R Environments for Gibbs Sampler State

I recently decided to revisit some R code that implements a Gibbs sampler in an attempt to decrease the iteration time. My strategy was to implement the sampler state as an R environment rather than a list. The rationale was that passing an environment to and from functions would reduce the amount of duplication (memory copying). In my experiments, this results in only marginal improvement in iteration time. The following is just a toy example to illustrate.

Consider the logistic regression model

 \begin{array}{r c l}<br />
y_i & \sim & B(\pi_i) \\<br />
\mathrm{logit}(\pi_i) & = & \beta x_i \\<br />
\beta & \sim & N(0, 10^{-3}),<br />
\end{array}


where y_i is a binary outcome, independently Bernoulli distributed with probability \pi_i, x_i is a scalar covariate, and the log odds \beta is a priori normal distributed with mean 0 and precision 10^{-3}.

For Gibbs sampling, it's convenient to store and manupulate a model state. In the past, I had represented this in R with a list, for example

> logit <- function(e) log(e) - log(1-e)
> expit <- function(l) exp(l) / (1+exp(l))
> # Simulate some data
> n <- 1e3
> x <- runif(n)
> y <- rbinom(n, 1, expit(5*x))
> iter <- 0
> itermax <- 1000
> state <- list(x=x, y=y, mu=0, tau=1e-3, beta=rep(0,itermax), iter=1)

Then, to perform a Gibbs update on beta, I would pass the old state to a function that would return the updated state. For example:

> Gibbs <- function(state) {
+     # Metropolis update
+     state$iter <- state$iter + 1
+
+     old_beta  <- state$beta[state$iter-1]
+     old_probs <- expit(old_beta*state$x)
+     old_logp  <- sum(log(ifelse(state$y,old_probs,1-old_probs))) +
+                  dnorm(old_beta, state$mu, 1/sqrt(state$tau), log=TRUE)
+
+     new_beta  <- rnorm(1, old_beta) # Proposal
+     new_probs <- expit(new_beta*state$x)
+     new_logp  <- sum(log(ifelse(state$y,new_probs,1-new_probs))) +
+                  dnorm(new_beta, state$mu, 1/sqrt(state$tau), log=TRUE)
+
+     if(new_logp - old_logp > log(runif(1)))
+         state$beta[state$iter] <- new_beta
+     else
+         state$beta[state$iter] <- old_beta
+
+     return(state)
+ }

Hence, the Gibbs sampler would be used as follows

> system.time({
+ while(state$iter < itermax)
+     state <- Gibbs(state)
+ })
   user  system elapsed
  2.708   0.012   2.717
> summary(state$beta)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
  0.000   4.414   4.564   4.515   4.792   5.417

Alternatively, the Gibbs sampler state may be stored and manipulated asan R environment. For example, to initialize the state:

> estate <- new.env()
> assign("x", x, envir=estate)
> assign("y", y, envir=estate)
> assign("mu", 0, envir=estate)
> assign("tau", 1e-3, envir=estate)
> assign("beta", rep(0,itermax), envir=estate)
> assign("iter", 1, envir=estate)

Then rewrite the Gibbs sampler function:

> GibbsExp <- expression({
+     # Metropolis update
+     iter <- iter + 1
+
+     old_beta  <- beta[iter-1]
+     old_probs <- expit(old_beta*x)
+     old_logp  <- sum(log(ifelse(y,old_probs,1-old_probs))) +
+                  dnorm(old_beta, mu, 1/sqrt(tau), log=TRUE)
+
+     new_beta  <- rnorm(1, old_beta) # Proposal
+     new_probs <- expit(new_beta*x)
+     new_logp  <- sum(log(ifelse(y,new_probs,1-new_probs))) +
+                  dnorm(new_beta, mu, 1/sqrt(tau), log=TRUE)
+
+     if(new_logp - old_logp > log(runif(1)))
+         beta[iter] <- new_beta
+     else
+         beta[iter] <- old_beta
+ })
>
> eGibbs <- function(estate) {
+     eval(GibbsExp, estate)
+ }

The Gibbs sampler is used in an almost identical manner:

> system.time({
+ while(get("iter",envir=estate) < itermax)
+     eGibbs(estate)
+ })
   user  system elapsed
  2.664   0.000   2.673
> summary(get("beta",envir=estate))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
  0.000   4.378   4.588   4.556   4.779   5.704

The percent decrease in iteration time is fairly constant for a variety of n and maxiter values. Presumably, this modest gain in speed results from less memory duplication. However, in some experiments with tracemem the story appears to be more complicated... For now, it doesn't appear that utilizing environments is worth the trouble when the sampling algorithm is much slower than memory allocation.

8 thoughts on “R Environments for Gibbs Sampler State

  1. Michael Bedward

    Thanks for this Matt. How about using the proto package rather than environments directly ? It probably won't make any difference to the running time but the semantic sugar is nice

  2. Michael Bedward

    In case it's of interest, here is your example reworked to use prototypes (proto package) rather than simple R environments.

    library(proto)
    Gibbs <- proto()
    
    Gibbs$new <- function(., x, y) {
      proto(., x=x, y=y)
    }
    
    Gibbs$run <- function(., niter, adapt=0, mu=0, tau=0.001) {
      .$beta <- numeric(niter)
      .$mu <- mu
      .$tau <- tau
      beta.old <- 0
    
      for (i in 1:adapt) {
            beta.old <- .$update(beta.old)
      }
    
      for (i in 1:niter) {
            beta.old <- .$update(beta.old)
            .$beta[i] <- beta.old
      }
    }
    
    Gibbs$update <- function(., beta.old) {
      probs.old <- plogis(beta.old * .$x)
      logp.old <- sum( log( ifelse(.$y , probs.old, 1-probs.old) ) ) +
                  dnorm(beta.old, .$mu, 1/sqrt(.$tau), log=TRUE)
    
      beta.new <- rnorm(1, beta.old) # Proposal
      probs <- plogis(beta.new * .$x)
      logp <- sum( log( ifelse( .$y, probs, 1 - probs ) ) ) +
              dnorm(beta.new, .$mu, 1/sqrt(.$tau), log=TRUE)
    
      ifelse (logp - logp.old > log(runif(1)), beta.new, beta.old)
    }
    
    #Using the sampler
    x <- rnorm(1000)
    y <- rbinom(1000, 1, plogis(5*x))
    
    gibbs <- Gibbs$new(x, y)
    gibbs$run(5000)
    
    summary(gibbs$beta)
       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
     0.5253  5.1030  5.3480  5.3570  5.6130  6.7160
    
  3. Gregor Gorjanc

    Which seems to be slower than the previous two solutions - I run it several times and timings were quite consistent.

    > system.time(gibbs$run(1000))
    user system elapsed
    3.19 0.00 3.19

  4. Michael Bedward

    Yes, it quite probably is slower. I wasn't suggesting proto for speed but for neat encapsulation. However, after posting the code I realized that there is an obvious improvement to be made that might reduce run time: rather than calculate probs.old and logp.old on each update, simply store the previous logp value. For example...

    
    Gibbs$update <- function(., beta.old) {
      beta.new <- rnorm(1, beta.old) # Proposal
      probs <- plogis(beta.new * .$x)
      logp  log(runif(1)) {
        .$logp.old <- logp
         return( beta.new )
      } else {
         return( beta.old )
      }
    }
    
  5. Gregor Gorjanc

    Michael, previously I did not looked at the maths of your implementation, but the results do not match with Matt. In addition, your R code in the last post is incomplete.

  6. Michael Bedward

    Sorry Gregor, I forgot to escape characters for html. Here is the code again...

    
    Gibbs$update <gt;- function(., beta.old) {
      beta.new <gt;- rnorm(1, beta.old) # Proposal
      probs <gt;- plogis(beta.new * .$x)
      logp  log(runif(1))) {
        .$logp.old >- logp
        return( beta.new )
      } else {
        return( beta.old )
      }
    }
    

    When I've run this with the same example that Matt uses I get a good recovery of the coefficient value (mean posterior) but I haven't checked to see the posterior density is different in other respects. I'll do that.

  7. Michael Bedward

    Hopefully this one works...

    Gibbs$update <- function(., beta.old) {
      beta.new <- rnorm(1, beta.old) # Proposal
      probs <- plogis(beta.new * .$x)
      logp <- sum( log( ifelse( .$y, probs, 1 - probs ) ) ) +
              dnorm(beta.new, .$mu, 1/sqrt(.$tau), log=TRUE)
    
      if (logp - .$logp.old > log(runif(1))) {
        .$logp.old <- logp
        return( beta.new )
      } else {
        return( beta.old )
      }
    }
    
    

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