Purpose: to compare drjacoby results for a challenging problem involving a multimodal posterior, both with and without temperature rungs.
We assume a single parameter mu
drawn from a double well
potential distribution, defined by the formula:
$$ \begin{aligned} \mu &\propto exp\left(-\gamma(\mu^2 - 1)^2\right) \end{aligned} $$ where γ is a parameter that defines the strength of the well (higher γ leads to a deeper valley and hence more challenging problem). NB, there is no data in this example, as the likelihood is defined exactly by these parameters.
Likelihood and prior:
Parameters dataframe:
mcmc <- run_mcmc(data = list(x = -1),
df_params = df_params,
loglike = "loglike",
logprior = "logprior",
burnin = 1e3,
samples = 1e5,
chains = 1,
rungs = 1,
silent = TRUE)
# extract posterior draws
output_sub <- subset(mcmc$output, phase == "sampling")
mu_draws <- output_sub$mu
# get analytical solution
x <- seq(-L, L, l = 1001)
fx <- exp(-gamma*(x^2 - 1)^2)
fx <- fx / sum(fx) * 1/(x[2]-x[1])
# overlay plots
hist(mu_draws, breaks = seq(-L, L, l = 201), probability = TRUE, main = "", col = "black")
lines(x, fx, col = 2, lwd = 2)
mcmc <- run_mcmc(data = list(x = -1),
df_params = df_params,
loglike = "loglike",
logprior = "logprior",
burnin = 1e3,
samples = 1e5,
chains = 1,
rungs = 11,
alpha = 2,
pb_markdown = TRUE)
## MCMC chain 1
## burn-in
## | |======================================================================| 100%
## acceptance rate: 21.7%
## sampling phase
## | |======================================================================| 100%
## acceptance rate: 22%
## chain completed in 2.994781 seconds
## total MCMC run-time: 2.99 seconds