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The VariationalBayes function is a numerical approximation
method for deterministically estimating the marginal posterior
distributions, target distributions, in a Bayesian model with
approximated distributions by minimizing the Kullback-Leibler
Divergence (KLD) between the target and its
approximation.
VariationalBayes(Model, parm, Data, Covar=NULL, Interval=1.0E-6,
Iterations=1000, Method="Salimans2", Samples=1000, sir=TRUE,
Stop.Tolerance=1.0E-5, CPUs=1, Type="PSOCK")This required argument receives the model from a
user-defined function. The user-defined function is where the model
is specified. VariationalBayes passes two arguments to
the model function, parms and Data. For more
information, see the LaplacesDemon function and
``LaplacesDemon Tutorial'' vignette.
This argument requires a vector of initial values equal in
length to the number of parameters. VariationalBayes will
attempt to optimize these initial values for the parameters, where
the optimized values are the posterior means, for later use with the
IterativeQuadrature, LaplacesDemon, or
PMC function. The GIV function may be
used to randomly generate initial values. Parameters must be
continuous.
This required argument accepts a list of data. The list of
data must include mon.names which contains monitored variable
names, and parm.names which contains parameter
names. VariationalBayes must be able to determine the
sample size of the data, and will look for a scalar sample size
variable n or N. If not found, it will look for
variable y or Y, and attempt to take its number of
rows as sample size. VariationalBayes needs to determine
sample size due to the asymptotic nature of this method. Sample size
should be at least
This argument defaults to NULL, but may otherwise
accept a Covar=NULL should be used. Once
VariationalBayes has finished updating, it may be desired
to continue updating where it left off, in which case the covariance
matrix from the last run can be input into the next run.
This argument receives an interval for estimating approximate gradients. The logarithm of the unnormalized joint posterior density of the Bayesian model is evaluated at the current parameter value, and again at the current parameter value plus this interval.
This argument accepts an integer that determines the
number of iterations that VariationalBayes will attempt
to maximize the logarithm of the unnormalized joint posterior
density. Iterations defaults to 1000.
VariationalBayes will stop before this number of
iterations if the tolerance is less than or equal to the
Stop.Tolerance criterion. The required amount of computer
memory increases with Iterations. If computer memory is
exceeded, then all will be lost.
This optional argument currently accepts only
Salimans2, which is the second algorithm in Salimans and
Knowles (2013).
This argument indicates the number of posterior samples
to be taken with sampling importance resampling via the
SIR function, which occurs only when
sir=TRUE. Note that the number of samples should increase
with the number and intercorrelations of the parameters.
This logical argument indicates whether or not Sampling
Importance Resampling (SIR) is conducted via the SIR
function to draw independent posterior samples. This argument
defaults to TRUE. Even when TRUE, posterior samples
are drawn only when VariationalBayes has
converged. Posterior samples are required for many other functions,
including plot.vb and predict.vb. The only
time that it is advantageous for sir=FALSE is when
VariationalBayes is used to help the initial values for
IterativeQuadrature, LaplacesDemon, or
PMC, and it is unnecessary for time to be spent on
sampling. Less time can be spent on sampling by increasing
CPUs, which parallelizes the sampling.
This argument accepts any positive number and
defaults to 1.0E-3. Tolerance is calculated each iteration, and the
criteria varies by algorithm. The algorithm is considered to have
converged to the user-specified Stop.Tolerance when the
tolerance is less than or equal to the value of
Stop.Tolerance, and the algorithm terminates at the end of
the current iteration. Often, multiple criteria are used, in
which case the maximum of all criteria becomes the tolerance. For
example, when partial derivatives are taken, it is commonly required
that the Euclidean norm of the partial derivatives is a criterion,
and another common criterion is the Euclidean norm of the
differences between the current and previous parameter
values. Several algorithms have other, specific tolerances.
This argument accepts an integer that specifies the number
of central processing units (CPUs) of the multicore computer or
computer cluster. This argument defaults to CPUs=1, in which
parallel processing does not occur. Parallelization occurs only for
sampling with SIR when sir=TRUE.
This argument specifies the type of parallel processing to
perform, accepting either Type="PSOCK" or
Type="MPI".
VariationalBayes returns an object of class vb
that is a list with the following components:
This is the matched call of VariationalBayes.
This is a logical indicator of whether or not
VariationalBayes converged within the specified
Iterations according to the supplied Stop.Tolerance
criterion. Convergence does not indicate that the global maximum has
been found, but only that the tolerance was less than or equal to
the Stop.Tolerance criterion.
This is the estimated covariance matrix. The
Covar matrix may be scaled and input into the Covar
argument of the LaplacesDemon or PMC
function for further estimation, or the diagonal of this matrix may
be used to represent the posterior variance of the parameters,
provided the algorithm converged and matrix inversion was
successful. To scale this matrix for use with Laplace's Demon or
PMC, multiply it by
This is a vector of the iterative history of the
deviance in the VariationalBayes function, as it sought
convergence.
This is an array of the iterative history of the
parameters in the VariationalBayes function, as it sought
convergence. The first matrix is for means and the second matrix is
for variances.
This is the vector of initial values that was
originally given to VariationalBayes in the parm
argument.
This is an approximation of the logarithm of the marginal
likelihood of the data (see the LML function for more
information). When the model has converged and sir=TRUE, the
NSIS method is used. When the model has converged and
sir=FALSE, the LME method is used. This is the
logarithmic form of equation 4 in Lewis and Raftery (1997). As a
rough estimate of Kass and Raftery (1995), the LME-based LML is
worrisome when the sample size of the data is less than five times
the number of parameters, and LML should be adequate in most
problems when the sample size of the data exceeds twenty times the
number of parameters (p. 778). The LME is inappropriate with
hierarchical models. However LML is estimated, it is useful
for comparing multiple models with the BayesFactor function.
This reports the final scalar value for the logarithm of the unnormalized joint posterior density.
This reports the initial scalar value for the logarithm of the unnormalized joint posterior density.
This is the number of minutes that
VariationalBayes was running, and this includes the
initial checks as well as drawing posterior samples and creating
summaries.
When sir=TRUE, a number of independent
posterior samples equal to Samples is taken, and the draws
are stored here as a matrix. The rows of the matrix are the samples,
and the columns are the monitored variables.
When sir=TRUE, a number of independent
posterior samples equal to Samples is taken, and the draws
are stored here as a matrix. The rows of the matrix are the samples,
and the columns are the parameters.
This is the final, scalar Step.Size
value at the end of the VariationalBayes algorithm.
This is the initial, scalar Step.Size.
This is a summary matrix that summarizes the point-estimated posterior means and variances. Uncertainty around the posterior means is estimated from the estimated covariance matrix. Rows are parameters. The following columns are included: Mean, SD (Standard Deviation), LB (Lower Bound), and UB (Upper Bound). The bounds constitute a 95% probability interval.
This is a summary matrix that summarizes the
posterior samples drawn with sampling importance resampling
(SIR) when sir=TRUE, given the point-estimated
posterior means and covariance matrix. Rows are parameters. The
following columns are included: Mean, SD (Standard Deviation),
LB (Lower Bound), and UB (Upper Bound). The bounds constitute a 95%
probability interval.
This is the last Tolerance of the
VariationalBayes algorithm.
This is the Stop.Tolerance criterion.
Variational Bayes (VB) is a family of numerical approximation algorithms that is a subset of variational inference algorithms, or variational methods. Some examples of variational methods include the mean-field approximation, loopy belief propagation, tree-reweighted belief propagation, and expectation propagation (EP).
Variational inference for probabilistic models was introduced in the field of machine learning, influenced by statistical physics literature (Saul et al., 1996; Saul and Jordan, 1996; Jaakkola, 1997). The mean-field methods in Neal and Hinton (1999) led to variational algorithms.
Variational inference algorithms were later generalized for conjugate exponential-family models (Attias, 1999, 2000; Wiegerinck, 2000; Ghahramani and Beal, 2001; Xing et al., 2003). These algorithms still require different designs for different model forms. Salimans and Knowles (2013) introduced general-purpose VB algorithms for Gaussian posteriors.
A VB algorithm deterministically estimates the marginal posterior
distributions (target distributions) in a Bayesian model with
approximated distributions by minimizing the Kullback-Leibler
Divergence (KLD) between the target and its
approximation. The complicated posterior distribution is approximated
with a simpler distribution. The simpler, approximated distribution is
called the variational approximation, or approximation distribution,
of the posterior. The term variational is derived from the calculus of
variations, and regards optimization algorithms that select the best
function (which is a distribution in VB), rather than merely selecting
the best parameters.
VB algorithms often use Gaussian distributions as approximating distributions. In this case, both the mean and variance of the parameters are estimated.
Usually, a VB algorithm is slower to convergence than a Laplace Approximation algorithm, and faster to convergence than a Monte Carlo algorithm such as Markov chain Monte Carlo (MCMC). VB often provides solutions with comparable accuracy to MCMC in less time. Though Monte Carlo algorithms provide a numerical approximation to the exact posterior using a set of samples, VB provides a locally-optimal, exact analytical solution to an approximation of the posterior. VB is often more applicable than MCMC to big data or large-dimensional models.
Since VB is deterministic, it is asymptotic and subject to the same limitations with respect to sample size as Laplace Approximation. However, VB estimates more parameters than Laplace Approximation, such as when Laplace Approximation optimizes the posterior mode of a Gaussian distribution, while VB optimizes both the Gaussian mean and variance.
Traditionally, VB algorithms required customized equations. The
VariationalBayes function uses general-purpose algorithms. A
general-purpose VB algorithm is less efficient than an algorithm
custom designed for the model form. However, a general-purpose
algorithm is applied consistently and easily to numerous model forms.
When Method="Salimans2", the second algorithm of Salimans and
Knowles (2013) is used. This requires the gradient and Hessian, which
is more efficient with a small number of parameters as long as the
posterior is twice differentiable. The step size is constant. This
algorithm is suitable for marginal posterior distributions that are
Gaussian and unimodal. A stochastic approximation algorithm is used
in the context of fixed-form VB, inspired by considering fixed-form VB
to be equivalent to performing a linear regression with the sufficient
statistics of the approximation as independent variables and the
unnormalized logarithm of the joint posterior density as the dependent
variable. The number of requested iterations should be large, since the
step-size decreases for larger requested iterations, and a small
step-size will eventually converge. A large number of requested
iterations results in a smaller step-size and better convergence
properties, so hope for early convergence. However convergence is
checked only in the last half of the iterations after the algorithm
begins to average the mean and variance from the samples of the
stochastic approximation. The history of stochastic samples is
returned.
Attias, H. (1999). "Inferring Parameters and Structure of Latent Variable Models by Variational Bayes". In Uncertainty in Artificial Intelligence.
Attias, H. (2000). "A Variational Bayesian Framework for Graphical Models". In Neural Information Processing Systems.
Ghahramani, Z. and Beal, M. (2001). "Propagation Algorithms for Variational Bayesian Learning". In Neural Information Processing Systems, p. 507--513.
Jaakkola, T. (1997). "Variational Methods for Inference and Estimation in Graphical Models". PhD thesis, Massachusetts Institute of Technology.
Salimans, T. and Knowles, D.A. (2013). "Fixed-Form Variational Posterior Approximation through Stochastic Linear Regression". Bayesian Analysis, 8(4), p. 837--882.
Neal, R. and Hinton, G. (1999). "A View of the EM Algorithm that Justifies Incremental, Sparse, and Other Variants". In Learning in Graphical Models, p. 355--368. MIT Press, 1999.
Saul, L. and Jordan, M. (1996). "Exploiting Tractable Substructures in Intractable Networks". Neural Information Processing Systems.
Saul, L., Jaakkola, T., and Jordan, M. (1996). "Mean Field Theory for Sigmoid Belief Networks". Journal of Artificial Intelligence Research, 4, p. 61--76.
Wiegerinck, W. (2000). "Variational Approximations Between Mean Field Theory and the Junction Tree Algorithm". In Uncertainty in Artificial Intelligence.
Xing, E., Jordan, M., and Russell, S. (2003). "A Generalized Mean Field Algorithm for Variational Inference in Exponential Families". In Uncertainty in Artificial Intelligence.
BayesFactor,
IterativeQuadrature,
LaplaceApproximation,
LaplacesDemon,
GIV,
LML,
PMC, and
SIR.
# NOT RUN {
# The accompanying Examples vignette is a compendium of examples.
#################### Load the LaplacesDemon Library #####################
library(LaplacesDemon)
############################## Demon Data ###############################
data(demonsnacks)
y <- log(demonsnacks$Calories)
X <- cbind(1, as.matrix(log(demonsnacks[,10]+1)))
J <- ncol(X)
for (j in 2:J) X[,j] <- CenterScale(X[,j])
######################### Data List Preparation #########################
mon.names <- "mu[1]"
parm.names <- as.parm.names(list(beta=rep(0,J), sigma=0))
pos.beta <- grep("beta", parm.names)
pos.sigma <- grep("sigma", parm.names)
PGF <- function(Data) {
beta <- rnorm(Data$J)
sigma <- runif(1)
return(c(beta, sigma))
}
MyData <- list(J=J, PGF=PGF, X=X, mon.names=mon.names,
parm.names=parm.names, pos.beta=pos.beta, pos.sigma=pos.sigma, y=y)
########################## Model Specification ##########################
Model <- function(parm, Data)
{
### Parameters
beta <- parm[Data$pos.beta]
sigma <- interval(parm[Data$pos.sigma], 1e-100, Inf)
parm[Data$pos.sigma] <- sigma
### Log-Prior
beta.prior <- sum(dnormv(beta, 0, 1000, log=TRUE))
sigma.prior <- dhalfcauchy(sigma, 25, log=TRUE)
### Log-Likelihood
mu <- tcrossprod(Data$X, t(beta))
LL <- sum(dnorm(Data$y, mu, sigma, log=TRUE))
### Log-Posterior
LP <- LL + beta.prior + sigma.prior
Modelout <- list(LP=LP, Dev=-2*LL, Monitor=mu[1],
yhat=rnorm(length(mu), mu, sigma), parm=parm)
return(Modelout)
}
############################ Initial Values #############################
#Initial.Values <- GIV(Model, MyData, PGF=TRUE)
Initial.Values <- rep(0,J+1)
#Fit <- VariationalBayes(Model, Initial.Values, Data=MyData, Covar=NULL,
# Iterations=1000, Method="Salimans2", Stop.Tolerance=1e-3, CPUs=1)
#Fit
#print(Fit)
#PosteriorChecks(Fit)
#caterpillar.plot(Fit, Parms="beta")
#plot(Fit, MyData, PDF=FALSE)
#Pred <- predict(Fit, Model, MyData, CPUs=1)
#summary(Pred, Discrep="Chi-Square")
#plot(Pred, Style="Covariates", Data=MyData)
#plot(Pred, Style="Density", Rows=1:9)
#plot(Pred, Style="Fitted")
#plot(Pred, Style="Jarque-Bera")
#plot(Pred, Style="Predictive Quantiles")
#plot(Pred, Style="Residual Density")
#plot(Pred, Style="Residuals")
#Levene.Test(Pred)
#Importance(Fit, Model, MyData, Discrep="Chi-Square")
#Fit$Covar is scaled (2.38^2/d) and submitted to LaplacesDemon as Covar.
#Fit$Summary[,1] is submitted to LaplacesDemon as Initial.Values.
#End
# }
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