Bayesian Simulation: Misspecified Likelihood

Project Overview

This project explores the effects of misspecified likelihood functions in Bayesian statistical analysis. Through simulation studies, I compare how different specifications of the likelihood function affect posterior distributions and survival curve estimates.

The simulation includes implementations of several posterior sampling techniques, including Gibbs sampling and the Metropolis-Hastings algorithm, to demonstrate their performance under various conditions.

Methodology

For this simulation study, I used the following approach:

1. Generated synthetic data from known distributions
2. Applied different likelihood specifications (correctly specified and misspecified)
3. Implemented both Gibbs sampling and Metropolis-Hastings algorithms
4. Compared posterior distributions across different specifications
5. Evaluated impacts on survival curve estimation

The primary metrics for comparison included posterior mean estimates, credible intervals, and mean squared error between the estimated and true survival curves.

R Code Implementation

Below is a sample of the R code used to implement the Metropolis-Hastings algorithm for one of the model specifications:

# Metropolis-Hastings implementation for exponential likelihood
metropolis_hastings <- function(data, n_iter, proposal_sd, prior_mean, prior_sd) {
  # Initialize
  theta_current <- mean(data)  # Starting value
  theta_samples <- numeric(n_iter)
  
  # Metropolis-Hastings algorithm
  for (i in 1:n_iter) {
    # Propose new value
    theta_proposal <- rnorm(1, mean = theta_current, sd = proposal_sd)
    
    # Skip negative proposals for rate parameter
    if (theta_proposal <= 0) {
      theta_samples[i] <- theta_current
      next
    }
    
    # Calculate log posterior ratios
    log_post_current <- sum(dexp(data, rate = theta_current, log = TRUE)) + 
                         dnorm(theta_current, mean = prior_mean, sd = prior_sd, log = TRUE)
    log_post_proposal <- sum(dexp(data, rate = theta_proposal, log = TRUE)) + 
                         dnorm(theta_proposal, mean = prior_mean, sd = prior_sd, log = TRUE)
    
    # Calculate acceptance probability
    log_accept_prob <- log_post_proposal - log_post_current
    
    # Accept or reject
    if (log(runif(1)) < log_accept_prob) {
      theta_current <- theta_proposal  # Accept
    }
    
    # Store current value
    theta_samples[i] <- theta_current
  }
  
  return(theta_samples)
}

The code for survival curve estimation:

# Calculate survival curves based on posterior samples
calculate_survival_curves <- function(theta_samples, time_points) {
  n_samples <- length(theta_samples)
  n_times <- length(time_points)
  survival_curves <- matrix(0, nrow = n_samples, ncol = n_times)
  
  # Calculate survival probability for each time point and each posterior sample
  for (i in 1:n_samples) {
    theta <- theta_samples[i]
    survival_curves[i, ] <- exp(-theta * time_points)
  }
  
  # Calculate mean and credible intervals
  mean_curve <- colMeans(survival_curves)
  lower_ci <- apply(survival_curves, 2, quantile, probs = 0.025)
  upper_ci <- apply(survival_curves, 2, quantile, probs = 0.975)
  
  return(list(
    time = time_points,
    mean = mean_curve,
    lower = lower_ci,
    upper = upper_ci
  ))
}

Results

The simulation study revealed several key findings:

• Misspecified likelihood functions led to biased posterior estimates
• The magnitude of bias increased with greater departure from the true data-generating process
• Survival curve estimates were more sensitive to likelihood misspecification than parameter estimates
• The Metropolis-Hastings algorithm demonstrated better mixing properties for complex likelihood specifications

Below is a visualization of posterior distributions under different likelihood specifications:

Comparison of estimated survival curves:

Conclusions

This simulation study demonstrates the importance of correctly specifying likelihood functions in Bayesian analysis. Key conclusions include:

• Misspecified likelihoods can lead to systematically biased inference
• The choice of sampling algorithm can impact the quality of posterior approximation
• When the true data-generating process is unknown, model checking and sensitivity analysis are essential
• Even with misspecified likelihoods, Bayesian methods can provide useful inference if the degree of misspecification is moderate

Future work could explore more robust approaches to handling model misspecification in Bayesian analysis, including the use of non-parametric priors or mixture models to accommodate a wider range of data-generating processes.