Bayesian Nonparametric Models in NIMBLE, Part 2: Nonparametric Random Effects

Bayesian nonparametrics in NIMBLE: Nonparametric random effects

Overview

NIMBLE is a hierarchical modeling package that uses nearly the same language for model specification as the popular MCMC packages WinBUGS, OpenBUGS and JAGS, while making the modeling language extensible — you can add distributions and functions — and also allowing customization of the algorithms used to estimate the parameters of the model.

Recently, we added support for Markov chain Monte Carlo (MCMC) inference for Bayesian nonparametric (BNP) mixture models to NIMBLE. In particular, starting with version 0.6-11, NIMBLE provides functionality for fitting models involving Dirichlet process priors using either the Chinese Restaurant Process (CRP) or a truncated stick-breaking (SB) representation of the Dirichlet process prior.

We will illustrate NIMBLE’s BNP capabilities using two examples. In a previous post, we showed how to use nonparametric mixture models with different kernels for density estimation. In this post, we will take a parametric generalized linear mixed model and show how to switch to a nonparametric representation of the random effects that avoids the assumption of normally-distributed random effects.

For more detailed information on NIMBLE and Bayesian nonparametrics in NIMBLE, see the NIMBLE User Manual.

Parametric meta analysis of Avandia myocardial infarctions (MIs)

We will illustrate the use of nonparametric mixture models for modeling random effects distributions in the context of a meta-analysis of the side effects of a formerly very popular drug for diabetes called Avandia. The data we analyze played a role in raising serious questions about the safety of this drug. The question is whether Avandia use increases the risk of myocardial infarction (heart attack). There are 48 studies (the 49th study in the data file is different in some ways and excluded here), each with treatment and control arms.

dat <- read.csv('https://rawgit.com/nimble-dev/nimble-demos/master/intro_bnp/avandia.csv')
head(dat)
##   trial nAvandia avandiaMI nControl controlMI
## 1     1      357         2      176         0
## 2     2      391         2      207         1
## 3     3      774         1      185         1
## 4     4      213         0      109         1
## 5     5      232         1      116         0
## 6     6       43         0       47         1
dat <- dat[-49, ]

Model formulation

We begin with a standard generalized linear mixed model (GLMM)-based meta analysis. The vectors n and x contain the total number of patients in the control and the number of patients suffering from myocardial infarctions in the control group of each study, respectively. Similarly, the vectors m and y contain similar information for patients receiving the drug Avandia. The model takes the form

  x_{i} \mid \theta, \gamma_i \sim \mbox{Bin} \left(n_i, \frac{\exp\left\{ \gamma_i \right\}}{1 + \exp\left\{ \gamma_i \right\}} \right) , \quad\quad y_{i} \mid \theta, \gamma_i \sim \mbox{Bin} \left(m_i, \frac{\exp\left\{ \theta + \gamma_i \right\}}{1 + \exp\left\{ \theta + \gamma_i \right\}} \right)

where the random effects, \gamma_i, follow a common normal distribution, \gamma_i \sim \mbox{N}(0, \tau^2), and the \theta and \tau^2 are given reasonably non-informative priors. The parameter \theta quantifies the difference in risk between the control and treatment arms, while the \gamma_i quantify study-specific variation.

This model can be specified in NIMBLE using the following code:

x <- dat$controlMI
n <- dat$nControl
y <- dat$avandiaMI
m <- dat$nAvandia

nStudies <- nrow(dat)
data <- list(x = x, y = y)
constants = list(n = n, m = m, nStudies = nStudies)

codeParam <- nimbleCode({
    for(i in 1:nStudies) {
        y[i] ~ dbin(size = m[i], prob = q[i]) # avandia MIs
        x[i] ~ dbin(size = n[i], prob = p[i]) # control MIs
        q[i] <- expit(theta + gamma[i])       # Avandia log-odds
        p[i] <- expit(gamma[i])               # control log-odds
        gamma[i] ~ dnorm(mu, var = tau2)      # study effects
    }
    theta ~ dflat()        # effect of Avandia
    # random effects hyperparameters
    mu ~ dnorm(0, 10)
    tau2 ~ dinvgamma(2, 1)
})

Running the MCMC

Let’s run a basic MCMC.

set.seed(9)
inits = list(theta = 0, mu = 0, tau2 = 1, gamma = rnorm(nStudies))

samples <- nimbleMCMC(code = codeParam, data = data, inits = inits,
                      constants = constants, monitors = c("mu", "tau2", "theta", "gamma"),
                      thin = 10, niter = 22000, nburnin = 2000, nchains = 1, setSeed = TRUE)
## defining model...
## building model...
## setting data and initial values...
## running calculate on model (any error reports that follow may simply reflect missing values in model variables) ... 
## checking model sizes and dimensions...
## checking model calculations...
## model building finished.
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
## runMCMC's handling of nburnin changed in nimble version 0.6-11. Previously, nburnin samples were discarded *post-thinning*.  Now nburnin samples are discarded *pre-thinning*.  The number of samples returned will be floor((niter-nburnin)/thin).
## running chain 1...
## |-------------|-------------|-------------|-------------|
## |-------------------------------------------------------|
par(mfrow = c(1, 4), cex = 1.1, mgp = c(1.8,.7,0))
ts.plot(samples[ , 'theta'], xlab = 'iteration', ylab = expression(theta))
hist(samples[ , 'theta'], xlab = expression(theta), main = 'effect of Avandia')

gammaCols <- grep('gamma', colnames(samples))
gammaMn <- colMeans(samples[ , gammaCols])
hist(gammaMn, xlab = 'posterior means', main = 'random effects distribution')
hist(samples[1000, gammaCols], xlab = 'single draw',
                   main = 'random effects distribution')

The results suggests there is an overall difference in risk between the control and treatment arms. But what about the normality assumption? Are our conclusions robust to that assumption? Perhaps the random effects distribution are skewed. (And recall that the estimates above of the random effects are generated under the normality assumption, which pushes the estimated effects to look more normal…)

DP-based random effects modeling for meta analysis

Model formulation

Now, we use a nonparametric distribution for the \gamma_is. More specifically, we assume that each \gamma_i is generated from a location-scale mixture of normal distributions:

  \gamma_i \mid \mu_i, \tau_i^2 \sim \mbox{N}(\mu_i, \tau_i^2), \quad\quad (\mu_i, \tau_i^2) \mid G \sim G, \quad\quad G \sim \mbox{DP}(\alpha, H),

where H is a normal-inverse-gamma distribution.

This specification induces clustering among the random effects. As in the case of density estimation problems, the DP prior allows the data to determine the number of components, from as few as one component (i.e., simplifying to the parametric model), to as many as n components, i.e., one component for each observation. This allows the distribution of the random effects to be multimodal if the data supports such behavior, greatly increasing its flexibility. This model can be specified in NIMBLE using the following code:

codeBNP <- nimbleCode({
    for(i in 1:nStudies) {
        y[i] ~ dbin(size = m[i], prob = q[i])   # avandia MIs
        x[i] ~ dbin(size = n[i], prob = p[i])   # control MIs
        q[i] <- expit(theta + gamma[i])         # Avandia log-odds
        p[i] <- expit(gamma[i])                 # control log-odds
        gamma[i] ~ dnorm(mu[i], var = tau2[i])  # random effects from mixture dist.
        mu[i] <- muTilde[xi[i]]                 # mean for random effect from cluster xi[i]
        tau2[i] <- tau2Tilde[xi[i]]             # var for random effect from cluster xi[i]
    }
    # mixture component parameters drawn from base measures
    for(i in 1:nStudies) {
        muTilde[i] ~ dnorm(mu0, var = var0)
        tau2Tilde[i] ~ dinvgamma(a0, b0)
    }
    # CRP for clustering studies to mixture components
    xi[1:nStudies] ~ dCRP(alpha, size = nStudies)
    # hyperparameters
    alpha ~ dgamma(1, 1)
    mu0 ~ dnorm(0, 10)
    var0 ~ dinvgamma(2, 1)
    a0 ~ dinvgamma(2, 1)
    b0 ~ dinvgamma(2, 1)
    theta ~ dflat()          # effect of Avandia
})

Running the MCMC

The following code compiles the model and runs a collapsed Gibbs sampler for the model

inits <- list(gamma = rnorm(nStudies), xi = sample(1:2, nStudies, replace = TRUE),
              alpha = 1, mu0 = 0, var0 = 1, a0 = 1, b0 = 1, theta = 0,
              muTilde = rnorm(nStudies), tau2Tilde = rep(1, nStudies))

samplesBNP <- nimbleMCMC(code = codeBNP, data = data, inits = inits,
               constants = constants,
               monitors = c("theta", "gamma", "alpha", "xi", "mu0", "var0", "a0", "b0"),
               thin = 10, niter = 22000, nburnin = 2000, nchains = 1, setSeed = TRUE)
## defining model...
## building model...
## setting data and initial values...
## running calculate on model (any error reports that follow may simply reflect missing values in model variables) ... 
## checking model sizes and dimensions...
## checking model calculations...
## model building finished.
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
## runMCMC's handling of nburnin changed in nimble version 0.6-11. Previously, nburnin samples were discarded *post-thinning*.  Now nburnin samples are discarded *pre-thinning*.  The number of samples returned will be floor((niter-nburnin)/thin).
## running chain 1...
## |-------------|-------------|-------------|-------------|
## |-------------------------------------------------------|
gammaCols <- grep('gamma', colnames(samplesBNP))
gammaMn <- colMeans(samplesBNP[ , gammaCols])
xiCols <- grep('xi', colnames(samplesBNP))

par(mfrow = c(1,5), cex = 1.1, mgp = c(1.8,.7,0))
ts.plot(samplesBNP[ , 'theta'], xlab = 'iteration', ylab = expression(theta),
   main = expression(paste('traceplot for ', theta)))
hist(samplesBNP[ , 'theta'], xlab = expression(theta), main = 'effect of Avandia')
hist(gammaMn, xlab = 'posterior means',
              main = "random effects distrib'n")
hist(samplesBNP[1000, gammaCols], xlab = 'single draw',
                   main = "random effects distrib'n")

# How many mixture components are inferred?
xiRes <- samplesBNP[ , xiCols]
nGrps <- apply(xiRes, 1, function(x) length(unique(x)))
ts.plot(nGrps, xlab = 'iteration', ylab = 'number of components',
   main = 'number of components')

The primary inference seems robust to the original parametric assumption. This is probably driven by the fact that there is not much evidence of lack of normality in the random effects distribution (as evidenced by the fact that the posterior distribution of the number of mixture components places a large amount of probability on exactly one component).

More information and future development

Please see our User Manual for more details.

We’re in the midst of improvements to the existing BNP functionality as well as adding additional Bayesian nonparametric models, such as hierarchical Dirichlet processes and Pitman-Yor processes, so please add yourself to our announcement or user support/discussion Google groups.

Bayesian Nonparametric Models in NIMBLE, Part 1: Density Estimation


Bayesian Nonparametric Models in NIMBLE, Part 1: Density Estimation

Bayesian nonparametrics in NIMBLE: Density estimation

Overview

NIMBLE is a hierarchical modeling package that uses nearly the same language for model specification as the popular MCMC packages WinBUGS, OpenBUGS and JAGS, while making the modeling language extensible — you can add distributions and functions — and also allowing customization of the algorithms used to estimate the parameters of the model.

Recently, we added support for Markov chain Monte Carlo (MCMC) inference for Bayesian nonparametric (BNP) mixture models to NIMBLE. In particular, starting with version 0.6-11, NIMBLE provides functionality for fitting models involving Dirichlet process priors using either the Chinese Restaurant Process (CRP) or a truncated stick-breaking (SB) representation of the Dirichlet process prior.

In this post we illustrate NIMBLE’s BNP capabilities by showing how to use nonparametric mixture models with different kernels for density estimation. In a later post, we will take a parametric generalized linear mixed model and show how to switch to a nonparametric representation of the random effects that avoids the assumption of normally-distributed random effects.

For more detailed information on NIMBLE and Bayesian nonparametrics in NIMBLE, see the NIMBLE User Manual.

Basic density estimation using Dirichlet Process Mixture models

NIMBLE provides the machinery for nonparametric density estimation by means of Dirichlet process mixture (DPM) models (Ferguson, 1974; Lo, 1984; Escobar, 1994; Escobar and West, 1995). For an independent and identically distributed sample y_1, \ldots, y_n, the model takes the form

  y_i \mid \theta_i \sim p(y_i \mid \theta_i), \quad\quad \theta_i \mid G \sim G, \quad\quad G \mid  \alpha, H \sim \mbox{DP}(\alpha, H), \quad\quad i=1,\ldots, n .

The NIMBLE implementation of this model is flexible and allows for mixtures of arbitrary kernels, p(y_i \mid \theta), which can be either conjugate or non-conjugate to the (also arbitrary) base measure H. In the case of conjugate kernel / base measure pairs, NIMBLE is able to detect the presence of the conjugacy and use it to improve the performance of the sampler.

To illustrate these capabilities, we consider the estimation of the probability density function of the waiting time between eruptions of the Old Faithful volcano data set available in R.

data(faithful)

The observations y_1, \ldots, y_n correspond to the second column of the dataframe, and n = 272.

Fitting a location-scale mixture of Gaussian distributions using the CRP representation

Model specification

We first consider a location-scale Dirichlet process mixture of normal distributionss fitted to the transformed data y_i^{*} = \log (y_i):

  y^{*}_i \mid \mu_i, \sigma^2_i \sim \mbox{N}(\mu_i, \sigma^2_i), \quad (\mu_i, \sigma^2_i) \mid G \sim G, \quad G \mid \alpha, H \sim \mbox{DP}(\alpha, H), \quad i=1,\ldots, n,

where H corresponds to a normal-inverse-gamma distribution. This model can be interpreted as providing a Bayesian version of kernel density estimation for y^{*}_i using Gaussian kernels and adaptive bandwidths. On the original scale of the data, this translates into an adaptive log-Gaussian kernel density estimate.

Introducing auxiliary variables \xi_1, \ldots, \xi_n that indicate which component of the mixture generates each observation, and integrating over the random measure G, we obtain the CRP representation of the model (Blackwell and MacQueen, 1973):

  y_i^{*} \mid \{ \tilde{\mu}_k \}, \{ \tilde{\sigma}_k^{2} \} \sim \mbox{N}\left( \tilde{\mu}_{\xi_i}, \tilde{\sigma}^2_{\xi_i} \right), \quad\quad \xi \mid \alpha \sim \mbox{CRP}(\alpha), \quad\quad (\tilde{\mu}_k, \tilde{\sigma}_k^2) \mid H \sim H, \quad\quad i=1,\ldots, n ,

where

  p(\xi \mid \alpha) = \frac{\Gamma(\alpha)}{\Gamma(\alpha + n)} \alpha^{K(\xi)} \prod_k  \Gamma\left(m_k(\xi)\right),

K(\xi) \le n is the number of unique values in the vector \xi, and m_k(\xi) is the number of times the k-th unique value appears in \xi. This specification makes it clear that each observation belongs to any of at most n normally distributed clusters, and that the CRP distribution corresponds to the prior distribution on the partition structure.

NIMBLE’s specification of this model is given by

code <- nimbleCode({
  for(i in 1:n) {
    y[i] ~ dnorm(mu[i], var = s2[i])
    mu[i] <- muTilde[xi[i]]
    s2[i] <- s2Tilde[xi[i]]
  }
  xi[1:n] ~ dCRP(alpha, size = n)
  for(i in 1:n) {
    muTilde[i] ~ dnorm(0, var = s2Tilde[i])
    s2Tilde[i] ~ dinvgamma(2, 1)
  }
  alpha ~ dgamma(1, 1)
})

Note that in the model code the length of the parameter vectors muTilde and s2Tilde has been set to n. We do this because the current implementation of NIMBLE requires that the length of vector of parameters be set in advance and does not allow for their number to change between iterations. Hence, if we are to ensure that the algorithm always performs as intended we need to work with the worst case scenario, i.e., the case where there are as many components as observations. While this ensures that the algorithm always works as intended, it is also somewhat inefficient, both in terms of memory requirements (when n is large a large number of unoccupied components need to be maintained) and in terms of computational burden (a large number of parameters that are not required for posterior inference need to be updated at every iteration). When we use a mixture of gamma distributions below, we will show a computational shortcut that improves the efficiency.

Note also that the value of \alpha controls the number of components we expect a priori, with larger values of \alpha corresponding to a larger number of components occupied by the data. Hence, by assigning a prior to \alpha we add flexibility to the model specification. The particular choice of a Gamma prior allows NIMBLE to use a data-augmentation scheme to efficiently sample from the corresponding full conditional distribution. Alternative prior specifications for \alpha are possible, in which case the default sampler for this parameter is an adaptive random-walk Metropolis-Hastings algorithm.

Running the MCMC algorithm

The following code sets up the data and constants, initializes the parameters, defines the model object, and builds and runs the MCMC algorithm. Because the specification is in terms of a Chinese restaurant process, the default sampler selected by NIMBLE is a collapsed Gibbs sampler (Neal, 2000).

set.seed(1)
# Model Data
lFaithful <- log(faithful$waiting)
standlFaithful <- (lFaithful - mean(lFaithful)) / sd(lFaithful)
data <- list(y = standlFaithful)
# Model Constants
consts <- list(n = length(standlFaithful))
# Parameter initialization
inits <- list(xi = sample(1:10, size=consts$n, replace=TRUE),
              muTilde = rnorm(consts$n, 0, sd = sqrt(10)),
              s2Tilde = rinvgamma(consts$n, 2, 1),
              alpha = 1)
# Model creation and compilation
rModel <- nimbleModel(code, data = data, inits = inits, constants = consts)
## defining model...
## building model...
## setting data and initial values...
## running calculate on model (any error reports that follow may simply reflect missing values in model variables) ... 
## checking model sizes and dimensions...
## model building finished.
cModel <- compileNimble(rModel)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
# MCMC configuration, creation, and compilation
conf <- configureMCMC(rModel, monitors = c("xi", "muTilde", "s2Tilde", "alpha"))
mcmc <- buildMCMC(conf)
cmcmc <- compileNimble(mcmc, project = rModel)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
samples <- runMCMC(cmcmc, niter = 7000, nburnin = 2000, setSeed = TRUE)
## running chain 1...
## |-------------|-------------|-------------|-------------|
## |-------------------------------------------------------|

We can extract the samples from the posterior distributions of the parameters and create trace plots, histograms, and any other summary of interest. For example, for the concentration parameter \alpha we have:

# Trace plot for the concentration parameter
ts.plot(samples[ , "alpha"], xlab = "iteration", ylab = expression(alpha))
# Posterior histogram
hist(samples[ , "alpha"], xlab = expression(alpha), main = "", ylab = "Frequency")
quantile(samples[ , "alpha"], c(0.5, 0.025, 0.975))
##        50%       2.5%      97.5% 
## 0.42365754 0.06021512 1.52299639

Under this model, the posterior predictive distribution for a new observation \tilde{y}, p(\tilde{y} \mid y_1, \ldots, y_n), is the optimal density estimator (under squared error loss). Samples for this estimator can be easily computed from the samples generated by our MCMC:

# posterior samples of the concentration parameter
alphaSamples <- samples[ , "alpha"]
# posterior samples of the cluster means
muTildeSamples <- samples[ , grep('muTilde', colnames(samples))]
# posterior samples of the cluster variances
s2TildeSamples <- samples[ , grep('s2Tilde', colnames(samples))]
# posterior samples of the cluster memberships
xiSamples <- samples [ , grep('xi', colnames(samples))]

standlGrid <- seq(-2.5, 2.5, len = 200) # standardized grid on log scale

densitySamplesStandl <- matrix(0, ncol = length(standlGrid), nrow = nrow(samples))
for(i in 1:nrow(samples)){
  k <- unique(xiSamples[i, ])
  kNew <- max(k) + 1
  mk <- c()
  li <- 1
  for(l in 1:length(k)) {
    mk[li] <- sum(xiSamples[i, ] == k[li])
    li <- li + 1
  }
  alpha <- alphaSamples[i]

  muK <-  muTildeSamples[i, k]
  s2K <-  s2TildeSamples[i, k]
  muKnew <-  muTildeSamples[i, kNew]
  s2Knew <-  s2TildeSamples[i, kNew]

  densitySamplesStandl[i, ] <- sapply(standlGrid,
                function(x)(sum(mk * dnorm(x, muK, sqrt(s2K))) +
                alpha * dnorm(x, muKnew, sqrt(s2Knew)) )/(alpha+consts$n))
}

hist(data$y, freq = FALSE, xlim = c(-2.5, 2.5), ylim = c(0,0.75), main = "",
     xlab = "Waiting times on standardized log scale")
## pointwise estimate of the density for standardized log grid
lines(standlGrid, apply(densitySamplesStandl, 2, mean), lwd = 2, col = 'black')
lines(standlGrid, apply(densitySamplesStandl, 2, quantile, 0.025), lty = 2, col = 'black')
lines(standlGrid, apply(densitySamplesStandl, 2, quantile, 0.975), lty = 2, col = 'black')

Recall, however, that this is the density estimate for the logarithm of the waiting time. To obtain the density on the original scale we need to apply the appropriate transformation to the kernel.

lgrid <- standlGrid*sd(lFaithful) + mean(lFaithful) # grid on log scale
densitySamplesl <- densitySamplesStandl / sd(lFaithful) # density samples for grid on log scale

hist(faithful$waiting, freq = FALSE, xlim = c(40, 100), ylim=c(0, 0.05),
     main = "", xlab = "Waiting times")
lines(exp(lgrid), apply(densitySamplesl, 2, mean)/exp(lgrid), lwd = 2, col = 'black')
lines(exp(lgrid), apply(densitySamplesl, 2, quantile, 0.025)/exp(lgrid), lty = 2,
      col = 'black')
lines(exp(lgrid), apply(densitySamplesl, 2, quantile, 0.975)/exp(lgrid), lty = 2,
      col = 'black')

In either case, there is clear evidence that the data has two components for the waiting times.

Generating samples from the mixing distribution

While samples from the posterior distribution of linear functionals of the mixing distribution G (such as the predictive distribution above) can be computed directly from the realizations of the collapsed sampler, inference for non-linear functionals of G requires that we first generate samples from the mixing distribution. In NIMBLE we can get posterior samples from the random measure G, using the getSamplesDPmeasure function. Note that, in order to get posterior samples from G, we need to monitor all the random variables involved in its computations, i.e., the membership variable, xi, the cluster parameters, muTilde and s2Tilde, and the concentration parameter, alpha.

The following code generates posterior samples from the random measure G. The cMCMC object includes the model and posterior samples from the parameters. The getSamplesDPmeasure function estimates a truncation level of G, namely truncG. The posterior samples are in a matrix with (\mbox{truncG} \cdot (p+1)) columns, where p is the dimension of the vector of parameters with distribution G (in this example p=2).

outputG <- getSamplesDPmeasure(cmcmc)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
## sampleDPmeasure: Approximating the random measure by a finite stick-breaking representation with and error smaller than 1e-10, leads to a truncation level of 33.
if(packageVersion('nimble') <= '0.6-12')
  samplesG <- outputG else samplesG <- outputG$samples

The following code computes posterior samples of P(\tilde{y} > 70) using the posterior samples from the random measure G. Note that these samples are computed based on the transformed model and a value larger than 70 corresponds to a value larger than 0.03557236 on the above defined grid.

if(packageVersion('nimble') >= '0.6.13')
     truncG <- outputG$trunc # truncation level for G

weightIndex <- grep('weight', colnames(samplesG))
muTildeIndex <- grep('muTilde', colnames(samplesG))
s2TildeIndex <- grep('s2Tilde', colnames(samplesG))

probY70 <- rep(0, nrow(samples))  # posterior samples of P(y.tilde > 70)
for(i in seq_len(nrow(samples))) {
  probY70[i] <- sum(samplesG[i, weightIndex] *
                    pnorm(0.03557236, mean = samplesG[i, muTildeIndex],
                      sd = sqrt(samplesG[i, s2TildeIndex]), lower.tail = FALSE))
}

hist(probY70,  xlab = "Probability", ylab = "P(yTilde > 70 | data)" , main = "" )

Fitting a mixture of gamma distributions using the CRP representation

NIMBLE is not restricted to using Gaussian kernels in DPM models. In the case of the Old Faithful data, an alternative to the mixture of Gaussian kernels on the logarithmic scale that we presented in the previous section is a (scale-and-shape) mixture of Gamma distributions on the original scale of the data.

Model specification

In this case, the model takes the form

  y_i \mid \{ \tilde{\beta}_k \}, \{ \tilde{\lambda}_k \} \sim \mbox{Gamma}\left( \tilde{\beta}_{\xi_i}, \tilde{\lambda}_{\xi_i} \right), \quad\quad \xi \mid \alpha \sim \mbox{CRP}(\alpha), \quad\quad (\tilde{\beta}_k, \tilde{\lambda}_k) \mid H \sim H ,

where H corresponds to the product of two independent Gamma distributions. The following code provides the NIMBLE specification for the model:

code <- nimbleCode({
  for(i in 1:n) {
    y[i] ~ dgamma(shape = beta[i], scale = lambda[i])
    beta[i] <- betaTilde[xi[i]]
    lambda[i] <- lambdaTilde[xi[i]]
  }
  xi[1:n] ~ dCRP(alpha, size = n)
  for(i in 1:50) { # only 50 cluster parameters
    betaTilde[i] ~ dgamma(shape = 71, scale = 2)
    lambdaTilde[i] ~ dgamma(shape = 2, scale = 2)
  }
  alpha ~ dgamma(1, 1)
})

Note that in this case the vectors betaTilde and lambdaTilde have length 50 \ll n = 272. This is done to reduce the computational and storage burdens associated with the sampling algorithm. You could think about this approach as truncating the process, except that it can be thought of as an *exact* truncation. Indeed, under the CRP representation, using parameter vector(s) with a length that is shorter than the number of observations in the sample will lead to a proper algorithm as long as the number of components instatiated by the sampler is strictly lower than the length of the parameter vector(s) for every iteration of the sampler.

Running the MCMC algorithm

The following code sets up the model data and constants, initializes the parameters, defines the model object, and builds and runs the MCMC algorithm for the mixture of Gamma distributions. Note that, when building the MCMC, a warning message about the number of cluster parameters is generated. This is because the lengths of betaTilde and lambdaTilde are smaller than n. Also, note that no error message is generated during execution, which indicates that the number of clusters required never exceeded the maximum of 50.

data <- list(y = faithful$waiting)
set.seed(1)
inits <- list(xi = sample(1:10, size=consts$n, replace=TRUE),
              betaTilde = rgamma(50, shape = 71, scale = 2),
              lambdaTilde = rgamma(50, shape = 2, scale = 2),
              alpha = 1)
rModel <- nimbleModel(code, data = data, inits = inits, constants = consts)
## defining model...
## building model...
## setting data and initial values...
## running calculate on model (any error reports that follow may simply reflect missing values in model variables) ... 
## checking model sizes and dimensions...
## model building finished.
cModel <- compileNimble(rModel)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
conf <- configureMCMC(rModel, monitors = c("xi", "betaTilde", "lambdaTilde", "alpha"))
mcmc <- buildMCMC(conf)
## Warning in samplerFunction(model = model, mvSaved = mvSaved, target = target, : sampler_CRP: The number of cluster parameters is less than the number of potential clusters. The MCMC is not strictly valid if ever it proposes more components than cluster parameters exist; NIMBLE will warn you if this occurs.
cmcmc <- compileNimble(mcmc, project = rModel)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
samples <- runMCMC(cmcmc, niter = 7000, nburnin = 2000, setSeed = TRUE)
## running chain 1...
## |-------------|-------------|-------------|-------------|
## |-------------------------------------------------------|

In this case we use the posterior samples of the parameters to construct a trace plot and estimate the posterior distribution of \alpha:

# Trace plot of the posterior samples of the concentration parameter
ts.plot(samples[ , 'alpha'], xlab = "iteration", ylab = expression(alpha))
# Histogram of the posterior samples for the concentration parameter 
hist(samples[ , 'alpha'], xlab = expression(alpha), ylab = "Frequency", main = "")

Generating samples from the mixing distribution

As before, we obtain samples from the posterior distribution of G using the getSamplesDPmeasure function.

outputG <- getSamplesDPmeasure(cmcmc)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
## sampleDPmeasure: Approximating the random measure by a finite stick-breaking representation with and error smaller than 1e-10, leads to a truncation level of 28.
if(packageVersion('nimble') <= '0.6-12')
  samplesG <- outputG else samplesG <- outputG$samples

We use these samples to create an estimate of the density of the data along with a pointwise 95% credible band:

if(packageVersion('nimble') >= '0.6.13')
     truncG <- outputG$trunc # truncation level for G
grid <- seq(40, 100, len = 200)

weightSamples <- samplesG[ , grep('weight', colnames(samplesG))]
betaTildeSamples <- samplesG[ , grep('betaTilde', colnames(samplesG))]
lambdaTildeSamples <- samplesG[ , grep('lambdaTilde', colnames(samplesG))]

densitySamples <- matrix(0, ncol = length(grid), nrow = nrow(samples))
for(iter in seq_len(nrow(samples))) {
  densitySamples[iter, ] <- sapply(grid, function(x)
    sum( weightSamples[iter, ] * dgamma(x, shape = betaTildeSamples[iter, ],
                scale = lambdaTildeSamples[iter, ])))
}


hist(faithful$waiting, freq = FALSE, xlim = c(40,100), ylim = c(0, .05), main = "",
   ylab = "", xlab = "Waiting times")
lines(grid, apply(densitySamples, 2, mean), lwd = 2, col = 'black')
lines(grid, apply(densitySamples, 2, quantile, 0.025), lwd = 2, lty = 2, col = 'black')
lines(grid, apply(densitySamples, 2, quantile, 0.975), lwd = 2, lty = 2, col = 'black')

Again, we see that the density of the data is bimodal, and looks very similar to the one we obtained before.

Fitting a DP mixture of Gammas using a stick-breaking representation

Model specification

An alternative representation of the Dirichlet process mixture uses the stick-breaking representation of the random distribution G (Sethuraman, 1994). NIMBLE allows us to specify an approximation that involves a truncation of the Dirichlet process to a finite number of atoms, L. The resulting model therefore reduces to a finite mixture with L components and a very particular prior on the weights of the mixture components.

Introducing auxiliary variables, z_1, \ldots, z_n, that indicate which component generated each observation, the corresponding model for the mixture of Gamma densities discussed in the previous section takes the form

  y_i \mid \{ {\beta}_k^{\star} \}, \{ {\lambda}_k^{\star} \}, z_i \sim \mbox{Gamma}\left( {\beta}_{z_i}^{\star}, {\lambda}_{z_i}^{\star} \right), \quad\quad \boldsymbol{z} \mid \boldsymbol{w} \sim \mbox{Discrete}(\boldsymbol{w}), \quad\quad ({\beta}_k^{\star}, {\lambda}_k^{\star}) \mid H \sim H ,

where H is again the product of two independent Gamma distributions,

  w_1=v_1, \quad\quad w_l=v_l\prod_{m=1}^{l-1}(1-v_m), \quad l=2, \ldots, L-1,\quad\quad w_L=\prod_{m=1}^{L-1}(1-v_m)

with v_l \mid \alpha\sim \mbox{Beta}(1, \alpha), l=1, \ldots, L-1. The following code provides the NIMBLE specification for the model:

code <- nimbleCode(
  {
    for(i in 1:n) {
      y[i] ~ dgamma(shape = beta[i], scale = lambda[i])
      beta[i] <- betaStar[z[i]]
      lambda[i] <- lambdaStar[z[i]]
      z[i] ~ dcat(w[1:Trunc])
    }
    for(i in 1:(Trunc-1)) { # stick-breaking variables
      v[i] ~ dbeta(1, alpha)
    }
    w[1:Trunc] <- stick_breaking(v[1:(Trunc-1)]) # stick-breaking weights
    for(i in 1:Trunc) {
      betaStar[i] ~ dgamma(shape = 71, scale = 2)
      lambdaStar[i] ~ dgamma(shape = 2, scale = 2)
    }
    alpha ~ dgamma(1, 1)
  }
)

Note that the truncation level L of G has been set to a value Trunc, which is to be defined in the constants argument of the nimbleModel function.

Running the MCMC algorithm

The following code sets up the model data and constants, initializes the parameters, defines the model object, and builds and runs the MCMC algorithm for the mixture of Gamma distributions. When a stick-breaking representation is used, a blocked Gibbs sampler is assigned (Ishwaran, 2001; Ishwaran and James, 2002).

data <- list(y = faithful$waiting)
set.seed(1)
consts <- list(n = length(faithful$waiting), Trunc = 50)
inits <- list(betaStar = rgamma(consts$Trunc, shape = 71, scale = 2),
              lambdaStar = rgamma(consts$Trunc, shape = 2, scale = 2),
              v = rbeta(consts$Trunc-1, 1, 1),
              z = sample(1:10, size = consts$n, replace = TRUE),
              alpha = 1)

rModel <- nimbleModel(code, data = data, inits = inits, constants = consts)
## defining model...
## building model...
## setting data and initial values...
## running calculate on model (any error reports that follow may simply reflect missing values in model variables) ... 
## checking model sizes and dimensions...
## model building finished.
cModel <- compileNimble(rModel)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
conf <- configureMCMC(rModel, monitors = c("w", "betaStar", "lambdaStar", 'z', 'alpha'))
mcmc <- buildMCMC(conf)
cmcmc <- compileNimble(mcmc, project = rModel)
## compiling... this may take a minute. Use 'showCompilerOutput = TRUE' to see C++ compiler details.
## compilation finished.
samples <- runMCMC(cmcmc, niter = 24000, nburnin = 4000, setSeed = TRUE)
## running chain 1...
## |-------------|-------------|-------------|-------------|
## |-------------------------------------------------------|

Using the stick-breaking approximation automatically provides an approximation, G_L, of the random distribution G. The following code computes posterior samples of G_L using posterior samples from the samples object, and from them, a density estimate for the data.

betaStarSamples <- samples[ , grep('betaStar', colnames(samples))]
lambdaStarSamples <- samples[ , grep('lambdaStar', colnames(samples))]
weightSamples <- samples[ , grep('w', colnames(samples))]

grid <- seq(40, 100, len = 200)

densitySamples <- matrix(0, ncol = length(grid), nrow = nrow(samples))
for(i in 1:nrow(samples)) {
  densitySamples[i, ] <- sapply(grid, function(x)
    sum(weightSamples[i, ] * dgamma(x, shape = betaStarSamples[i, ],
                                    scale = lambdaStarSamples[i, ])))
}

hist(faithful$waiting, freq = FALSE,  xlab = "Waiting times", ylim=c(0,0.05),
     main = '')
lines(grid, apply(densitySamples, 2, mean), lwd = 2, col = 'black')
lines(grid, apply(densitySamples, 2, quantile, 0.025), lwd = 2, lty = 2, col = 'black')
lines(grid, apply(densitySamples, 2, quantile, 0.975), lwd = 2, lty = 2, col = 'black')

As expected, this estimate looks identical to the one we obtained through the CRP representation of the process.

More information and future development

Please see our User Manual for more details.

We’re in the midst of improvements to the existing BNP functionality as well as adding additional Bayesian nonparametric models, such as hierarchical Dirichlet processes and Pitman-Yor processes, so please add yourself to our announcement or user support/discussion Google groups.

References

Blackwell, D. and MacQueen, J. 1973. Ferguson distributions via Polya urn schemes. The Annals of Statistics 1:353-355.

Ferguson, T.S. 1974. Prior distribution on the spaces of probability measures. Annals of Statistics 2:615-629.

Lo, A.Y. 1984. On a class of Bayesian nonparametric estimates I: Density estimates. The Annals of Statistics 12:351-357.

Escobar, M.D. 1994. Estimating normal means with a Dirichlet process prior. Journal of the American Statistical Association 89:268-277.

Escobar, M.D. and West, M. 1995. Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association 90:577-588.

Ishwaran, H. and James, L.F. 2001. Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96: 161-173.

Ishwaran, H. and James, L.F. 2002. Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information. Journal of Computational and Graphical Statistics 11:508-532.

Neal, R. 2000. Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics 9:249-265.

Sethuraman, J. 1994. A constructive definition of Dirichlet prior. Statistica Sinica 2: 639-650.


Version 0.6-12 of NIMBLE released

We’ve released the newest version of NIMBLE on CRAN and on our website. Version 0.6-12 is primarily a maintenance release with various bug fixes.

Changes include:

  • a fix for the bootstrap particle filter to correctly calculate weights when particles are not resampled (the filter had been omitting the previous weights when calculating the new weights);
  • addition of an option to print MCMC samplers of a particular type;
  • avoiding an overly-aggressive check for ragged arrays when building models; and
  • avoiding assigning a sampler to non-conjugacy inverse-Wishart nodes (thereby matching our handling of Wishart nodes).

Please see the NEWS file in the installed package for more details.

Version 0.6-11 of NIMBLE released

We’ve released the newest version of NIMBLE on CRAN and on our website.

Version 0.6-11 has important new features, notably support for Bayesian nonparametric mixture modeling, and more are on the way in the next few months.

New features include:

  • support for Bayesian nonparametric mixture modeling using Dirichlet process mixtures, with specialized MCMC samplers automatically assigned in NIMBLE’s default MCMC (See Chapter 10 of the manual for details);
  • additional resampling methods available with the auxiliary and bootstrap particle filters;
  • user-defined filtering algorithms can be used with NIMBLE’s particle MCMC samplers;
  • MCMC thinning intervals can be modified at MCMC run-time;
  • both runMCMC() and nimbleMCMC() now drop burn-in samples before thinning, making their behavior consistent with each other;
  • increased functionality for the ‘setSeed’ argument in nimbleMCMC() and runMCMC();
  • new functionality for specifying the order in which sampler functions are executed in an MCMC; and
  • invalid dynamic indexes now result in a warning and NaN values but do not cause execution to error out, allowing MCMC sampling to continue.

Please see the NEWS file in the installed package for more details

Two day workshop: Flexible programming of MCMC and other methods for hierarchical and Bayesian models

We’ll be giving a two day workshop at the 43rd Annual Summer Institute of Applied Statistics at Brigham Young University (BYU) in Utah, June 19-20, 2018.

Abstract is below, and registration and logistics information can be found here.

This workshop provides a hands-on introduction to using, programming, and sharing Bayesian and hierarchical modeling algorithms using NIMBLE (r-nimble.org). In addition to learning the NIMBLE system, users will develop hands-on experience with various computational methods. NIMBLE is an R-based system that allows one to fit models specified using BUGS/JAGS syntax but with much more flexibility in defining the statistical model and the algorithm to be used on the model. Users operate from within R, but NIMBLE generates C++ code for models and algorithms for fast computation. I will open with an overview of creating a hierarchical model and fitting the model using a basic MCMC, similarly to how one can use WinBUGS, JAGS, and Stan. I will then discuss how NIMBLE allows the user to modify the MCMC – changing samplers and specifying blocking of parameters. Next I will show how to extend the BUGS syntax with user-defined distributions and functions that provide flexibility in specifying a statistical model of interest. With this background we can then explore the NIMBLE programming system, which allows one to write new algorithms not already provided by NIMBLE, including new MCMC samplers, using a subset of the R language. I will then provide examples of non-MCMC algorithms that have been programmed in NIMBLE and how algorithms can be combined together, using the example of a particle filter embedded within an MCMC. We will see new functionality in NIMBLE that allows one to fit Bayesian nonparametric models and spatial models. I will close with a discussion of how NIMBLE enables sharing of new methods and reproducibility of research. The workshop will include a number of breakout periods for participants to use and program MCMC and other methods, either on example problems or problems provided by participants. In addition, participants will see NIMBLE’s flexibility in action in several real problems.

Version 0.6-10 of NIMBLE released

We’ve released the newest version of NIMBLE on CRAN and on our website. Version 0.6-10 primarily contains updates to the NIMBLE internals that may speed up building and compilation of models and algorithms, as well as a few bug fixes.

Changes include:

  • some steps of model and algorithm building and compilation are faster;
  • compiled execution with multivariate distributions or function arguments may be faster;
  • data can now be provided as a numeric data frame rather than a matrix;
  • to run WAIC, a user now must set ‘enableWAIC’ to TRUE, either in NIMBLE’s options or as an argument to buildMCMC();
  • if ‘enableWAIC’ is TRUE, buildMCMC() will now check to make sure that the nodes monitored by the MCMC algorithm will lead to a valid WAIC calculation; and
  • the use of identityMatrix() is deprecated in favor of diag().

Please see the NEWS file in the installed package for more details

NIMBLE webinar Friday April 13

We’ll be presenting a webinar on NIMBLE, hosted by the Eastern North America Region of the International Biometric Society. Details are as follows.
Programming with hierarchical statistical models: An introduction to the BUGS-compatible NIMBLE system for MCMC and more
Friday, April 13, 2018
11:00 a.m. – 1:00 p.m. EST
Must register before April 12. You can register here. (You’ll need to create an account on the ENAR website and there is a modest fee – from $25 for ENAR student members up through $85 for non-IBS members.)
This webinar will introduce attendees to the NIMBLE system for programming with hierarchical models in R. NIMBLE (r-nimble.org) is a system for flexible programming and dissemination of algorithms that builds on the BUGS language for declaring hierarchical models. NIMBLE provides analysts with a flexible system for using MCMC, sequential Monte Carlo and other techniques on user-specified models. It provides developers and methodologists with the ability to write algorithms in an R-like syntax that can be easily disseminated to users. C++ versions of models and algorithms are created for speed, but these are manipulated from R without any need for analysts or algorithm developers to program in C++. 

While analysts can use NIMBLE as a drop-in replacement for WinBUGS or JAGS, NIMBLE provides greatly enhanced functionality in a number of ways. The webinar will first show how to specify a hierarchical statistical model using BUGS syntax (including user-defined function and distributions) and fit that model using MCMC (including user customization for better performance). We will demonstrate the use of NIMBLE for biostatistical methods such as semiparametric random effects models and clustering models. We will close with a discussion of how to use the system to write algorithms for use with hierarchical models, including building and disseminating your own methods.

Presenter:
Chris Paciorek
Adjunct Professor, Statistical Computing Consultant
Department of Statistics, University of California, Berkeley

Version 0.6-9 of NIMBLE released

We’ve released the newest version of NIMBLE on CRAN and on our website. Version 0.6-9 is primarily a maintenance release with various bug fixes and fixes for CRAN packaging issues.

New features include:

  • dimensions in a model will now be determined from either ‘inits’ or ‘data’ if not otherwise available;
  • one can now specify “nBootReps = NA” in the runCrossValidate() function, which will prevent the Monte Carlo error from being calculated;
  • runCrossValidate() now returns the averaged loss over all k folds, instead of the summed loss;
  • We’ve added the besselK function to the NIMBLE language;
  • and a variety of bug fixes.

Please see the NEWS file in the installed package for more details

NIMBLE has a post-doc or software developer position open

The NIMBLE statistical software project at the University of California, Berkeley is looking for a post-doc or statistical software developer. NIMBLE is a tool for writing hierarchical statistical models and algorithms from R, with compilation via code-generated C++. Major methods currently include MCMC and sequential Monte Carlo, which users can customize and extend. More information can be found at https://R-nimble.org. Currently we seek someone with experience in computational statistical methods such as MCMC and excellent software development skills in R and C++. This could be someone with a Ph.D. in Statistics, Computer Science, or an applied statistical field in which they have done relevant work. Alternatively it could be someone with relevant experience in computational statistics and software engineering. The scope of work can include both core development of NIMBLE and development and application of innovative methods using NIMBLE, with specific focus depending on the background of the successful candidate. Applicants must have either a Ph.D. in a relevant field or have a proven record of relevant work. Please send cover letter, CV, and the names and contact information for three references to nimble.stats@gmail.com. Applications will be considered on a rolling basis starting 30 January, 2018.

Version 0.6-8 of NIMBLE released

We’ve released the newest version of NIMBLE on CRAN and on our website a week ago. Version 0.6-8 has a few new features, and more are on the way in the next few months.

New features include:

  • the proper Gaussian CAR (conditional autoregressive) model can now be used in BUGS code as dcar_proper, which behaves similarly to BUGS’ car.proper distribution;
  • a new nimbleMCMC function that provides one-line invocation of NIMBLE’s MCMC engine, akin to usage of JAGS and WinBUGS through R;
  • a new runCrossValidate function that will conduct k-fold cross-validation of NIMBLE models fit by MCMC;
  • dynamic indexing in BUGS code is now allowed by default;
  • and a variety of bug fixes and efficiency improvements.

Please see the NEWS file in the installed package for more details.