the marginal likelihood of analytical result is different from that of bridge_sampler

Question

I try to calculate the marginal likelihood of the example in the article " a tutorial on bridge sampling", which is estimating the marginal likelihood for a binomial model assuming a uniform prior on the rate parameter θ (i.e., the beta-binomial model). Consider a single participant who answered k = 2 out of n = 10 true/false questions correctly. The analytical result is 1/(n+1), that is 1/11.

And Then I get a stanfit with rstan, and use bridge_sampler(stanfit) to get the log marginal likelihood. After transforming the log marginal likelihood to the marginal likelihood, I find it is totally different from 1/11. I don't know the reason for this. besides, there is a warning message: “effective sample size cannot be calculated, has been replaced by number of samples.”

stanmodelcode <- "
data {
    int<lower=0> N; 
    int y;
}

parameters {
    real<lower=0,upper=1> theta; 
}

model {
    theta ~ beta(1,1);
    y ~ binomial(N,theta) ;
}"

y <- 2
dat <- list(N = 15, y = y);

stanfit <- stan(model_code = stanmodelcode,data=dat,chains=1 , iter=50000 , warmup=500)

library(bridgesampling)
(Lik_marginal <- bridge_sampler(stanfit))
exp(logml(Lik_marginal))

Answer 1

The model specifications should all be in the form "target += X" instead of "y ~ X" because the latter drops constants.

So you should have "target += binomial_lpmf(y | N, theta);" instead of "y~binomial(N,theta);". And the same with the theta distribution.

see: https://rdrr.io/cran/bridgesampling/f/vignettes/bridgesampling_stan_ttest.Rmd