buildMCEM: Builds an MCEM algorithm for a given NIMBLE model

In nimble: MCMC, Particle Filtering, and Programmable Hierarchical Modeling

Builds an MCEM algorithm for a given NIMBLE model

Description

Takes a NIMBLE model (with some missing data, aka random effects or latent state nodes) and builds a Monte Carlo Expectation Maximization (MCEM) algorithm for maximum likelihood estimation. The user can specify which latent nodes are to be integrated out in the E-Step, or default choices will be made based on model structure. All other stochastic non-data nodes will be maximized over. The E-step is done with a sample from a nimble MCMC algorithm. The M-step is done by a call to optim.

Usage

buildMCEM(
  model,
  paramNodes,
  latentNodes,
  calcNodes,
  calcNodesOther,
  control = list(),
  ...
)

Arguments

model	a NIMBLE model object, either compiled or uncompiled.
paramNodes	a character vector of names of parameter nodes in the model; defaults are provided by setupMargNodes. Alternatively, paramNodes can be a list in the format returned by setupMargNodes, in which case latentNodes, calcNodes, and calcNodesOther are not needed (and will be ignored).
latentNodes	a character vector of names of unobserved (latent) nodes to marginalize (sum or integrate) over; defaults are provided by setupMargNodes (as the randomEffectsNodes in its return list).
calcNodes	a character vector of names of nodes for calculating components of the full-data likelihood that involve latentNodes; defaults are provided by setupMargNodes. There may be deterministic nodes between paramNodes and calcNodes. These will be included in calculations automatically and thus do not need to be included in calcNodes (but there is no problem if they are).
calcNodesOther	a character vector of names of nodes for calculating terms in the log-likelihood that do not depend on any latentNodes, and thus are not part of the marginalization, but should be included for purposes of finding the MLE. This defaults to stochastic nodes that depend on paramNodes but are not part of and do not depend on latentNodes. There may be deterministic nodes between paramNodes and calcNodesOther. These will be included in calculations automatically and thus do not need to be included in calcNodesOther (but there is no problem if they are).
control	a named list for providing additional settings used in MCEM. See control section below.
...	provided only as a means of checking if a user is using the deprecated interface to 'buildMCEM' in nimble versions < 1.2.0.

Details

buildMCEM is a nimbleFunction that creates an MCEM algorithm for a model and choices (perhaps default) of nodes in different roles in the model. The MCEM can then be compiled for fast execution with a compiled model.

Note that buildMCEM was re-written for nimble version 1.2.0 and is not backward-compatible with previous versions. The new version is considered to be in beta testing.

Denote data by Y, latent states (or missing data) by X, and parameters by T. MCEM works by the following steps, starting from some T:

1. Draw a sample of size M from P(X | Y, T) using MCMC.

2. Update T to be the maximizer of E[log P(X, Y | T)] where the expectation is approximated as a Monte Carlo average over the sample from step(1)

3. Repeat until converged.

The default version of MCEM is the ascent-based MCEM of Caffo et al. (2015). This attempts to update M when necessary to ensure that step 2 really moves uphill given that it is maximizing a Monte Carlo approximation and could accidently move downhill on the real surface of interest due to Monte Carlo error. The main tuning parameters include alpha, beta, gamma, Mfactor, C, and tol (tolerance).

If the model supports derivatives via nimble's automatic differentiation (AD) (and buildDerivs=TRUE in nimbleModel), the maximization step can use gradients from AD. You must manually set useDerivs=FALSE in the control list if derivatives aren't supported or if you don't want to use them.

In the ascent-based method, after maximization in step 2, the Monte Carlo standard error of the uphill movement is estimated. If the standardized uphill step is bigger than 0 with Type I error rate alpha, the iteration is accepted and the algorithm continues. Otherwise, it is not certain that step 2 really moved uphill due to Monte Carlo error, so the MCMC sample size M is incremented by a fixed factor (e.g. 0.33 or 0.5, called Mfactor in the control list), the additional samples are added by continuing step 1, and step 2 is tried again. If the Monte Carlo noise still overwhelms the magnitude of uphill movement, the sample size is increased again, and so on. alpha should be between 0 and 0.5. A larger value than usually used for inference is recommended so that there is an easy threshold to determine uphill movement, which avoids increasing M prematurely. M will never be increased above maxM.

Convergence is determined in a similar way. After a definite move uphill, we determine if the uphill increment is less than tol, with Type I error rate gamma. (But if M hits a maximum value, the convergence criterion changes. See below.)

beta is used to help set M to a minimal level based on previous iterations. This is a desired Type II error rate, assuming an uphill move and standard error based on the previous iteration. Set adjustM=FALSE in the control list if you don't want this behavior.

There are some additional controls on convergence for practical purposes. Set C in the control list to be the number of times the convergence criterion mut be satisfied in order to actually stop. E.g setting C=2 means there will always be a restart after the first convergence.

One problem that can occur with ascent-based MCEM is that the final iteration can be very slow if M must become very large to satisfy the convergence criterion. Indeed, if the algorithm starts near the MLE, this can occur. Set maxM in the control list to set the MCMC sample size that should never be exceeded.

If M==maxM, a softer convergence criterion is used. This second convergence criterion is to stop if we can't be sure we moved uphill using Type I error rate delta. This is a soft criterion because for small delta, Type II errors will be common (e.g. if we really did move uphill but can't be sure from the Monte Carlo sample), allowing the algorithm to terminate. One can continue the algorithm from where it stopped, so it is helpful to not have it get stuck when having a very hard time with the first (stricter) convergence criterion.

All of alpha, beta, delta, and gamma are utilized based on asymptotic arguments but in practice must be chosen heuristically. In other words, their theoretical basis does not exactly yield practical advice on good choices for efficiency and accuracy, so some trial and error will be needed.

It can also be helpful to set a minimum and maximum of allowed iterations (of steps 1 and 2 above). Setting minIter>1 in the control list can sometimes help avoid a false convergence on the first iteration by forcing at least one more iteration. Setting maxIter provides a failsafe on a stuck run.

If you don't want the ascent-based method at all and simply want to run a set of iterations, set ascent=FALSE in the control list. This will use the second (softer) convergence criterion.

Parameters to be maximized will by default be handled in an unconstrained parameter space, transformed if necessary by a parameterTransform object. In that case, the default optim method will be "BFGS" and can can be changed by setting optimMehod in the control list. Set useTransform=FALSE in the control list if you don't want the parameters transformed. In that case the default optimMethod will be "L-BFGS-B" if there are any actual constraints, and you can provide a list of boxConstraints in the control list. (Constraints may be determined by priors written in the model for parameters, even though their priors play no other role in MLE. E.g. sigma ~ halfflat() indicates sigma > 0).

Most of the control list elements can be overridden when calling the findMLE method. The findMLE argument continue=TRUE results in attempting to continue the algorithm where the previous call finished, including whatever settings were in use.

See setupMargNodes (which is called with the given arguments for paramNodes, calcNodes, and calcNodesOther; and with allowDiscreteLatent=TRUE, randomEffectsNodes=latentNodes, and check=check) for more about how the different groups of nodes are determined. In general, you can provide none, one, or more of the different kinds of nodes and setupMargNodes will try to determine the others in a sensible way. However, note that this cannot work for all ways of writing a model. One key example is that if random (latent) nodes are written as top-level nodes (e.g. following N(0,1)), they appear structurally to be parameters and you must tell buildMCEM that they are latentNodes. The various "Nodes" arguments will all be passed through model$expandNodeNames, allowing for example simply "x" to be provided when there are many nodes within "x".

Estimating the Monte Carlo standard error of the uphill step is not trivial because the sample was obtained by MCMC and so likely is autocorrelated. This is done by calling whatever function in R's global environment is called "MCEM_mcse", which is required to take two arguments: samples (which will be a vector of the differences in log(P(Y, X | T)) between the new and old values of T, across the sample of X) and m, the sample size. It must return an estimate of the standard error of the mean of the sample. NIMBLE provides a default version (exported from the package namespace), which calls mcmcse::mcse with method "obm". Simply provide a different function with this name in your R session to override NIMBLE's default.

Control list details

The control list accepts the following named elements:

• initM initial MCMC sample size, M. Default=1000.

• Mfactor Factor by which to increase MCMC sample size when step 2 results in noise overwhelming the uphill movement. The new M will be 1+Mfactor)*M (rounded up). Mfactor is 1/k of Caffo et al. (2015). Default=1/3.

• maxM Maximum allowed value of M (see above). Default=initM*20.

• burnin Number of burn-in iterations for the MCMC in step 1. Note that the initial states of one MCMC will be the last states from the previous MCMC, so they will often be good initial values after multiple iterations. Default=500.

• thin Thinning interval for the MCMC in step 1. Default=1.

• alpha Type I error rate for determining when step 2 has moved uphill. See above. Default=0.25.

• beta Used for determining a minimal value of $M$ based on previous iteration, if adjustM is TRUE. beta is a desired Type II error rate for determining uphill moves. Default=0.25.

• delta Type I error rate for the soft convergence approach (second approach above). Default=0.25.

• gamma Type I error rate for determining when step 2 has moved less than tol uphill, in which case ascent-based convergence is achieved (first approach above). Default=0.05.

• buffer A small amount added to lower box constraints and substracted from upper box constraints for all parameters, relevant only if useTransform=FALSE and some parameters do have boxConstraints set or have bounds that can be determined from the model. Default=1e-6.

• tol Ascent-based convergence tolerance. Default=0.001.

• ascent Logical to determine whether to use the ascent-based method of Caffo et al. Default=TRUE.

• C Number of convergences required to actually stop the algorithm. Default = 1.

• maxIter Maximum number of MCEM iterations to run.

• minIter Minimum number of MCEM iterations to run.

• adjustM Logical indicating whether to see if M needs to be increased based on statistical power argument in each iteration (using beta). Default=TRUE.

• verbose Logical indicating whether verbose output is desired. Default=TRUE.

• MCMCprogressBar Logical indicating whether MCMC progress bars should be shown for every iteration of step 1. This argument is passed to configureMCMC, or to config if provided. Default=TRUE.

• derivsDelta If AD derivatives are not used, then the method vcov must use finite difference derivatives to implement the method of Louis (1982). The finite differences will be delta or delta/2 for various steps. This is the same for all dimensions. Default=0.0001.

• mcmcControl This is passed to configureMCMC, or config if provided, as the control argument. i.e. control=mcmcControl.

• boxContrainst List of box constraints for the nodes that will be maximized over, only relevant if useTransform=FALSE and forceNoConstraints=FALSE (and ignored otherwise). Each constraint is a list in which the first element is a character vector of node names to which the constraint applies and the second element is a vector giving the lower and upper limits. Limits of -Inf or Inf are allowed. Any nodes that are not given constrains will have their constraints automatically determined by NIMBLE. See above. Default=list().

• forceNoConstraints Logical indicating whether to force ignoring constraints even if they might be necessary. Default=FALSE.

• useTransform Logical indicating whether to use a parameter transformation (see parameterTransform) to create an unbounded parameter space for the paramNodes. This allows unconstrained maximization algorithms to be used. Default=TRUE.

• check Logical passed as the check argument to setupMargNodes. Default=TRUE.

• useDerivs Logical indicating whether to use AD. If TRUE, the model must have been build with 'buildDerivs=TRUE'. It is not automatically determined from the model whether derivatives are supported. Default=TRUE.

• config Function to create the MCMC configuration used for step 1. The MCMC configuration is created by calling

  config(model, nodes = latentNodes, monitors = latentNodes,
  thin = thinDefault, control = mcmcControl, print = FALSE) 

  The default for config (if it is missing) is configureMCMC, which is nimble's general default MCMC configuration function.

Methods in the returned algorithm

The object returned by buildMCEM is a nimbleFunction object with the following methods

• findMLE is the main method of interest, launching the MCEM algorithm. It takes the following arguments:

  • pStart. Vector of initial parameter values. If omitted, the values currently in the model object are used.

  • returnTrans. Logical indicating whether to return parameters in the transformed space, if a parameter transformation is in use. Default=FALSE.

  • continue. Logical indicating whether to continue the MCEM from where the last call stopped. In addition, if TRUE, any other control setting provided in the last call will be used again. If FALSE, all control settings are reset to the values provided when buildMCEM was called. Any control settings provided in the same call as continue=TRUE will over-ride these behaviors and be used in the continued run.

  • All run-time control settings available in the control list for buildMCEM (except for buffer, boxConstraints, forceNoConstraints, useTransform, and useDerivs) are accepted as individual arguments to over-ride the values provided in the control list.

  findMLE returns on object of class optimResultNimbleList with the results of the final optimization of step 2. The par element of this list is the vector of maximum likelihood (MLE) parameters.

• vcov computes the approximate variance-covariance matrix of the MLE using the method of Louis (1982). It takes the following arguments:

  • params. Vector of parameters at which to compute the Hessian matrix used to obtain the vcov result. Typically this will be MLE$par, if MLE is the output of findMLE.

  • trans. Logical indicating whether params is on the transformed prameter scale, if a parameter transformation is in use. Typically this should be the same as the returnTrans argument to findMLE. Default=FALSE.

  • returnTrans. Logical indicting whether the vcov result should be for the transformed parameter space. Default matches trans.

  • M. Number of MCMC samples to obtain if resetSamples=TRUE. Default is the final value of M from the last call to findMLE. It can be helpful to increase M to obtain a more accurate vcov result (i.e. with less Monte Carlo noise).

  • resetSamples. Logical indicating whether to generate a new MCMC sample from P(X | Y, T), where T is params. If FALSE, the last sample from findMLE will be used. If MLE convergence was reasonable, this sample can be used. However, if the last MCEM step made a big move in parameter space (e.g. if convergence was not achieved), the last MCMC sample may not be accurate for obtaining vcov. Default=FALSE.

  • atMLE. Logical indicating whether you believe the params represents the MLE. If TRUE, one part of the computation will be skipped because it is expected to be 0 at the MLE. If there are parts of the model that are not connected to the latent nodes, i.e. of calcNodesOther is not empty, then atMLE will be ignored and set to FALSE. Default=FALSE. It is not really worth using TRUE unless you are confident and the time saving is meaningful, which is not very likely. In other words, this argument is provided for technical completeness.

  vcov returns a matrix that is the inverse of the negative Hessian of the log likelihood surface, i.e. the usual asymptotic approximation of the parameter variance-covariance matrix.

• doMCMC. This method runs the MCMC to sample from P(X | Y, T). One does not need to call this, as it is called via the MCEM algorithm in findMLE. This method is provided for users who want to use the MCMC for latent states directly. Samples should be retrieved by as.matrix(MCEM$mvSamples), where MCEM is the (compiled or uncompiled) MCEM algorithm object. This method takes the following arguments:

  • M. MCMC sample size.

  • thin. MCMC thinning interval.

  • reset. Logical indicating whether to reset the MCMC (passed to the MCMC run method as reset).

• transform and inverseTransform. Convert a parameter vector to an unconstrained parameter space and vice-versa, if useTransform=TRUE in the call to buildDerivs.

• resetControls. Reset all control arguments to the values provided in the call to buildMCEM. The user does not normally need to call this.

Author(s)

Perry de Valpine, Clifford Anderson-Bergman and Nicholas Michaud

References

Caffo, Brian S., Wolfgang Jank, and Galin L. Jones (2005). Ascent-based Monte Carlo expectation-maximization. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(2), 235-251.

Louis, Thomas A (1982). Finding the Observed Information Matrix When Using the EM Algorithm. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 44(2), 226-233.

Examples

## Not run: 
pumpCode <- nimbleCode({
 for (i in 1:N){
     theta[i] ~ dgamma(alpha,beta);
     lambda[i] <- theta[i]*t[i];
     x[i] ~ dpois(lambda[i])
 }
 alpha ~ dexp(1.0);
 beta ~ dgamma(0.1,1.0);
})

pumpConsts <- list(N = 10,
              t = c(94.3, 15.7, 62.9, 126, 5.24,
                31.4, 1.05, 1.05, 2.1, 10.5))

pumpData <- list(x = c(5, 1, 5, 14, 3, 19, 1, 1, 4, 22))

pumpInits <- list(alpha = 1, beta = 1,
             theta = rep(0.1, pumpConsts$N))
pumpModel <- nimbleModel(code = pumpCode, name = 'pump', constants = pumpConsts,
                         data = pumpData, inits = pumpInits,
                         buildDerivs=TRUE)

pumpMCEM <- buildMCEM(model = pumpModel)

CpumpModel <- compileNimble(pumpModel)

CpumpMCEM <- compileNimble(pumpMCEM, project=pumpModel)

MLE <- CpumpMCEM$findMLE()
vcov <- CpumpMCEM$vcov(MLE$par)


## End(Not run)

nimble documentation built on Sept. 11, 2024, 7:10 p.m.