discrete choice models form the basis for understanding the behavioural process that results in a choice made by a decision maker ( or agent ) among a finite set of alternatives .they are highly flexible and can be applied in a wide variety of choice situations .for example , the agents can be consumers choosing between different brands in a product category ; or households selecting among different types of heating systems .discrete choice models are widely used to predict demand for new systems in transportation planning , develop pricing policies in marketing research , elicit preferences for healthcare products and programmes in health economics and in many other applications .the mixed multinomial logit ( mmnl ) model is a popular discrete choice model that captures heterogeneity in the preferences of decision makers through random coefficients ( see , e.g. * ? ? ?choice probabilities of the mmnl model are expressed in the form where denotes the number of alternatives , is a vector of observed variables relating to alternative and agent , and is a vector of random coefficients that vary over agents in the population with mixing distribution . the distribution may be discrete or continuous ( e.g. normal , lognormal , triangular and uniform ) . the parameters of , and in some cases , the values of ( which represent the preferences of individual decision makers ) are of interest .the importance of the mmnl model and its ability to accommodate heterogeneity is well established ( e.g. * ? ? ? * ; * ? ? ? showed that any random utility model can be approximated to an arbitrary degree of accuracy by a mmnl model with appropriate choice of variables and mixing distribution .the classical approach to estimating mmnl models is via maximization of the simulated likelihood function ( see , e.g. * ? ? ? * ) .this procedure can be difficult numerically ; the algorithm may not converge due to various reasons and there is a risk of getting trapped in local maxima .bayesian procedures avoid some of these issues , and consistency and efficiency in estimation can be attained under fewer restrictions ( see * ? ? ? * chap .markov chain monte carlo ( mcmc ) methods provide the bayesian analogue to classical procedures for estimating mmnl models . in the hierarchical bayesian approach , draws from the posterior distribution can be obtained using gibbs sampling and the metropolis - hastings algorithm ( see , e.g. * ? ? ?however , computations can be prohibitively expensive for large datasets , which are increasingly common .an alternative is to obtain approximate bayesian inference via variational approximation methods . showed that predictive inference for mmnl models can be obtained using variational methods at a lower computational cost but with accuracy close to that of mcmc .  in this paper, we develop variational methods for estimating mmnl models that allow the coefficients in to be correlated in the posterior .previously , considered mmnl models where is normally distributed and derived a variational bayes procedure for posterior approximation .the approximating density was assumed to be of a factorized parametric form and the reverse kullback - leibler divergence between the true posterior and the variational approximation was minimized .this is equivalent to maximizing a lower bound on the log marginal likelihood .the intractable lower bound was approximated using the multivariate delta method for moments .maximization of the lower bound over individual - level variational parameters was performed using standard unconstrained convex optimization techniques , and the covariance matrix of was assumed to be diagonal in the variational posterior .this is due likely to the high computational cost of optimizing a full covariance matrix .we explore three alternatives that allow the posterior independence assumption among random coefficients to be relaxed at low computational cost : ( 1 ) laplace variational inference , ( 2 ) nonconjugate variational message passing and ( 3 ) stochastic linear regression . the performances of these approaches are compared using real and simulated datasets .laplace variational inference was first considered by . formalized the approach for a class of nonconjugate models .this method uses laplace approximations within the optimal density update in variational bayes , and results in a gaussian approximation of the posterior .nonconjugate variational message passing is an algorithm proposed by to extend variational bayes to nonconjugate models .the variational posterior is assumed to be some member of the exponential family and variational parameters can be obtained using fast fixed point updates .we continue to use the delta method to approximate the intractable lower bound when using nonconjugate variational message passing .it is important to note that convergence is not guaranteed when the delta method is used , as the objective function being optimized is no longer a proper bound on the log marginal likelihood .we have experienced divergence in a small number of experiments .stochastic linear regression is useful in such cases as it makes the same assumptions as in nonconjugate variational message passing , but does not require expectations to be evaluated analytically . instead , updates are obtained stochastically using weighted monte carlo by simulating variates from the variational posterior . an alternate stochastic optimization method for optimizing the intractable lower bound , which uses control variates ( functions with tractable expectations that are highly correlated with the intractable function in the lower bound ) to reduce the variance of the stochastic search gradient .  to accelerate convergence for large datasets, we develop stochastic variational inference for mmnl models using each of the above three alternatives . in stochastic variational inference, a random subset of agents is selected at each iteration and local variational parameters specific to these agents are optimized .global variational parameters are then updated using stochastic gradient ascent , where the gradients are computed based only on the minibatch of optimized local variational parameters .stochastic variational inference was developed for conjugate - exponential models and has been applied to latent dirichlet allocation and the hierarchical dirichlet process in topic modeling . extended stochastic variational inference to logistic and poisson mixed models using nonconjugate variational message passing .here , we further extend stochastic variational inference to nonconjugate models via laplace variational inference and stochastic linear regression , with applications to mmnl models . as large choice sets ( e.g. scanner panel data in marketing )become more readily available , stochastic variational inference can play an important role in deriving inference efficiently from large - scale discrete choice models .another contribution of this article is the proposal of a novel strategy to increase minibatch sizes adaptively within stochastic variational inference . at the beginning of the procedure , estimates of the global variational parameters are far from the optimum and only a small minibatch is required to compute the appropriate direction to move in . as the estimates move closer towards the optimum , a more accurate definition of the direction in which to move is required and this can be supplied through using larger minibatches .the idea of adapting batch sizes has been studied by , and in machine learning problems .the results of are of theoretical interest and they suggest that the best adaptive batch schedule is exponential . and construct frequentist hypothesis tests using the central limit theorem for large sums of random variables , and propose increasing batch sizes if the probability of updating parameters in the wrong direction is large .we develop a new criterion based on ratio of progress and path " while using constant step sizes within the stochastic approximation .minibatch sizes are increased when the ratio falls beneath a critical value .this paper is organized as follows .section [ sec_mmnl_model ] defines the mmnl model .section [ sec_vi ] develops variational inference for the mmnl model using three different approaches .section [ sec_svi ] develops stochastic variational inference for the mmnl model and describes the proposal for increasing minibatch sizes adaptively .section [ sec_assessment ] outlines measures for assessing the accuracy of proposed variational methods .sec_examples ] considers examples including real and simulated datasets and section [ conclusion ] concludes .the mixed multinomial logit ( mmnl ) model considered in this paper is defined as follows .suppose choice events are observed for each agent , , and the agent selects from among alternatives at each choice event .let the utility that agent obtains from alternative at the choice event be here , is a vector of observed variables that relate to alternative and agent at the choice event , is a random vector of coefficients for agent representing the agent s preferences and is a random error term representing unobserved utility .coefficients in are assumed to be distributed as let ^t ] be a matrix of covariates .assuming that the random error terms are iid extreme value ( see * ? ? ? * ) , the choice probabilities become for , and   we adopt a full bayesian approach to inference and assume the priors : ^t ,  \label{iwprior}\\ a_k| a_k \stackrel{\text{iid}}{\sim } ig(1/2,\ ; 1/a_k^2 ) , \ ; a_k > 0 \ ; \text{for } \ ; k=1,\dots , k .\label{igprior}\end{gathered}\ ] ] the hyperparameters , , and are considered known .the prior distributions for are marginally noninformative . showed that and induce half- distributions on the standard deviation terms in , and a large leads to weakly informative priors on these terms .moreover , setting leads to marginal uniform distributions for all correlation terms in .the set of unknown parameters in the mmnl model is , where ^t ] and  ] .the gradient and hessian of are given by general numerical optimization methods can be used to find .we use the bfgs algorithm via the optim function in r for optimizing .an approximation of the variational lower bound can be obtained using and is given in [ appendix_laplace ] .the second approach optimizes using nonconjugate variational message passing . besides assuming for some partition , each further assumed to belong to some exponential family such that where is the vector of natural parameters and are the sufficient statistics .the condition that when is maximized leads to the fixed point update : ^{-1}\ ; \nabla_{\lambda_i } e_q\{\log p(y,\theta)\}\ ] ] for , where  ] .it can be shown that with , updates for and in can be evaluated in closed form and these are given in algorithm 1 .an approximation of the variational lower bound can also be obtained using and details are given in [ appendix_delta ] .we observe that the delta method leads to good posterior estimation generally .however , this algorithm is not guaranteed to converge as is not a proper lower bound to the marginal log likelihood .an example of such divergence is given in section [ eg_electricity ] .in such cases , the third approach described next will be helpful . present a stochastic linear regression algorithm that allows fixed - form vb to be applied to any posterior ( available in closed form up to the proportionality constant ) without having to evaluate integrals analytically .suppose we make the same assumptions as in nonconjugate variational message passing .the fixed point update in can be expressed as ( see [ appendix_lb_grad ] ) ^{-1 } \;\text{cov}_{q_i }  \left [ t_i(\theta_i ) ,  e_{-q_i } \ { \log p(y,\theta)\ }  \right].\ ] ] instead of evaluating  ] directly , approximate these terms iteratively using weighted monte carlo by generating random samples from .when , they showed that implies that where , \\ m_i = e_{q_i }  \ { \theta_i  \ } , \\g_i = e_{q_i } [ \nabla_{\theta_i } e_{-q_i } \ { \log p(y,\theta)\ } ] , \end{gathered}\ ] ] and denotes the hessian matrix of in . present a proof for the univariate case by considering a linear transformation and the identities derived by and .these identities can be restated as ,\end{gathered}\ ] ] where is any function in .we provide a proof for the multivariate case by substituting into .details are given in [ slrmg ] .the quantities , and are approximated stochastically using weighted monte carlo .the procedure is described in figure [ slrproc ] .  in figure [ slrproc ] , is updated continually and the weights help to diminish effects from earlier iterations when was less accurate .following , we adopt fixed weights and average iterates over the second half of the iterations to reduce variability ( see algorithm 2 ) . in setting ( the total number of iterations ) , it is important to balance accuracy and efficiency .the accuracy of stochastic linear regression deteriorates if is too small and are not sufficiently close to convergence .however , setting to a very large value can be inefficient as well .when generating draws from , it is computationally more efficient to consider transformation of standard normal random variables using a cholesky decomposition of instead of evaluating and explicitly at each iteration .for the mmnl model , the expectation of the log - sum - exp term in can not be evaluated in closed form and the delta method was used to approximate this term in the previous section .stochastic linear regression , on the other hand , does not require expectations to be evaluated analytically and is well suited to the mmnl model .it can also overcome convergence issues in nonconjugate variational message passing , as choosing to be sufficiently small ensures convergence .  instead of updating all variational parameters using stochastic linear regression , a combined approachis considered in this paper .we update for using stochastic linear regression while , and are updated using explicit variational parameter updates .this approach allows for a straightforward extension to stochastic variational inference , which is discussed in section [ sec_svi ] .note that defined in .the gradient and hessian of are given by and in .this result highlights a close connection between stochastic linear regression and laplace variational inference .while both approximates by a gaussian distribution , an important distinction is that stochastic linear regression optimizes both and , while laplace variational inference optimizes only , the location of the gaussian variational posterior ( is set as the negative inverse hessian at this point ) . in the examples ,we observe that this procedure in laplace variational inference often results in underestimation of the standard deviation terms in .we present the algorithm for computing the variational approximation . in algorithm [ alg1 ] , may be updated using ( 1 ) laplace variational inference , ( 2 ) nonconjugate variational message passing or ( 3 ) stochastic linear regression. the computational complexity of these algorithms are , and respectively , where denotes the number of iterations in bfgs .  in variational algorithms ,the lower bound is commonly used to check for convergence . however , for stochastic linear regression , it is not easy to compute at each iteration . for laplace approximation and nonconjugate variational message passing , we can only compute approximations of which are not guaranteed to increase after each cycle of updates .we consider the following stopping criterion instead .let ^t l \geq m 5  < l < m l \geq m$}.   \end{array } \right .\end{aligned}\ ] ] the minibatch size is increased by a factor when the minimum value of the ratios falls beneath a critical value .we allow to vary according to the minibatch size . for a small , a smaller is required as the path of the algorithm can be quite erratic even though progress is being made due to the greater randomness present between iterations .the proposed algorithm for implementing stochastic variational inference using adaptive batch sizes is outlined in algorithm 2 .  in stochastic variational inference, the local variational parameters should be optimized as a function of the global variational parameters at their current setting ( step 3 of algorithm 2 ) .laplace variational inference optimizes only , the location of the gaussian variational posterior and sets as the inverse of the negative hessian at this point . using our adaptive batch size approach , convergence is ensured , however , as the entire dataset is used eventually . for nonconjugate variational message passing , the updates for and are recursive and they have to be performed repeatedly until convergence is reached in order for and to be optimized .let be a concatenation of the vectors for at the iteration . in the examples ,we terminate the number of iterations in nonconjugate variational message passing when where denotes the euclidean norm , or when the number of iterations hit a maximum of three .we have used a loose stopping criterion here for greater computational efficiency . in the case of stochastic linear regression, we fix the number of iterations at for simplicity and assume that this number of iterations is sufficient for to be sufficiently close to convergence .as described in section [ sec_proposed strategy ] , constant stepsizes are used within each minibatch size , and we allow the stepsize to increase with the minibatch size . intuitively , smaller stepsizes are required at the beginning as we are less confident in the direction of gradient ascent computed based on the small minibatch of optimized variational parameters . as the minibatch size increase , our confidence level increases .the stepsize is 1 when the algorithm transits to batch mode ( when ) . in the examples , we start with a minibatch size of and an initial stepsize .we let the stepsize increase linearly with the minibatch size , and is computed similarly . ] until it reaches 1 when . we find that the performance of minibatches with sizes smaller than 25tend to be more erratic .the critical value for the `` ratio of progress and path '' is also initialized at 0.4 for the initial minibatch size of 25 , and allowed to increased linearly with the minibatch size until it reaches 1 when .we have used linear increments as this is a straightforward option .it is possible to experiment with other settings .our experiments indicate that minor variations from these settings do not result in much changes in the performance of algorithm 2 .the standard bayesian procedure for obtaining inference from mmnl models is via mcmc methods . how posterior samples for a mmnl model can be obtained using a metropolis - hastings within gibbs algorithm . proposed an improved random walk metropolis algorithm for drawing using a fractional likelihood approach .they demonstrate that the improved random walk metropolis exhibits better mixing and dissipates initial conditions in a shorter time than a random walk metropolis and an independence metropolis sampler .this algorithm in implemented in the r package bayesm via the function rhiermnlrwmixture .we modify this function slightly to accommodate the marginally noninformative priors for defined in and , and use it to compare mcmc with proposed variational methods .   for assessing the accuracy of proposed variational methods, we use the measures discussed in , which are based on the predictive choice distribution .the true predictive choice distribution of a vector of outcomes given the matrix of observed variables is defined as for data simulated artificially from the mmnl model , the true predictive choice distribution can be computed using monte carlo integration as and are known . in section [ eg_sim ] ,we use 1000,000 draws of from to compute the true predictive choice distribution for simulated data . showed that variability arising from monte carlo integration is not noticeable if this many draws of are used .a point estimate of the predictive choice distribution can be obtained by taking the mean of under the posterior of and : the estimated predictive choice distribution can be computed using monte carlo integration for both variational and mcmc methods . for variational methods , we approximate the posterior density with the fitted variational posterior density .we use 500 draws of from for variational methods and 10,000 draws for mcmc .more samples are used in the case of mcmc as there is some autocorrelation among the draws . for the estimated predictive choice distributions , we use 10,000 iid draws of .following , we use the total variation ( tv ) metric to compute the distance between two predictive choice distributions ( see , e.g. * ? ? ?* ) . for the simulated datasets ,the tv distance between the estimated and true predictive choice distributions at the attribute matrix can be computed using \\ = \frac{1}{2}\sum_{j=1}^j |p_{true } ( y_{\text{new}}^j=1| x_{\text{new } } ) - \hat{p } ( y_{\text{new}}^j=1| x_{\text{new}})|.\end{gathered}\ ] ] for real datasets , the true predictive choice distribution is unknown and we compute the tv distances between the predictive choice distribution estimated using mcmc and the variational methods instead .this provides a means of assessing the degree of agreement between mcmc and variational methods .in the following examples , the performances of laplace approximation ( laplace ) , nonconjugate variational message passing ( ncvmp ) and stochastic linear regression ( slr ) are compared with that of mcmc in terms of the predictive choice distribution .we set and for slr in both algorithms 1 and 2 .the runtime of slr in algorithm 1 varies slightly between runs and we report the mean runtime and standard deviation over 5 runs .estimates of the variational parameters are almost identical in each run of slr .we use the run with runtime closest to the mean runtime to compute the predictive choice distribution . for algorithm 2, there is greater variation and we repeat runs for each alternative ten times .the mean runtime and standard deviation over the ten runs are reported .  for mcmc, 4 independent chains were run in each example and the first half of each chain was discarded as burn - in . in each example , 10,000 draws remained after thinning and the gelman - rubin diagnostics were used to check that these draws are a good approximation of the posterior distribution .these draws were then used to compute the estimated predictive choice distribution for mcmc .we note that there are some inherent difficulties in comparing runtimes of variational methods with mcmc .mcmc is a simulation - based method and runtimes depend on the thinning factor , length of burn - in and number of sampling iterations .these are problem dependent ; a higher thinning factor or longer burn - in may be required when mixing is poor . on the other hand ,variational bayes is deterministic and the time to convergence ( which is also problem dependent ) depends on the initialization and stopping rule . due to these concerns , we only present the results of a predictive log - likelihood experiment in section 6.1 , where the variational algorithms and mcmc are given an equal computational budget .we use a vague for and set , for . all code was written in r and computationswere carried out on a 64-bit 3.20 ghz intel core i5 processor with 8 gb of memory .our r code is available as supplementary materials for this article .the first example considers data representing consumers choices among vehicles in stated preference experiments .this dataset consists of respondents and comes from a study for toyota and general motors on the marketability of electric and hybrid vehicles .it is available at http://elsa.berkeley.edu/users/train/ec244.html and is a subset of the full dataset of 500 respondents ( see * ? ? ?each respondent faced up to 15 experiments and chose among different vehicles in each experiment .the vehicles are described in terms of the following attributes : negative of price ( in ten thousand dollars ) , negative of operating cost per month ( in ten dollars ) , engine type ( gas , electric , or hybrid ) , range in hundreds of miles between recharging ( if engine is electric ) and performance level ( high , medium , or low ) .the range of a vehicle is set to zero for all non - electric vehicles .an indicator variable for hybrid vehicle is included as a covariate .performance level is represented using two dummy variables with low performance as the base .we have covariates and as some respondents did not complete all 15 experiments .we run algorithm 1 using laplace , ncvmp and slr and runtimes are given in table [ vehicle_runtime ] .mcmc was run for 4 chains , each with 25000 iterations and a thinning factor of 5 was applied .mixing is poor for parameters corresponding to the indicator variable for medium performance and so a larger number of iterations and higher thinning factor were used ..[vehicle_runtime ] vehicles example : cpu times ( seconds ) for algorithm 1 ( laplace , ncvmp and slr ) .standard deviation over repeated runs given in brackets .[ cols="<,^,^,^,^",options="header " , ]   barplot shows the average number of iterations spent by algorithm 2 ( laplace , ncvmp , slr ) at each minibatch size for the electricity data.,scaledwidth=22.0% ]in this paper , we have developed three different approaches for fitting mmnl models using variational bayes : ( 1 ) laplace approximation , ( 2 ) nonconjugate variational message passing with a delta method approximation and ( 3 ) stochastic linear regression .we also proposed a novel adaptive batch size strategy for implementing stochastic variational inference using these approaches .the performances of these variational methods were investigated for a wide range of datasets , both real and simulated . across all examples ,predictive choice distributions computed using stochastic linear regression were closest to that of mcmc , with nonconjugate variational message passing close behind .the discrepancy between laplace approximation and mcmc is much more pronounced . in terms of stability ,stochastic linear regression and laplace approximation are very stable and we did not encounter any convergence issues in all our experiments . while nonconjugate variational message passing failed to converge in one of the examples , the failure is due to the delta method approximation rather than nonconjugate variational message passing itself . in the rest of the examples ,nonconjugate variational message passing performed very well and converged in the shortest time .stochastic variational inference further accelerates convergence for large scale data sets . with our adaptive batch size strategy ,algorithm 2 is nearly automatic and we recommend increasing proportionately with the number of agents .significant speedups can be obtained using algorithm 2 for datasets as small as a few hundreds .variational methods provide an important alternative as well as complement to mcmc methods for fitting mmnl models , yielding high computational efficiency with competitive accuracy .investigating how variational methods can be used for performing model selection with large - scale discrete choice datasets is another important area for future research . withthe availability of large scale data sets in marketing and other applications , variational methods enable predictive inference to be obtained in a timely manner .the author is grateful to cathy trower , jordan louviere and the institute for choice at the university of south australia for sharing data from the project on faculty appointments .the author also wish to thank kenneth train for sharing data from the vehicle choice stated preference experiments , david nott for his comments on the 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( 1996 ) .bayesian methods for mixtures of experts . in _ advances in neural information procesing systems 8 _ ( eds .touretzky , d. s. , mozer , m. c. and hasselmo , m. e. ) , 351357 .usa : mit press .optimal densities for , and can be derived using . where   \\       & = n(\mu_\zeta , \sigma_\zeta ) , \end{aligned}\ ] ]  where   \\ & = iw ( \omega , \upsilon )   \end{aligned}\ ] ]  where    \\ & \quad + \sum_{h=1}^h  \left\{- \frac{1}{2 } \log |\omega| - \frac{1}{2 } ( \beta_h -\zeta)^t \omega^{-1 } ( \beta_h -\zeta )  \right\ }  \\ & \quad  -\frac{(h+1)k}{2 } \log ( 2\pi ) - \frac{1}{2 } ( \zeta -\mu_0)^t \sigma_0^{-1 } ( \zeta -\mu_0 ) \\ & \quad + \frac{(\nu+k-1)k}{2 } \log \nu - \sum_{k=1}^k \log \gamma \left ( \frac{\nu+k - k}{2 } \right ) \\ & \quad - \frac{\nu+k-1}{2}\sum_{k=1}^k \log a_k  - \frac{1}{2 } \log |\sigma_0|- \nu \sum_{k=1}^k \frac{\omega_{kk}^{-1}}{a_k}\\ & \quad + \sum_{k=1}^k\left\ { \log a_k - \log \gamma\left(\frac{1}{2}\right ) -\frac{3}{2 } \log a_k -\frac{1}{a_k^2 a_k } \right\ } \\ & \quad - \frac{\nu+2k}{2 } \log |\omega|  - \frac{k(k-1)}{4 } \log \pi .\end{aligned}\ ] ]  .\end{aligned}\ ] ]    where is as defined in .noting that the updates for and are deterministic , the variational lower bound can be simplified as    from , since with and , using this approximation in the variational lower bound gives us an estimate of .   from , is approximately given by \\- \frac { \omega}{2 } ( \mu_h -\mu_\zeta)^t \upsilon^{-1 } ( \mu_h -\mu_\zeta ) - \frac { \omega}{2 } \text{tr } ( \sigma_h \upsilon^{-1}).\end{gathered}\ ] ] using vector differential calculus , it can be shown that - \omega\upsilon^{-1 } ( \mu_h-\mu_\zeta).\end{gathered}\ ] ]the gradient of the variational lower bound with respect to is the first term can be written as .alternatively , since .the second term is \lambda_i + 0 \\ & = \text{cov}_{q_i } [ t_i(\theta_i ) ] \lambda_i . \end{aligned}\ ] ] when is a member of the exponential family , the fisher information matrix is given by \\ & = e_{q_i } \left [ \left\ {  t_i(\theta_i)-\nabla_{\lambda_i } h_i(\lambda_i ) \right\ } \left\ { t_i(\theta_i)-\nabla_{\lambda_i}h_i(\lambda_i ) \right\ } ^t\right ]  \\ & = \text{cov}_{q_i } [ t_i(\theta_i)].\end{aligned}\ ] ]when , the updates of and from nonconjugate variational message passing are ^{-1 } \;\ ; \text{and } \\\mu_i \leftarrow \mu_i + \sigma_i \ ;  \nabla\negthinspace_{\mu_i } e_q\{\log p(y,\theta)\}.\end{gathered}\ ] ] using the identities in , we have \right)\\ & =  -2 \text{vec}^{-1}\left ( \frac{1}{2 } \text{vec } \left\ { e_{q_i } [ \nabla_{\theta_i}^2 e_{-q_i}\{\log p(y,\theta)\ } ] \right\ } \right)\\ & =  - e_{q_i } [ \nabla_{\theta_i}^2 e_{-q_i}\{\log p(y,\theta)\ } ] \\\end{aligned}\ ] ] and \\  & = e_{q_i}(\theta_i ) + \sigma_i \ ; e_{q_i } [ \nabla_{\theta_i } e_{-q_i}\{\log p(y,\theta)\ } ] .\end{aligned}\ ] ] hence , we have derived the updates in from nonconjugate variational message passing updates for multivariate gaussian .we provide a proof below for the identities in , which were first derived by and . for simplicity ,we omit the subscript . for and any real function ,we show that .\label{vr}\end{gathered}\ ] ]     the element of is applying integration by parts twice .therefore \\ & = \frac{1}{2}\text{vec}\left [ \int \nabla_\theta^2 q(\theta ) \ ; v(\theta )  \;d\theta \right ] \\  & = \frac{1}{2}\text{vec}\bigg [ \int  q(\theta ) \left\ { \sigma^{-1 } ( \theta - \mu)(\theta - \mu)^t \sigma^{-1 } - \sigma^{-1 }  \right\ }  \\ & \quad v(\theta )  \;d\theta \bigg ]  \\ & = \frac{1}{2 } \int q(\theta ) [ ( \sigma^{-1 } \otimes \sigma^{-1 } ) \text{vec } \{(\theta-\mu)(\theta-\mu)^t\ } \\ & \quad - \text{vec}(\sigma^{-1 } ) ] v(\theta ) \;d\theta . \end{aligned}\ ] ]

Abstract: discrete choice models describe the choices made by decision makers among alternatives and play an important role in transportation planning , marketing research and other applications . the mixed multinomial logit ( mmnl ) model is a popular discrete choice model that captures heterogeneity in the preferences of decision makers through random coefficients . while markov chain monte carlo methods provide the bayesian analogue to classical procedures for estimating mmnl models , computations can be prohibitively expensive for large datasets . approximate inference can be obtained using variational methods at a lower computational cost with competitive accuracy . in this paper , we develop variational methods for estimating mmnl models that allow random coefficients to be correlated in the posterior and can be extended easily to large - scale datasets . we explore three alternatives : ( 1 ) laplace variational inference , ( 2 ) nonconjugate variational message passing and ( 3 ) stochastic linear regression . their performances are compared using real and simulated data . to accelerate convergence for large datasets , we develop stochastic variational inference for mmnl models using each of the above alternatives . stochastic variational inference allows data to be processed in minibatches by optimizing global variational parameters using stochastic gradient approximation . a novel strategy for increasing minibatch sizes adaptively within stochastic variational inference is proposed .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore


since the fundamental work on random graphs by erds and rnyi , especially the seminal works on the small - world phenomenon by watts and strogatz and the scale - free property by barabsi and albert , the study of complex networks has received extensive attention .scientists have found that most real networks are neither completely regular nor completely random , but usually display a small - world effect and/or a scale - free behavior .the research results show that complex network theory has been proven to be a powerful tool for the analysis in various fields , such as the world wide web , social networks , ecological systems , traffic systems , and so on .  in biological systems , elements that interact or regulate each othercan be represented by a network , namely , a collection of nodes and edges ( links ) . at the simplest level, the individual elements are described as nodes and their interactions are reduced to edges connecting between pairs of nodes . here , the network - based approach is an effective way to discover the collective action of the individual parts , namely , the systems - level behavior , which has been widely used in biochemical and medical research and offers a conceptual framework to understand molecular mechanisms and disease pathologies .hypertension is a major risk factor for cardiovascular morbidity and mortality ; the worldwide prevalence of hypertension is  . over the years, a great deal of research shows that the arterial blood pressure of many essential hypertensive patients exhibits an increased sensitivity to dietary salt intake , which is known as the salt - sensitive ( ss ) hypertension .these people account for about of hypertensive patients .therefore , the studies about ss hypertension contribute to a decrease in the level of blood pressure and in the incidence of mortality due to cardiovascular complications , and thus improve human health .existing research for the dahl ss rat and consomic rat strains reveals that salt and genetic factors have a major impact on hypertension . during the last 30 years or so ,the clinical research and treatment of hypertension have improved dramatically .however , its molecular mechanisms and pathologies involved remain intricate and elusive .  in the present work, we attempt to study the genes that are involved in ss hypertension using the complex network approach. we will propose a simple rule to construct the network model of hypertension - related genes , where the nodes are individual genes and the connections are derived from the expression correlations that are based on microarray data . through calculating several statistical indices and analyzing topological characteristics of the network ,we find out key hub genes that play significant roles in hypertension and describe their functions .based on both biological knowledge and network theory , this study confirms the role of key genes in hypertension from another perspective and provides an idea for studying the relation between genes associated with hypertension .this paper is organized as follows . in section [ sec:2 ]we introduce the construction of the network model of hypertension - related genes . in section [ sec:3 ], we analyze the statistical and topological characteristics of the gene network , and determine key hub genes by introducing an integrated centrality .the biological descriptions of hub genes are presented in section [ sec:4 ] , while section [ sec:5 ] gives concluding remarks .the dahl ss rat is a widely used genetic model of human hypertension , which was proposed by dahl et al . in the early 1960s . it develops hypertension upon exposure to a high - salt intake .the consomic rat strains , used as the normotensive control for the dahl ss rat , are generated by substituting a chromosome or a part of a chromosome from a normal rat strain for the corresponding genomic region of the ss rat .previous research showed that substitution of chromosome 13 or 18 could significantly attenuate hypertension .the two consomic rat strains ( generated by substituting chromosomes 13 and 18 , respectively ) have genetic homology with the dahl ss rat , but lead to the amelioration of hypertension ; and the segregation of hypertensive phenotype is regarded as the result of different gene expression patterns among the three rat strains caused by substituting chromosomes . therefore , our study focuses on the analysis of gene expression data ( ged ) among the dahl ss rat and two consomic rat strains : ( ged for ss rat with high blood pressure ) , ( ged for substitution of chromosome 13 ) and ( ged for substitution of chromosome 18 ) , where all three strains of rats were maintained on a high sodium intake for two weeks and gene expression profiles were examined using microarrays ( see http://pga.mcw.edu ) .all data of , and are given in three columns ( ss2wk , ss13 2wk , and ss18 2wk ) in table s2 of supplemental figures and tables of ref . . in table [ dataformat ] we list detailed data of some randomly selected genes and feature genes calculated in section [ sec:3 ] .let us consider the network , where  is the set of nodes , and the set of edges or connections between nodes .we will use the following notation : indicates that there is an edge between nodes and ; and otherwise .our gene network model is constructed from the correlations based on ged in three rat strains in two steps , which is designed to explore the relationship between genetic change trend and hypertensive phenotype .calculation of change ratios of ged : _the hypertension - related genes are served as nodes of the network , each node represents an individual gene shown as gene symbol or cloneid ( for the gene without a gene symbol ) . for each of the nodes, we define the change ratios and of ged between and as follows : where . then we can calculate the change ratios and separately ( these data are shown as and in table [ dataformat ] ) .it is easy to obtain the averages of two groups of change ratios and , respectively : , and ; which can be used as thresholds to determine whether a connection should be made between two nodes in the next step .it should be indicated that there are many ways of choosing the threshold , a higher threshold would result in a higher average degree and a greater influence or disturbance among nodes ( genes ) , whereas a lower threshold would result in fewer connections but a greater possibility of the actual correlation of genes .so the average change ratio would be a reasonable choice of the threshold to make reliable connections and a moderate scale of the network .indeed , setting the threshold a little higher or lower than the average would not disturb the clustered ( not paired ) relationship of genes considered in this paper ..some ged ( , and ) of three rat strains and change ratios ( and ) of two groups . in this paper , we use `` _ hist1h2ai _ '' as the abbreviation of gene symbol `` _ hist1h2ai_predicted ///hist1h4a_predicted _ '' .[ cols="<,>,>,>,>,>",options="header " , ]   the gene _col4a1 _ encodes the major type slowromancap4@ alpha collagen chain of basement membranes .it has been indicated that mutations in _col4a1 _ result in a complex vascular phenotype encompassing defects in maintenance of vascular tone , endothelial cell function and blood pressure regulation .other studies have suggested that a snp ( single - nucleotide polymorphism ) in the _ col4a1 _gene is strongly associated with pwv ( pulse wave velocity ) , an established independent predictor of adverse cardiovascular outcomes .the gene _ lcn2 _ , which has the most connections in the gene network , is a recently identified adipokine that belongs to the superfamily of lipocalins .it is recognized as a biomarker of obesity and inflammation , which are both risk factors for hypertension .recent findings suggest that genetic variants in _lcn2 _ may affect blood pressure , and adiponectin has multiple protective effects on vascular endothelium .  the gene _cdk4 _ is a positive regulator of the cell cycle , which plays a role in cell cycle progression .angiotensin slowromancap2@ , an essential factor for hypertension , is an important modulator of cell growth through at receptors .studies have reported when at receptors are stimulated _ in vivo _, dna synthesis is enhanced in blood vessels by activation of cyclin d1 and _ cdk4 _ .  the gene_ usp48 _ encodes a protein containing domains that associate it with the peptidase family c19 .it is also known as the family 2 of ubiquitin carboxyl - terminal hydrolases , whose family members function as de - ubiquitinating enzymes , recognizing and hydrolyzing the peptide bond at the c - terminal glycine of ubiquitin .recent data demonstrate that via the inhibition of _usp48 _ , agonist activation of d ( dopamine d receptor gene ) promotes the degradation of nhe3 ( sodium / hydrogen exchanger isoform 3 ) , and thus engenders natriuresis and regulates blood pressure .  the gene _fzd2 _ belongs to a class of highly conserved genes , which acts as cell surface receptors for wnt proteins .there is evidence that wnt / fzd signaling is involved in the formation and remodeling of the vasculature .some existing data also show that _fzd2 _ expressed in aortic smooth muscle cells is modulated by ang slowromancap2@ , and ang slowromancap2@ plays a crucial role in blood pressure regulation and cardiovascular homeostasis , both _ in vitro _ and _ in vivo _ .the gene _ ctsd _ encodes a lysosomal aspartyl protease .this proteinase has a variety of biological functions , such as degradating hemoglobin , serum albumin and myoglobin in endosome .it has been reported that genetic variation of the aspartic proteinases may have an effect on specific clinical diseases such as hypertension and ulcers .other research suggests that _ ctsd _ significantly positively correlates with the arterial hypertension stage as well as with histological grading of atherosclerotic lesions .  the gene _timp1 _ is linked to extracellular matrix fibrosis and is elevated in hypertension .hypertension results in structural changes to the cardiac and vascular extracellular matrix ( ecm ) .matrix metalloproteinases ( mmp ) , and their inhibitors ( timp ) may play a central role in the modulation of this matrix .existing observations suggest a possible role for these surrogate markers of tissue ecm composition and the prognosis of cardiovascular events in hypertension .fbn1 _ encodes a member of the fibrillin family .recently , it is reported that _fbn1 _ plays an important role in maintaining the physiological arterial stiffness of essential hypertension .meanwhile , _fbn1 _ is a major component of the microfibrils that form a sheath surrounding the amorphous elastin .defects in the _ fbn1 _ gene are associated with an increased risk of prevalent hypertension .the investigation also reveals that _ fbn1 _ may contribute to glomerular damage in hypertensive and diabetic kidney disease .since our network model is constructed by considering the correlations of ged , hub genes which have similar expression patterns with most of the genes with changed expression levels are typical representatives of the genes with strong correlation . from relevant research literature and the biological functions of these hub genes shown in table [ genefunction ] , we know that these hub genes are key ( feature ) genes that play important roles in hypertension .this shows that our construction of the gene network is suitable .hypertension has the complexity and diversity of the genetic factors and pathogenesis . however , a comprehensive understanding of hypertension is still extremely deficient . in this study , based on microarray data ,we attempt to reversely obtain the relationships between the hypertension - related genes , rebuild the structure of the gene network by visualization technology , and try to abstract the complex interactions between genes through calculating statistical characteristics of the gene network .the network of hypertension - related genes is sparse , which has the following characteristics : ( 1 ) the gene network shows a scale - free property with power - law degree distribution . a few nodes with large values of three centrality indices ( or equivalently with large values of integrated centrality ) correspond to hub genes ; they are key ( feature ) genes involved in the formation of hypertension .( 2 ) the gene network has a small average path length ( ) and a large clustering coefficient ( ) , i.e. , the small - world property , indicating that the local disturbance to hubs would rapidly transfer to the whole network .this property reveals the direct influence of these hub genes on hypertension from another perspective , and implies possible novel molecular genetic signals .( 3 ) the gene network of hypertension is a description of an abnormal state ( illness ) , which exhibits assortative feature ( assortativity coefficient ) unlike most biological networks .the nodes with many connections tend to be connected to those with the similar type , showing that besides being affected by a few feature genes , hypertension is the consequence of accumulation of genetic changes and interaction caused by a variety of factors ; this has been demonstrated by the existing research findings .therefore , it is necessary to focus on further study of the interplay between genes in biological experiments .  in this paper , we construct the network model of hypertension - related genes in genetic level based on both biological knowledge and network theory .the hub genes ( _ col4a1 _ , _ lcn2 _ , _ cdk4 _ , etc . ) in our network have been confirmed by biological / medical research to play important roles in hypertension .furthermore , the network can also be analyzed based on actual functional correlation metrics with other threshold selection methods ( e.g. , ) to get more biological information about hypertension in the future ; it is believable that the results derived from those methods would be consistent and harmonic in comparison with the theoretical analysis of this paper , and they would complement each other .this study provides another perspective on expounding the molecular genetic mechanism , prevention , and individualization treatment of salt - sensitive hypertension .meanwhile , the research may hopefully shed light on the development of network - based models ( e.g. , directed and/or weighted networks ) of hypertension and other serious diseases , and explore the mutual regulatory relationships between genes of complex diseases , as well as contribute to finding new drug targets and developing novel ideas .at last , we expect that the complex network approach can provide an effective tool for analyzing the pathogenesis of critical illness .this work was supported in part by the national natural science foundation of china ( nsfc ) ( grant nos .10565004 and 11365023 ) and the specialized research fund for the doctoral program of higher education of china ( srfdp ) ( grant no .20050673001 ) .our special thanks go to the authors of ref . for providing the online ged of the ss rat and two consomic rat strains , which are the basis of construction of our network model .the authors would like to thank prof .l . peng and dr . j .- j .hao for their helpful discussions and suggestions , and the anonymous reviewer for calling our attention to ref . .burt , j.a .cutler , m. higgins , m.j .horan , d. labarthe , p. whelton , c. brown , e.j .roccella , trends in the prevalence , awareness , treatment , and control of hypertension in the adult us population : data from the health examination surveys , 1960 to 1991 , hypertension 26 ( 1 ) ( 1995 ) 6069 .dahl , m. heine , l. tassinari , effects of chronic excess salt ingestion : evidence that genetic factors play an important role in susceptibility to experimental hypertension , j. exp . med .115 ( 6 ) ( 1962 ) 11731190 .cowley jr . , r.j .roman , m.l .kaldunski , p. dumas , j.g .dickhout , a.s .greene , h.j .jacob , brown norway chromosome 13 confers protection from high salt to consomic dahl s rat , hypertension 37 ( 2 ) ( 2001 ) 456461 .b. joe , n.e .letwin , m.r .garrett , s. dhindaw , b. frank , r. sultana , k. verratti , j.p .rapp , n.h .lee , transcriptional profiling with a blood pressure qtl interval - 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Abstract: in this paper , a network of hypertension - related genes is constructed by analyzing the correlations of gene expression data among the dahl salt - sensitive rat and two consomic rat strains . the numerical calculations show that this sparse and assortative network has small - world and scale - free properties . further , key hub genes ( _ col4a1 _ , _ lcn2 _ , _ cdk4 _ , etc . ) are determined by introducing an integrated centrality and have been confirmed by biological / medical research to play important roles in hypertension .   complex network ; hypertension ; integrated centrality ; hub gene


molecular dynamics is the computer simulation of a material at the atomic level . in principlethe only inputs to a simulation are the characteristics of a set of particles and a description of the forces between them . an initial condition is chosen and from these first principles the evolution of the system in time is simulated using newton s laws and a simple numerical integrator .molecular dynamics is a very prevalent computational practice , as a glance at an issue of the journal of chemical physics will show .it does have its limitations : the motion of only a relatively small number of particles can be simulated over a short time interval .however , most of the mesoscopic models that have been suggested to overcome these difficulties still rely on molecular dynamics as a form of calibration .it is likely that molecular dynamics will continue to be important in the future .  given its scientific importance there is very little rigourous justification of molecular dynamics simulation . from the viewpoint of numerical analysisit is surprising that it works at all .the problem is that individual trajectories computed by molecular dynamics simulations are accurate for only small time intervals . as we will see in section [ sec : problem ] , numerical trajectories diverge rapidly from true trajectories given the step - lengths used in practice .no one disputes this fact , and no one is particularly concerned with it either .the reason is that practitioners are never interested in particular trajectories to begin with .they are interested in ensembles of trajectories . as long as the numerical trajectories are representative of a particular ensemble of true trajectories , researchers are content . however , that this statistical information is computed accurately has yet to be rigourously demonstrated in representative cases .the goal of this article is to present a concise mathematical conjecture that encapsulates this fundamental difficulty .we present a model system that is representative of systems commonly simulated in molecular dynamics .we present the results of numerical simulations of this system using the strmer - verlet method , the work - horse of molecular dynamics . in each simulationa random initial condition is generated , an approximate trajectory for the system is computed and the net displacement of one particle over the duration of the simulation is recorded .we show that even for step - sizes that are far too large to accurately compute the position of the particle , the distribution of the particle s displacement over the many initial conditions appears to be accurate . from the numerical data we conjecture a rate of convergence for this particular statistical property .we believe that if this conjectured rate of convergence ( or one like it ) can be rigourously established , even for this single system , then we will understand significantly better why molecular dynamics works .  the problem of explaining the accuracy of molecular dynamics simulation is well - known both in the physical sciences ( for example ) and in the mathematics community .this latter reference is a survey of the relation between computation and statistics for initial value problems in general .there has been plenty of excellent mathematical work that has done much to explain various features of this type of simulation , but has not resolved the issue we consider here .see for surveys .one body of work that has addressed the statistical accuracy of under - resolved trajectories in a special case is by a. stuart and co - workers . in they have explored some linear test systems with provable statistical properties in the limit of large numbers of particles .they are able to show that if the systems are simulated with appropriate methods the statistical features of numerical trajectories are accurate in the same limit even when the step - lengths are too large to resolve trajectories .though these results are interesting since they are the only ones of their kind now known , for the highly nonlinear problems of practical molecular dynamics very different arguments will be required .one subproblem that has been attacked more successfully is that of the computation of ergodic averages .these are averages of functions along very long trajectories .all that numerical trajectories have to do to get these correct is sample the entire phase space evenly .this is a much weaker property than getting all statistical features correct .the most striking work on this question is by s. reich which establishes rapid convergence of ergodic averages for hamiltonian systems which are uniformly hyperbolic on sets of constant energy .unfortunately , this property has never been established for realistic systems , and is unlikely to hold for them .the work established similar results for systems with much weaker properties but requires radically small time steps for convergence to occur .the contribution of this work is to precisely specify a simple problem which encapsulates all the essential difficulties of the more general problem . in section [ sec : system ]we present the system we will study .section [ sec : problem ] shows the results of some numerical experiments on this system .there we state our conjecture based on the results . in section[ sec : approaches ] we will discuss two possible approaches to proving the conjecture . finally , in section [sec : discussion ] we will discuss prospects for the eventual resolution of the conjecture .the system consists of point particles interacting on an 11.5 by 11.5 square periodic domain .we let and denote the positions and velocities of the particles , with denoting the position and velocity of particle .the motion of the system is described by a system of hamiltonian differential equations : with hamiltonian here denotes the famous lennard - jones potential . in our simulations we use a truncated version : figure [ fig : movie ] shows the positions of the particles on the periodic domain for one state of the system . though the particles are only points , in the figure each is represented by a circle of radius 1/2 .we take our initial conditions to be randomly distributed according to the probability density function where is chosen so that the function integrates to one .this is known as the canonical distribution ( or ensemble ) for the system at temperature .there is a simple physical interpretation of this distribution : if the system is weakly connected to another very large system at temperature , this is the distribution we will find the original system in after a long period of time . in our units , and we choose .there are many ways of sampling from the canonical distribution at a given temperature . for our experiments we generated initial conditions using langevin dynamics .see for an explanation of this technique and a comparison with other methods . if done correctly , the precise method of sampling from the canonical distribution will have no bearing on the results of the experiments we will present subsequently .the numerical method we use for integrating our system is the strmer - verlet scheme .given an initial and a it generates a sequence of states such that .the version of the algorithm we use is this is a second - order explicit method .it is symplectic , and as a consequence conserves phase space volume .finally we have to decide upon our step - length .if is too large the energy of the computed solution will increase rapidly and explode . in practice, it is observed that for small enough step lengths energy remains within a narrow band of the true energy for very long time intervals .( there is extensive theoretical justification for this phenomenon , see section [ subsec : bea ] ) .practitioners tend to pick a as large as possible while still maintaining this long - term stability on their time interval of interest . for the system and initial conditionswe describe here yields good approximate energy conservation on the time interval  ] , we expect that reducing the time step by a factor of a thousand would not yield a significantly different curve .however , we see that the two curves for and very quickly diverge .they are distinguishable to the eye almost immediately and completely diverge around time units .  reducing the step length to a curve that agrees with the line longer , but still diverges around time units .similarly , even with trajectory is not accurate over the whole interval depicted .  from these numerical results , we might conjecture that reducing the step - length by a constant factor only extends the duration for which the simulation is accurate by a constant amount of time .this is consistent with theoretical results about the convergence of numerical methods for ordinary differential equations .what is surprising in this case is that the time - scale on which the trajectories are valid appears to be miniscule compared to the time - scale on which computation are actually performed .it seems that the trajectories we compute here with stepsize even as small as are not accuarate over the whole interval  ] , for some constants .we will explain the reasons for hypothesizing this particular dependence in the next section . herewe will briefly note what dependence the classical theory of convergence for numerical odes gives : for  ] , for some appropriate constants .( this alone is not useful for analysing molecular dynamics since and are both large . ) secondly , the modified hamiltonian is conserved extremely well by the numerical method for long time intervals : for  ] .we chose the bound in conjecture 1 in analogy with this last result .suppose we wanted to bound the error between and using these estimates .the fact that the initial conditions are random adds an extra level of complication to the problem .we have been using to denote the average with respect to the canonical distribution for the hamiltonian .the perturbed hamiltonian has a different canonical distribution .we denote averages with respect to it by .we let denote the net displacement of the tracer particle under the new flow given by .we might try bounding the error in the following way : we discuss each of the three terms in turn .the first term is due to the numerical trajectory not agreeing with the exact trajectory of the modified system with hamiltonian .according to ( [ eqn : smaller ] ) we can bound this term by for a duration of .the studies in suggest that this is a tight estimate for typical molecular dynamics simulations .the second term is the difference in the expectation of due to a perturbation in the measure .since the two measures are proportional to and respectively , and , we expect this term to be on the order of for all .this probably can be rigourously controlled without much difficulty .the third term is just the difference in between the original system and the perturbed system .this is likely to be extremely difficult to bound .however , showing that it is small is not a question about computation but about statistical physics . fornow let us assume that it is for all for now .already we can see that this approach will not get us the result that we want , even assuming we can bound the third term .the best estimate we have so far is that the error is bounded by for  ] .this means that ( assuming we can obtain reasonable bounds on and ) that for $ ] .here we have let be the composition of the time flow map of the hamiltonian system with the 2-norm .now the first term above is bounded by by ( [ eqn : hardthing ] ) and the second term is by ( [ eqn : preserve ] ) , thus establishing the conjecture . simultaneously proving ( [ eqn : preserve ] ) and ( [ eqn : hardthing ] ) for some shadowing map may not be easy , but it may be much easier than proving the usual stronger shadowing result .despite the ideas presented in the previous section , the conjecture we have presented is probably not open to attack by existing techniques .the problem is that there is no rigourous mathematical theory of how statistical regularities emerge from the dynamics of generic high - dimensional hamiltonian systems .consequently , there is no theory of how perturbations in the hamiltonian dynamics leads to perturbation in the statistics .a numerical analyst has three choices when faced with this situation :  1 .  *take up mathematical physics . *if we are to make progress on the conjecture these entirely non - numerical problems need to be tackled first . mathematical physicists are interested in proving things like ergodicity and decay of correlations for hamiltonian systems such as presented here , and it is conceivable that eventually there will a robust body of theory that we can apply to our problem .so one possibility is to work on developing such a theory .this likely will not have much to do with computation .2 .  * relax standards of rigour .* theoretical physicists , as opposed to mathematical physicists , have accepted that much reliable information can be obtained through calculations that can not be rigourously justified . typically theoreticalphysicists study systems about which nothing interesting can be proved ; to do otherwise would be far too restrictive .there is no reason why this informal yet highly fruitful style of reasoning should be restricted to systems themselves and not numerical discretizations of systems .a combination of non - rigourous arguments and careful numerical experiments could do a lot to clarify how the strmer - verlet method is able to compute statistics so accurately for our system .3 .  * abandon the whole pursuit . * for many ,the purpose of numerical analysis is to provide reliable , efficient algorithms .if one is pursuing a theoretical question , it is hoped that it will lead to better algorithms eventually . sadly , even a complete resolution of the conjecture we have presented in unlikely to have much effect on computational practice .many people have tried for years to devise an integrator that is more efficient than the strmer - verlet method for computing statistically accurate trajectories in molecular dynamics .they have only been successful for hamiltonian systems with special structure .( the prime example of this is the multiple time stepping methods , see ( * ? ? ?viii.4 ) . ) in fact , we state another conjecture which is not formulated rigourously .+ [ conj : solver ] no integration scheme can improve the efficiency by more than a factor of two with which strmer - verlet computes statistically accurate trajectories for systems like that in section [ sec : system ] .+ here even a clear mathematical formulation would be a challenge .obviously if we already know a lot about a system we can contrive an algorithm which will give correct statistics for a tracer particle , but this does not count .the conjecture is intended to capture the idea that strmer - verlet is a very general purpose method ; we do not need to know anything about a system to apply it .g. bennetin and a. giorgilli , _ on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms _ , j. stat .74 * , ( 1994 ) 11171143 .h. sigurgeirsson and a. m. stuart , _ statistics from computations ._ foundations of computational mathematics ( oxford , 1999 ) , 323344 , london math .lecture note ser ., 284 , cambridge univ . press , cambridge , 2001 .

Abstract: an open problem in numerical analysis is to explain why molecular dynamics works . the difficulty is that numerical trajectories are only accurate for very short times , whereas the simulations are performed over long time intervals . it is believed that statistical information from these simulations is accurate , but no one has offered a rigourous proof of this . in order to give mathematicians a clear goal in understanding this problem , we state a precise mathematical conjecture about molecular dynamics simulation of a particular system . we believe that if the conjecture is proved , we will then understand why molecular dynamics works .