lemmaourbackdoorstrongerthanpearls ] conditions ( p - i ) and ( b - i ) are trivially equivalent for dags . we therefore only show that ( p - ii ) implies ( b - ii ) , by contradiction . thus , suppose that blocks all back - door paths between and in , but there exist and such that there is a back - door path from to that is not blocked by .this means that : ( i ) no noncollider on is in , ( ii ) all colliders on have a descendant in , ( iii ) there is at least one collider on that has a descendant in but not in . among all colliders satisfying ( iii ) , let be the one that is closest to on , and let denote a descendant of in .then the directed path from to is m - connecting given , since it is a path consisting of noncolliders and none of its vertices are in . moreover , the sub - path of is m - connecting given by construction . butthis means that is a back - door path from to that is m - connecting given .this contradicts ( p - ii ) .we first give several lemmas , starting with a result about m - connection in mags .this result basically says that replacing condition ( b ) in definition [ defm - connection ] by `` every collider on the path is an ancestor of some member of '' does not change the m - separation relations in a mag .[ lemmarichardsoncorollary ] let and be two distinct vertices and be a subset of vertices in a mixed graph , with .if there is a path between and in on which no noncollider is in and every collider is in , then there is a path ( not necessarily the same path ) m - connecting and given in .proof of lemma [ lemmad - sep ] let be an ancestral graph .first , we note that ( iii ) trivially implies ( i ) .next , we show that ( i ) implies ( ii ) , or equivalently , that not ( ii ) implies not ( i ) . thus , suppose that . then there is a collider path between and such that every vertex on the path is an ancestor of in .this path is m - connecting given any subset of the remaining vertices , by lemma [ lemmarichardsoncorollary ] .next , we show that ( ii ) implies ( iii ) .suppose that .if there is no path between and in , then and are trivially m - separated by any subset of the remaining vertices .thus , assume that there is at least one path between and .consider an arbitrary such path , and call it .since , we have .hence the length of must be at least two .we will show that is blocked by .suppose starts with .then , since .since is a noncollider on , this implies that is blocked by .suppose is of the form . then , so that .since is a noncollider on , this implies that is blocked by .suppose starts with and the sub - path of contains at least one collider .let be the collider closest to on . then .if , then is blocked by . hence , suppose .since any vertex in is an ancestor of in , this implies and hence and .since is a noncollider on , is blocked by .suppose is a collider path of the form .then at least one of the colliders is not in , since otherwise .let be the collider closest to on that is not in . then .hence , is blocked by .suppose is of the form , with ( was treated in the previous case ) and the sub - path is allowed to be of length one ( i.e. , ) .if , then is blocked by .so suppose that .then there does not exist a collider path between and such that each vertex on the path is in .this implies that there is a collider on the sub - path of that is not in . among such vertices ,let be the one that is closest to on . then .hence , is blocked by .  finally , if is a mag , two vertices are adjacent if and only if no subset of the remaining variables can m - separate them .hence , ( i ) and ( iv ) are equivalent for mags .the following lemma says that we can check the existence of m - connecting definite status back - door paths in by checking the existence of m - connecting paths in , where is any graph satisfying definition [ defr ] .this lemma is closely related to lemma 5.1.7 of and lemmas 26 and 27 of .[ lemmam - seppag2mag ] let and be two distinct vertices and be a subset of vertices in , where is a dag , cpdag , mag or pag .let be any graph satisfying definition [ defr ] .then there is a definite status m - connecting back - door path from to given in if and only if there is an m - connecting path between and given in .let and satisfy definition [ defr ] .we first prove the `` only if '' statement .suppose there is a definite status m - connecting back - door path from to given in .let and be the corresponding paths in and , consisting of the same sequence of vertices .( note that exists by the definition of and the fact that is a back - door path in . )then the path is m - connecting given in .the path , however , is not necessarily m - connecting in , since it may happen that there is a collider on the path such that but . butthis can only occur if .hence , satisfies the following properties : no noncollider on is in and every collider on is in .it then follows from lemma [ lemmarichardsoncorollary ] that there is an m - connecting path between and given in .we now prove the `` if '' statement .suppose that there is an m - connecting path between and given in .let be the corresponding path in , consisting of the same sequence of vertices .then is also m - connecting given in .moreover , does not start with a visible edge out of in , because exists in . by lemma 2 in the proof of lemma 5.1.7 of , it then follows that there exists an m - connecting definite status back - door path between and given in .the next lemma is used several times to derive a contradiction .[ lemmapossiblydirectedpathandedgeinto ] let and be two distinct vertices in , where denotes a dag , cpdag , mag or pag .then can not have both a possibly directed path from to and an edge of the form .this lemma is trivial for dags and mags , since they can not contain ( almost ) directed cycles .so we only show the result for cpdags and pags .let denote the cpdag or pag , and suppose that contains an edge of the form as well as a possibly directed path from to in .then there is also a possibly directed definite status path from to in  , by lemma [ lemmapossibledescendantsdefinitestatuspath ] .the path has the following properties : if for some , then for all , and the length of must be at least two , because of the edge .if is fully directed , there is an ( almost ) directed cycle in any dag or mag in the markov equivalence class described by , which violates the ancestral property .  otherwise , if contains a directed sub - path , let be the longest directed sub - path. then the sub - path must be of the form or . in either case ,the edge and repeated applications of lemma [ lemmapropertycp1forcpdagsandpags ] imply the edge .this gives an ( almost ) directed cycle together with the directed path in any dag or mag in the markov equivalence class described by .this again contradicts the ancestral property .otherwise , does not contain a directed sub - path .let be the vertex preceding on the path .then the path has one of the following two forms : or .the edge and repeated applications of lemma [ lemmapropertycp1forcpdagsandpags ] yield the edge , which contradicts or .theorem [ thbackdoorsetforgeneralgraph ] requires a dag or mag in ; see definition [ defr ] .the following lemma establishes such a dag or mag exists , since is always nonempty .this result is closely related to constructions in , theorem 2 of and lemma 27 of .[ lemmamagwithsameindegreeaspag ] let be a pag ( cpdag ) with edges into , .then there exists at least one mag ( dag ) in the markov equivalence class represented by that has edges into .building on the work of , theorem 2 of gives a procedure to create a mag ( dag ) in the markov equivalence class represented by a pag ( cpdag ) .one first replaces all partially directed ( ) edges in by directed ( ) edges .next , one considers the circle component of , that is , the sub - graph of consisting of nondirected ( ) edges and orients this into a directed graph without directed cycles and unshielded colliders .the first step of this procedure only creates tail marks , and hence can not yield an additional edge into . for the second step , we will argue that we can construct such a graph that does not have any edges into .first , we note that is chordal ; that is , any cycle of length four or more has a chord , which is an edge joining two vertices that are not adjacent in the cycle ; see the proof of lemma 4.1 of .any chordal graph with more than one vertex has two simplicial vertices , that is , vertices such that all vertices adjacent to are also adjacent to each other [ e.g. , ] .hence , must have at least one simplicial vertex that is different from .we choose such a vertex and orient any edges incident to into .since is simplicial , this does not create unshielded colliders .we then remove and these edges from the graph .the resulting graph is again chordal [ e.g. , ] and therefore again has at least one simplicial vertex that is different from .choose such a vertex , and orient any edges incident to into .we continue this procedure until all edges are oriented .the resulting ordering is called a perfect elimination scheme for . by construction, this procedure yields an acyclic directed graph without unshielded colliders .moreover , since is chosen as the last vertex in the perfect elimination scheme , we do not orient any edges into .  [ lemmaancestorofy ] let and be two distinct vertices in , where is a dag , cpdag , mag or pag .let be any graph satisfying definition [ defr ] .if , then .let satisfy definition [ defr ] , and let .this means that there is a collider path between and in such that every vertex on the path is an ancestor of or in .in particular , .  we first show that leads to a contradiction .thus , suppose there is an edge in . by construction of , then contains an edge of the form or , but this forms a contradiction together with , by lemma [ lemmapossiblydirectedpathandedgeinto ] .we now show that leads to a contradiction .thus suppose there is a directed path from to in of the form , where and . by construction of , the edge must also be into in , so that contains or .since , there is a possibly directed path from to in .since contains a directed path from to , must also contain a possibly directed path from  to .this implies that is a possibly directed path from to in , so that .but this forms a contradiction with or in , by lemma [ lemmapossiblydirectedpathandedgeinto ] .hence , we must have .  we can now prove the main result in section [ secfindingbackdoorset ] .proof of theorem [ thbackdoorsetforgeneralgraph ] let satisfy definition [ defr ] . we first show that or implies that there does not exist a generalized back - door set relative to and , since no set can satisfy conditions ( b - i ) and ( b - ii ) in definition [ defbackdoorgraphical ] .thus suppose that .then there is a definite status back - door path of length one in that can not be blocked . hence condition ( b - ii )can not be satisfied by any set .next , suppose that there exists some vertex . then there is a collider path between and in such that every vertex on the path is in .moreover , by lemma [ lemmaancestorofy ] , there is a directed path from to in . now consider .all nonendpoint vertices on that are not on are colliders on and in .the remaining nonendpoint vertices on are noncolliders and in [ since , so that including them in violates condition ( b - i ) .it then follows by lemma [ lemmarichardsoncorollary ] that for any subset satisfying condition ( b - i ) , there exists an m - connecting path between and given in . by lemma [ lemmam - seppag2mag ] , this means that we can not block all definite status back - door paths from to in without violating condition ( b - i ) .we now prove the other direction .thus suppose that and .then we need to show that satisfies conditions ( b - i ) and ( b - ii ) of definition [ defbackdoorgraphical ] .condition ( b - i ) is satisfied trivially , since . to prove that condition ( b - ii ) is satisfied as well , we first show , by contradiction . thus ,suppose .by lemma [ lemmad - sep ] , this implies . since , this implies that in with a visible edge .but this means that , so that .this is a contradiction , which implies .hence m - separates and in by lemma [ lemmad - sep ] ( we use here that is ancestral ) . by lemma [ lemmam - seppag2mag ], this implies that blocks all definite status back - door paths from to in , so that condition ( b - ii ) is satisfied .proof of corollary [ corbackdoorsetdag ] although this result for dags is well known , we show how one can derive this from theorem [ thbackdoorsetforgeneralgraph ] .note that is the graph obtained by removing all directed edges out of from .moreover , and .now the condition is equivalent to .the other condition reduces to , and this is fulfilled automatically by the acyclicity of .hence theorem [ thbackdoorsetforgeneralgraph ] reduces to the given statement .proof of corollary [ corbackdoorsetcpdag ] let be a dag in the markov equivalence class represented by , constructed without orienting additional edges into .let be obtained from by removing all directed edges out of that were directed out of in .let be obtained from by removing all directed edges out of .we first show that or imply or .thus suppose . then .next , suppose .it can be easily shown that satisfies the basic property of lemma [ lemmapropertycp1forcpdagsandpags ] , that implies ( since all edges that are removed are directed edges out of ) .hence , lemma [ lemmapossibledescendantsdefinitestatuspath ] applies to , and it follows that there is a possibly directed definite status path from to in .all nonendpoint vertices on this path must be definite noncolliders . by construction of , the first edge on this pathmust be nondirected in , and by construction of , this edge must be oriented out of in .this implies that the entire path must be directed from to in , since all nonendpoint vertices are noncolliders .let be the vertex adjacent to on the path. then .moreover , .hence .we now show that or imply or .thus suppose that imply or .thus suppose .then either or in .this implies that or .next , suppose that there exists a vertex .note that implies : ( i ) or ( ii ) or ( iii ) . by construction of ,case ( i ) implies .but this is in contradiction with , by lemma [ lemmapossiblydirectedpathandedgeinto ] . in case ( ii ), we have and a directed path from to in , so that .similarly , we can obtain in case ( iii ) .this implies in cases ( ii ) and ( iii ) .the above shows the following : if or , then it is impossible to satisfy the generalized back - door criterion relative to and . on the other hand ,if and , then satisfies the generalized back - door criterion relative to and .it is left to show that in the latter case , we can replace by . since , it is clear that satisfies condition ( b - i ) of definition [ defbackdoorgraphical ] .we will now show that it also satisfies condition ( b - ii ) .thus , suppose that and .consider a definite status back - door path from to in .since is a back - door path , it must start with or .moreover , the length of is at least two . if , then it is clear that blocks .if , then can not have a sub - path of the form , , because is of a definite status .moreover , can not be possibly directed , because then .hence , there must be at least one collider on .let be the collider on that is closest to .then the sub - path is a possibly directed path from to in .suppose that is an ancestor of some vertex in .then there is a possibly directed path from to in , as well as an edge .but this is impossible by lemma [ lemmapossiblydirectedpathandedgeinto ] .hence , can not be an ancestor of any member of in .this implies that is blocked by .we are very grateful to markus kalisch , thomas richardson and two anonymous referees for their comments and suggestions that have significantly improved the paper .

Abstract: we generalize pearl s back - door criterion for directed acyclic graphs ( dags ) to more general types of graphs that describe markov equivalence classes of dags and/or allow for arbitrarily many hidden variables . we also give easily checkable necessary and sufficient graphical criteria for the existence of a set of variables that satisfies our generalized back - door criterion , when considering a single intervention and a single outcome variable . moreover , if such a set exists , we provide an explicit set that fulfills the criterion . we illustrate the results in several examples . r - code is available in the ` pcalg ` .  ./style / arxiv - general.cfg


unsupervised neural networks assume unlabeled data to be generated from a neural network structure , and have been applied extensively to pattern analysis and recognition .the most basic one is the restricted boltzmann machine ( rbm ) [ 12 ] , an energy - based model with a layer of hidden nodes and a layer of visible nodes .with such a basic structure , we can stack multiple layers of rbms to create a unsupervised deep neural network structure , such as the deep belief network ( dbn ) and the deep boltzmann machine ( dbm ) [ 7 , 13 ] .these models can be calibrated with a combination of the stochastic gradient descent and the contrastive divergence ( cd ) algorithm or the pcd algorithm [ 12 , 19 ] .once we learn the parameters of a model , we can retrieve the values of the hidden nodes from the visible nodes , thus applying unsupervised neural networks for feature selection .alternatively , we may consider applying the parameters obtained from a unsupervised deep neural network to initialize a deep feedforward neural network ( ffnn ) , thus improving supervised learning .one essential question for such models is to adjust for the high - dimensionality of their parameters and avoid overfitting . in ffnns ,the simplest regularization is arguably the early stopping method , which stops the gradient descent algorithm before the validation error rate goes up .the weight decay method , or regularization , is also commonly used [ 23 ] .recently dropout is proposed , which optimizes the parameters over an average of exponentially many models with a subset of all nodes [ 17 ] .it has been shown to outperform weight decay regularization in many situations .  for regularizing unsupervised neural networks ,sparse - rbm - type models encourage a smaller proportion of -valued hidden nodes [ 4 , 9 ] .dbns are regularized in [ 5 ] with outcome labels .while these works tend to be goal - specific , we consider regularization for unsupervised neural networks in a more general setting . our work and contributionsare as follows : ( 1 ) we extend common regularization methods to unsupervised deep neural networks , and explain their underlying mechanisms ; ( 2 ) we propose partial dropout / dropconnect which can improve the performance of dropout / dropconnect ; ( 3 ) we compare the performance of different regularization methods on real data sets , thus providing suggestions on regularizing unsupervised neural networks .we note that this is the very first study illustrating the mechanisms of various regularization methods for unsupervised neural nets with model convergence and likelihood bounds , including the effective newly proposed partial dropout / dropconnect .section 2 reviews recent works for regularizing neural networks , and section 3 exhibits rbm regularization as a basis for regularizing deeper networks .section 4 discusses the model convergence of each regularization method .section 5 extends regularization to unsupervised deep neural nets .section 6 presents a numerical comparison of different regularization methods on rbm , dbn , dbm , rsm [ 14 ] and gaussian rbm [ 12 ] .section 7 discusses potential future research and concludes the paper .to begin with , we consider a simple ffnn with a single layer of input and a single layer of output .the weight matrix is of size .we assume the relation where is the activation function , such as the sigmoid function applied element - wise .equation ( 1 ) has the modified form in [ 17 ] , where denotes element - wise multiplication , thereby achieving the dropout ( do ) regularization for neural networks . in dropout, we minimize the objective function ,\ ] ] which can be achieved by a stochastic gradient descent algorithm , sampling a different mask per data example and per iteration .we observe that this can be readily extended to deep ffnns .dropout regularizes neural networks because it incorporates prediction based on any subset of all the nodes , therefore penalizing the likelihood .a theoretical explanation is provided in [ 20 ] for dropout , noting that it can be viewed as feature noising for glms , and we have the relation here for simplicity , and , where is the log - partition function of a glm .therefore , dropout can be viewed approximately as the adaptive regularization [ 2 , 20 ] .a recursive approximation of dropout is provided in [ 2 ] using normalized weighted geometric means to study its averaging properties .an intuitive extension of dropout is dropconnect ( dc ) [ 21 ] , which has the form below and thus masks the weights rather than the nodes .the objective has the same form as in ( 3 ) .there are a number of related model averaging regularization methods , each of which averages over subsets of the original model .for instance , standout varies dropout probabilities for different nodes which constitute a binary belief network [ 1 ] .shakeout adds additional noise to dropout so that it approximates elastic - net regularization [ 8 ] .drop - in considers the skipping of layers for deep ffnns [ 16 ] .fast dropout accelerates dropout with gaussian approximation [ 22 ] .we note that while dropout has been discussed for rbms [ 17 ] , to the best of our knowledge , there is no literature extending common regularization methods to rbms and unsupervised deep neural networks ; for instance , adaptive regularization and dropconnect as mentioned .therefore , we discuss their implementations and examine their empirical performance below .in addition to studying model convergence and likelihood bounds , we propose partial dropout / dropconnect which randomly drops a subset of nodes or edges based on a first calibrated model , therefore improving robustness in many situations .for a restricted boltzmann machine , we assume denotes the visible vector , and denotes the hidden vector .each , is a visible node and each , is a hidden node .the joint probability is we let , i.e. a vector containing all components of , , and . to calibratethe model is to find .an rbm is a neural network because we have the following relations where and represent , respectively , the -th row and -th column of . the gradient descent algorithm is applied to calibration , noting that where is the free energy .the right - hand side of ( 8) is approximated by contrastive divergence with steps of gibbs sampling ( cd- ) [ 12 ] .weight decay , or regularization , adds the term to the negative log - likelihood of an rbm .the most commonly used is ( ridge regression ) , or ( lasso ) . in all situations, we do not regularize biases for simplicity .here we consider a more general form .suppose we have a trained set of weights from cd with no regularization . instead of adding the term , we add the term to the negative log - likelihood .apparently this adjusts for the different scales of the components of .we refer to this approach as adaptive .we note that adaptive is the adaptive lasso [ 24 ] , and adaptive plus is the elastic - net [ 25 ] .we consider the performance of regularization plus adaptive regularization ( ) below .   as discussed in [ 17 ] , to characterize a dropout ( do ) rbm , we simply need to apply the following conditional distributions therefore , given a fixed mask , we actually have an rbm with all visible nodes and hidden nodes . hidden nodes are fixed to zeroso they have no influence on the conditional rbm .apart from replacing ( 7 ) with ( 9 ) , the only other change needed is to replace with . in terms of training , we suggest sampling a different mask per data example and per iteration as in [ 17 ] .  a dropconnect ( dc ) rbm is closely related ; given a mask on weights , in a plain rbm is replaced by everywhere .we suggest sampling a different mask per mini - batch since it is usually much larger than a mask in a dropout rbm .there are typically many nodes or weights which are of little importance in a neural network . in networkpruning , such unimportant nodes or weights are discarded , and the neural network is retrained .this process can be conducted iteratively [ 11 ] .now we consider two variants of network pruning for rbms .for an trained set of weights with no regularization , we consider implementing a fixed mask where i.e. is the -th left percentile of all , and is some fixed proportion of retained weights .we then recalibrate the weights and biases fixing mask , leading to a simple network pruning ( snp ) procedure which deletes of all weights .we may also consider deleting of all weights at a time , and conduct the above process times , leading to an iterative network pruning ( inp ) procedure .we may consider combining some of the above approaches .for instance , [ 17 ] considered a combination of and dropout .we introduce two such hybrid approaches , namely partial dropconnect ( pdc ) and partial dropout ( pdo ) , which generalizes dropconnect and dropout , and borrows from network pruning .the rationale comes from some of the model convergence results below .  again , suppose we have a trained set of weights with no regularization .instead of implementing a fixed mask , we perform dropconnect regularization with different retaining probabilities for each weight .we let , and   therefore , we sample a different per mini - batch , which means we always keep of all the weights , and randomly drop the remaining weights with probability . the mask can be adjusted iteratively .intuitively , we are trying to maximize the following .\ ] ] + we propose this technique because we hypothesize that some weights could be more important than others _ a posteriori _ , so dropping them could cause much variation among the models being averaged .thus , it could be preferable not to drop them .we demonstrate this point empirically later .we also consider partial dropout which is analogous to partial dropconnect and does not drop some nodes rather than weights .we let a mask for nodes , , where this algorithm protects more important hidden nodes from being dropped in order to reduce variation .we also evaluate its empirical performance later .here we discuss the model convergence properties of different regularization methods when the number of data examples .we mark all regularization coefficients and parameter estimates with when there are data examples .we assume , which is compact , , is unique for each , and are i.i.d . generated from an rbm with a `` true '' set of parameters .we denote each regularized calibrated set of parameters as .let and .[ 24 ] showed that guarantees asymptotic normality and identification of set for linear regression .we demonstrate that similar results hold for for rbms .we let and be the and regularization coefficients for each component .we have the following ._ proposition 1_. ( a ) if , as , then the estimate ; ( b ) if also , , , then , where is the fisher information matrix ; , where .  _ proof_. for all proofs , see appendix .  for dropout and dropconnect rbms , we also assume that the data is generated from a plain rbm structure .we assume is of size as in ( 12 ) for dropconnect and of length as in ( 13 ) for dropout , therefore covering the cases of both original and partial dropout / dropconnect . with a decreasing dropping rate with , the following result holds ._ proposition 2_. if as , then .  for network pruning , we show that as the number of data examples increase , if the retained proportion of parameters can cover all nonzero components of , we will not miss any important component ._ proposition 3_. assume . then for simple network pruning , as , ( a ) ; ( b ) for sufficiently large , there exists such that .  _ corollary 1_. the above results also hold for iterative network pruning. note that for all regularization methods , under the above conditions , the calibrated weights converge to the `` true '' set of parameters , which indicates consistency . also , adding regularization guarantees that we can identify components of zero value with infinitely many examples .the major benefits of dropout come from the facts that it makes regularization adaptive , and also encourages more confident prediction of the outcomes [ 22 ] .we propose partial dropconnect also base on proposition 3 , i.e. we do not drop the more important components of , therefore possibly reducing variation caused by dropping influential weights .partial dropout follows from the same reasoning .we consider the multilayer network below , where each probability on the right - hand side is from an rbm .to train the weights of , , , we only need to carry out a greedy layer - wise training approach , i.e. we first train the weights of , and then use to train , etc . the weights of the rbms are used to initialize a deep ffnn which is finetuned with gradient descent .apparently rbm regularization can be extended to each layer of a dbn .  herewe show that adding layers to a dropout / drop - connect dbn improves the likelihood given symmetry of the weights of two adjacent layers .similar results for plain dbn are in [ 3 , 7 ] .we demonstrate this by using likelihood bounds .we let denote an -layer dbn and denote an -layer dbn with the first layers being the same as in . for a data example of a visible vector , the log - likelihood is bounded as follows , \\ & \ge e_m[h_{p_{dbn_l}(h^l|v , m ) } ] + \sum_{h^l } e_{m , m^\ast}\{p_{dbn_l}(h^l|v , m)\\  & { \hspace{1em } } \cdot [ \log p_{rbm_{l+1}}(h^l|m^\ast)+\log p_{dbn_l}(v|h^l , m)]\}. { \refstepcounter{equation}\tag{\theequation}}\end{aligned}\ ] ]  here , is the entropy function , and the derivation is analygous to section 11 in [ 3 ] .mask is for , and mask is for the new -th layer .note that after we have trained the first layers , and initialized the -th layer symmetric to the -th layer , assuming a constant dropping probability , = e_{m}[\log p_{dbn_l}(h^l|m)],\ ] ] so has the same log - likelihood bound as .training , $ ] is guaranteed to increase , and therefore the likelihood of is expected to improve . therefore , for regularized unsupervised deep neural nets , adding layers also tend to elevate the explanatory power of the network . adding nodes has the same effect , providing a rationale for deep and large - scale networks .we present the following preposition ._ proposition 4_. adding nodes or layers ( preserving weight symmetry ) to a dropout / dropconnect dbn continually improves the likelihood ; also , adding layers of size continually improves the likelihood .   unlike a dbn which trains model ( 14 ) ,a dbm [ 13 ] trains an undirected graphical model .the major difference is that in the training procedure for a dbm , both the visible and hidden nodes in the deepest hidden layer are doubled .the weights for the doubled nodes are tied to the original weights .  for a replicated softmax model ( rsm ) [ 14 ] , the energy function and for a gaussian rbm[ 12 ] , the energy function where is element - wise .they are simple extensions of an rbm to count outcomes and real - valued outcomes .rbm regularization can be easily extended to all these situations .we compare the empirical performance of the aforementioned regularization methods on the following data sets : mnist , norb ( image recognition ) ; 20 newsgroups , reuters21578 ( text classification ) ; isolet ( speech recognition ) .all results are obtained using geforce gtx titan x in theano .we consider the following unsupervised neural network structures : dbn / dbm for mnist ; dbn for norb ; rsm plus logistic regression for 20 newsgroups and reuters21578 ; grbm for isolet .cd- is performed for the rest of the paper .the following regularization methods are considered : none ( no regularization ) ; do ; dc ; ; ; snp ; inp( ) ; pdo ; pdc .the number of pretraining epochs is per layer and the number of finetuning epochs is , with a finetuning learning rate of . for , snp , and inp which need re - calibration , we cut the epochs into two halves ( quarters for inp ) .for regularization parameters , we apply the following ranges : for do / dc / snp / inp ; for , similar to [ 6 ] ; for ; , or the reverse for pdo / pdc .we only make one update to the `` partial '' dropping rates to keep simple . from the results, we note that unsupervised neural networks tend to need less regularization than ffnns .we choose the best iteration and regularization parameters over a fixed set of parameter values according to the validation error rates .the http://yann.lecun.com/exdb/mnist/[mnist ] data set consists of pixels of handwritten - digits .there are training examples , validation and testing examples .we first consider the likelihood of the testing data of an rbm with nodes for mnist .there are two model fitting evaluation criteria : pseudo - likelihood and ais - likelihood [ 15 ] .the former is a sum of conditional likelihoods , while the latter directly estimates with ais .figure 1 is using log - scale .pretraining epochs .right : ais - likelihood of the rbm over pretraining epochs . ]here , for do , and for .these figures tend to be representative of the model fitting process .pseudo - likelihood is a more optimistic estimate of the model fitting .we observe that dropout outperforms the other two after about epochs , and does not improve pseudo - likelihood . in terms of ais - likelihood , which is a much more conservative estimate of the model fitting , the fitting process seems to have three stages : ( 1 ) initial fitting ; ( 2 ) overfitting ; ( 3 ) re - fitting .we observe that improves the likelihood significantly , while dropout catches up at about epochs .therefore , both dropout and can improve model fitting according to the likelihood .classification error rates tend to be a more practical measure .we first consider a -hidden - layer dbn with nodes per layer , pretraining learning rate , and batch size ; see table 1 .we tried dbns of , , and hidden layers and found the aforementioned structure to perform best with none as baseline .the same was done for all other structures .we calculate the means of the classification errors for each regularization method averaged over random replicates and their standard deviations . in each table, we stress in bold the top performers with ties broken by deviation .we note that most of the regularization methods tend to improve the classification error rates , with dc and pdo yielding slightly higher error rates than no regularization ..classification errors for a -layer dbn for the mnist data set .[ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ]   from the above results , we observe that regularization does improve the structure of unsupervised deep neural networks and yields lower classification error rates for each data set studied herein .the most robust methods which yield improvements for all six instances are , , and pdc .snp is also acceptable , and preferable over inp .pdo can yield improvements for dropout when dropout is unsuitable for the network structure .pdc turns out to be the most stable method of all , and thus the recommended choice .regularization for deep learning has aroused much interest , and in this paper , we extend regularization to unsupervised deep learning , i.e. for dbns and dbms .we proposed several approaches , demonstrated their performance , and empirically compared the different techniques .for the future , we suggest that it would be of interest to consider more variants of model averaging regularization for supervised deep learning as well as better hybrid regularization methods .

Abstract: unsupervised neural networks , such as restricted boltzmann machines ( rbms ) and deep belief networks ( dbns ) , are powerful tools for feature selection and pattern recognition tasks . we demonstrate that overfitting occurs in such models just as in deep feedforward neural networks , and discuss possible regularization methods to reduce overfitting . we also propose a `` partial '' approach to improve the efficiency of dropout / dropconnect in this scenario , and discuss the theoretical justification of these methods from model convergence and likelihood bounds . finally , we compare the performance of these methods based on their likelihood and classification error rates for various pattern recognition data sets .   content areas : deep learning , neural networks , unsupervised learning


the transmission rate of an infectious disease is the rate of which susceptible individuals become infected . in section 3.4.9 of , the authors state that _... the direct measurement of the transmission rate is essentially impossible for most infections .but if we wish to predict the changes wrought by public health programmes , we need to know the transmission rate ... . _ the transmission rate of many acute infectious diseases varies significantly in time and frequently exhibits significant seasonal dependence : influenza , pneumococcus , and rotavirus cases peak in winter ; rsv and measles cases peak in spring ; and polio cases peak in summer .measles outbreaks in several uk cities ( pre - vaccination ) followed a two - year cycle ( see figure 3(a)(b ) ) , and many investigators have attempted to explain this two - year cycle using mathematical models . since measles cases , as well as cases of several other childhood viral diseases such as pertussis , seem to strongly correlate with school terms and breaks , most previous modelers have assumed that the measles transmission rate can be represented by a simple sinusoidal or haar function with one - year period corresponding to the school year .this assumption is based on little empirical evidence and ignores the seasonal changes of the environment , immune system , etc . on the transmission rate .for example , since measles spreads through respiration directly or indirectly through aerosol , the high absolute humidity indoors and outdoors during winter may constrain both transmission efficiency and the virus survival rate .most investigators estimate the transmission rate of an infectious disease using the formula where , for instance , are the fractions of susceptible and infected individuals during week  .this formula requires knowledge of , which is usually very difficult to estimate .our new algorithm obviates this requirement .  toward obtaining our new transmission rate recovery algorithm, we first consider the simplest sir transmission model ( ) and allow the transmission rate to be a time - dependent function , _i.e. _ , there is a positive function such that where , and are the fractions of susceptible , infected , and removed individuals at time .we begin by asking the mathematical question :  _ given smooth infection data on time interval  ] to compute on the given  interval  ] represents the continuous infection data , and figure 1(a ) contains plots of both ( solid ) and its associated transmission rate function ( dashed ) .the second data set simulates an infectious disease with periodic outbreaks that decays in time , as observed in influenza .the periodic function \exp(-0.1 t) ]  }. ] .: :  apply the formula  \right. ] .  with variable birth rate and constant death rate ,then the seir model becomes in this case , the formula in step 4 should be }.\ ] ] all other steps in the algorithm remain the same .beggs cb , shepherd sj , kerr kg ( 2010 ) potential for airborne transmission of infection in the waiting areas of healthcare premises : stochastic analysis using a monte carlo model .bmc infectious diseases 10 : 247 - 254 .ellner sp , bailey ba , bobashev gv , gallant ar , grenfell bt , nychka dw ( 1998 ) noise and nonlinearity in measles epidemics : combining mechanistic and statistical approaches to population modeling .the american naturalist 151 : 425 - 440 .fujinami rs , sun x , howell jm , jenkin jc , burns jb ( 1998 ) modulation of immune system function by measles virus infection : role of soluble factor and direct infection .journal of virology 72 : 9421 - 9427 .van den hof s , meffre cm , conyn - van spaendonck ma , woonink f , de melker he , van binnendijk rs ( 2001 ) measles outbreak in a community with very low vaccine coverage , the netherlands .emerg infect dis .7 : 593 - 597 .wichmann o , siedler a , sagebiel d , hellenbrand w , santibanez s , mankertz a , vogt g , treeck u , krause g ( 2009 ) further efforts needed to achieve measles elimination in germany : results of an outbreak investigation .bull world health organ 87 : 108 - 115 .( a ) we extract equally spaced data points from the periodic function  ] ; the dashed curve is recovered from using ( [ equ8 ] ) .( b ) these transmission functions are estimated using spline and trigonometric interpolations on the data points .( a ) aggregated monthly measles data from england and wales in .( b ) fourier transform of filtered and smoothly interpolated aggregated monthly data showing the dominant frequency components ( normalized modulus ) .note : we filter and remove the artificial peak at zero frequency in fourier transform .( c ) uk birth rates during .( d ) fourier transform of smoothly interpolated uk birth data showing the dominant frequency component ( normalized modulus ) .the transmission rate function recovered from our extended algorithm with historic birth rates .( a ) the recovered with : fast increasing peaks .( b ) fourier transform of filtered showing the dominant frequency component , per year .( c ) the recovered with : slowly increasing peaks .( d ) fourier transform of filtered showing two comparable dominant frequencies components , and per year .( e ) the recovered with : stationary peaks .( f ) fourier transform of filtered showing the dominant frequency component , per year .( g ) the recovered with : stationary peaks .( h ) fourier transform of filtered showing the dominant frequency component , per year ; per year peak is large and comparable to per year peak .we test our estimations of with data correction .the moduli of fourier transform of with the correction factor have identical spectral peaks as those without data correction in figure 4 .( a ) the recovered with .( c ) the recovered with .( e ) the recovered with .( g ) the recovered with .panels ( b)(d)(f)(h ) plot moduli of fourier transform of corresponding in ( a)(c)(e)(g ) , respectively . { \includegraphics[width=2.1in]{measlesmonthly.eps } } \subfigure [ ] { \includegraphics[width=2.2in]{monthlydataft1.eps}}\\ \subfigure [ ] { \includegraphics[width=2.3in]{birthdata.eps } } \subfigure [ ] { \includegraphics[width=2.1in]{birthdataspectrogram.eps } } \end{array} ]   { \includegraphics[width=2.5in]{ini120.eps } } \subfigure [ ] { \includegraphics[width=2in]{ini120spectrum.eps}}\\ \subfigure [ ] { \includegraphics[width=2.5in]{ini100.eps } } \subfigure [ ] { \includegraphics[width=2in]{ini100spectrum.eps}}\\ \subfigure [ ] { \includegraphics[width=2.5in]{ini60.eps } } \subfigure [ ] { \includegraphics[width=2in]{ini60spectrum.eps } } \\ \subfigure [ ] { \includegraphics[width=2.5in]{ini30.eps } } \subfigure [ ] { \includegraphics[width=2in]{ini30spectrum.eps } } \end{array}$ ]

Abstract: * abstract *  the transmission rate of many acute infectious diseases varies significantly in time , but the underlying mechanisms are usually uncertain . they may include seasonal changes in the environment , contact rate , immune system response , etc . the transmission rate has been thought impossible to measure directly . we present an algorithm to recover the time - dependent transmission rate directly from prevalence data , which makes no assumptions about the number of susceptibles , vital rates , etc .   the algorithm is derived from the complete and explicit solution of a mathematical inverse problem for sir - type transmission models . we illustrate the algorithm with the historic ( pre - vaccination ) uk measles data . fourier analysis of the recovered transmission rate yields robust spectral peaks with - and -year periods . many modelers have assumed that measles transmission is entirely driven by school contacts and can be represented by a simple sinusoidal or haar function with one - year period . our algorithm also provides a new method to estimate the initial transmission rate via stabilizing the recovered transmission rate function .   the main objective of this work is to provide a new algorithm to recover the transmission rate function directly from infection data . our algorithm also yields that almost any infection profile can be perfectly fitted by an sir - type model with variable transmissibility . this clearly illustrates a serious danger of overfitting an sir transmission model with time - dependent transmission rate .