In this vignette, we explain how one can compute marginal likelihoods, Bayes factors, and posterior model probabilities using a simple hierarchical normal model implemented in Stan
. This vignette uses the same models and data as the Jags
vignette.
The model that we will use assumes that each of the \(n\) observations \(y_i\) (where \(i\) indexes the observation, \(i = 1,2,...,n\)) is normally distributed with corresponding mean \(\theta_i\) and a common known variance \(\sigma^2\): \(y_i \sim \mathcal{N}(\theta_i, \sigma^2)\). Each \(\theta_i\) is drawn from a normal group-level distribution with mean \(\mu\) and variance \(\tau^2\): \(\theta_i \sim \mathcal{N}(\mu, \tau^2)\). For the group-level mean \(\mu\), we use a normal prior distribution of the form \(\mathcal{N}(\mu_0, \tau^2_0)\). For the group-level variance \(\tau^2\), we use an inverse-gamma prior of the form \(\text{Inv-Gamma}(\alpha, \beta)\).
In this example, we are interested in comparing the null model \(\mathcal{H}_0\), which posits that the group-level mean \(\mu = 0\), to the alternative model \(\mathcal{H}_1\), which allows \(\mu\) to be different from zero. First, we generate some data from the null model:
library(bridgesampling)
### generate data ###
set.seed(12345)
<- 0
mu <- 0.5
tau2 <- 1
sigma2
<- 20
n <- rnorm(n, mu, sqrt(tau2))
theta <- rnorm(n, theta, sqrt(sigma2)) y
Next, we specify the prior parameters \(\mu_0\), \(\tau^2_0\), \(\alpha\), and \(\beta\):
### set prior parameters ###
<- 0
mu0 <- 1
tau20 <- 1
alpha <- 1 beta
Next, we implement the models in Stan
. Note that to compute the (log) marginal likelihood for a Stan
model, we need to specify the model in a certain way. Instad of using "~"
signs for specifying distributions, we need to directly use the (log) density functions. The reason for this is that when using the "~"
sign, constant terms are dropped which are not needed for sampling from the posterior. However, for computing the marginal likelihood, these constants need to be retained. For instance, instead of writing y ~ normal(mu, sigma)
we would need to write target += normal_lpdf(y | mu, sigma)
. The models can then be specified and compiled as follows (note that it is necessary to install rstan
for this):
library(rstan)
# models
<- 'data {
stancodeH0 int<lower=1> n; // number of observations
vector[n] y; // observations
real<lower=0> alpha;
real<lower=0> beta;
real<lower=0> sigma2;
}
parameters {
real<lower=0> tau2; // group-level variance
vector[n] theta; // participant effects
}
model {
target += inv_gamma_lpdf(tau2 | alpha, beta);
target += normal_lpdf(theta | 0, sqrt(tau2));
target += normal_lpdf(y | theta, sqrt(sigma2));
}
'
<- 'data {
stancodeH1 int<lower=1> n; // number of observations
vector[n] y; // observations
real mu0;
real<lower=0> tau20;
real<lower=0> alpha;
real<lower=0> beta;
real<lower=0> sigma2;
}
parameters {
real mu;
real<lower=0> tau2; // group-level variance
vector[n] theta; // participant effects
}
model {
target += normal_lpdf(mu | mu0, sqrt(tau20));
target += inv_gamma_lpdf(tau2 | alpha, beta);
target += normal_lpdf(theta | mu, sqrt(tau2));
target += normal_lpdf(y | theta, sqrt(sigma2));
}
'
# compile models
<- stan_model(model_code = stancodeH0, model_name="stanmodel")
stanmodelH0 <- stan_model(model_code = stancodeH1, model_name="stanmodel") stanmodelH1
Now we can fit the null and the alternative model in Stan
. One usually requires a larger number of posterior samples for estimating the marginal likelihood than for simply estimating the model parameters. This is the reason for using a comparatively large number of samples for these simple models.
# fit models
<- sampling(stanmodelH0, data = list(y = y, n = n,
stanfitH0 alpha = alpha,
beta = beta,
sigma2 = sigma2),
iter = 50000, warmup = 1000, chains = 3, cores = 1)
<- sampling(stanmodelH1, data = list(y = y, n = n,
stanfitH1 mu0 = mu0,
tau20 = tau20,
alpha = alpha,
beta = beta,
sigma2 = sigma2),
iter = 50000, warmup = 1000, chains = 3, cores = 1)
Computing the (log) marginal likelihoods via the bridge_sampler
function is now easy: we only need to pass the stanfit
objects which contain all information necessary. We use silent = TRUE
to suppress printing the number of iterations to the console:
# compute log marginal likelihood via bridge sampling for H0
<- bridge_sampler(stanfitH0, silent = TRUE)
H0.bridge
# compute log marginal likelihood via bridge sampling for H1
<- bridge_sampler(stanfitH1, silent = TRUE) H1.bridge
We obtain:
print(H0.bridge)
## Bridge sampling estimate of the log marginal likelihood: -37.53183
## Estimate obtained in 5 iteration(s) via method "normal".
print(H1.bridge)
## Bridge sampling estimate of the log marginal likelihood: -37.79683
## Estimate obtained in 4 iteration(s) via method "normal".
We can use the error_measures
function to compute an approximate percentage error of the estimates:
# compute percentage errors
<- error_measures(H0.bridge)$percentage
H0.error <- error_measures(H1.bridge)$percentage H1.error
We obtain:
print(H0.error)
## [1] "0.143%"
print(H1.error)
## [1] "0.164%"
To compare the null model and the alternative model, we can compute the Bayes factor by using the bf
function. In our case, we compute \(\text{BF}_{01}\), that is, the Bayes factor which quantifies how much more likely the data are under the null versus the alternative model:
# compute Bayes factor
<- bf(H0.bridge, H1.bridge)
BF01 print(BF01)
## Estimated Bayes factor in favor of H0.bridge over H1.bridge: 1.30343
In this case, the Bayes factor is close to one, indicating that there is not much evidence for either model. We can also compute posterior model probabilities by using the post_prob
function:
# compute posterior model probabilities (assuming equal prior model probabilities)
<- post_prob(H0.bridge, H1.bridge)
post1 print(post1)
## H0.bridge H1.bridge
## 0.5658657 0.4341343
When the argument prior_prob
is not specified, as is the case here, the prior model probabilities of all models under consideration are set equal (i.e., in this case with two models to 0.5). However, if we had prior knowledge about how likely both models are, we could use the prior_prob
argument to specify different prior model probabilities:
# compute posterior model probabilities (using user-specified prior model probabilities)
<- post_prob(H0.bridge, H1.bridge, prior_prob = c(.6, .4))
post2 print(post2)
## H0.bridge H1.bridge
## 0.6616079 0.3383921