Classical vs. Bayesian Regression

Project Overview

This project investigates the similarities and differences between classical (frequentist) and Bayesian approaches to regression analysis through simulation studies. While both approaches aim to estimate relationships between variables, they differ fundamentally in their philosophical foundations and practical implementations.

The study explores how these differences manifest in parameter estimation, uncertainty quantification, and prediction under various data conditions. Through controlled simulations, I evaluated the performance of both approaches across different sample sizes, noise levels, and model misspecifications.

Methodology

The methodology for this project included:

Generation of synthetic datasets with known parameters under different conditions
Implementation of classical regression using ordinary least squares (OLS)
Implementation of Bayesian regression with various prior specifications
Comparison of point estimates, interval estimates, and predictive performance
Analysis of robustness to violations of assumptions
Assessment of computational efficiency and practical considerations

For the Bayesian implementation, I explored both conjugate priors (which allow for analytical solutions) and more flexible prior specifications that required Markov Chain Monte Carlo (MCMC) sampling. This allowed for a comprehensive comparison of different approaches within the Bayesian framework itself.

R Code Implementation

The following code demonstrates the implementation of classical and Bayesian regression models:

# Load necessary libraries
library(rstanarm)  # For Bayesian regression
library(bayesplot)
library(ggplot2)
library(dplyr)

# Function to generate synthetic data
generate_data <- function(n_samples, beta_true, sigma, x_range = c(-3, 3)) {
  # Generate predictor variables
  X <- matrix(runif(n_samples * length(beta_true), 
                   min = x_range[1], max = x_range[2]), 
             nrow = n_samples)
  
  # Generate response variable with noise
  y <- X %*% beta_true + rnorm(n_samples, mean = 0, sd = sigma)
  
  # Return as data frame
  data.frame(y = y, X)
}

# Simulation parameters
set.seed(123)
n_samples <- 100
beta_true <- c(2.5, 1.5, -0.5)  # Intercept, X1, X2
sigma_true <- 2
variable_names <- c("Intercept", "X1", "X2")

# Generate data
sim_data <- generate_data(n_samples, beta_true, sigma_true)
colnames(sim_data) <- c("y", "X1", "X2")

# Fit classical regression model
classical_model <- lm(y ~ X1 + X2, data = sim_data)
classical_summary <- summary(classical_model)

# Extract classical estimates and confidence intervals
classical_coef <- coef(classical_model)
classical_se <- classical_summary$coefficients[, "Std. Error"]
classical_ci <- confint(classical_model, level = 0.95)
classical_sigma <- classical_summary$sigma

# Fit Bayesian regression model with weakly informative priors
bayesian_model <- stan_glm(y ~ X1 + X2, 
                          data = sim_data,
                          family = gaussian(),
                          prior = normal(0, 10),
                          prior_intercept = normal(0, 10),
                          prior_aux = cauchy(0, 5),
                          chains = 4,
                          iter = 2000,
                          seed = 123)

# Extract Bayesian estimates and credible intervals
bayesian_summary <- summary(bayesian_model)
bayesian_coef <- bayesian_summary$coefficients[, "mean"]
bayesian_ci <- posterior_interval(bayesian_model, prob = 0.95)
bayesian_sigma <- mean(as.matrix(bayesian_model)[, "sigma"])

# Compare results
comparison_df <- data.frame(
  Parameter = c(variable_names, "Sigma"),
  True_Value = c(beta_true, sigma_true),
  Classical_Estimate = c(classical_coef, classical_sigma),
  Classical_Lower = c(classical_ci[, 1], NA),
  Classical_Upper = c(classical_ci[, 2], NA),
  Bayesian_Estimate = c(bayesian_coef, bayesian_sigma),
  Bayesian_Lower = c(bayesian_ci[, 1], NA),
  Bayesian_Upper = c(bayesian_ci[, 2], NA)
)

# Print comparison
print(comparison_df)

The code for comparing performance across different data conditions:

# Function to run simulation for different conditions
run_simulation <- function(n_samples_list, n_reps, beta_true, sigma) {
  results <- data.frame()
  
  for (n in n_samples_list) {
    for (rep in 1:n_reps) {
      # Generate data
      sim_data <- generate_data(n, beta_true, sigma)
      colnames(sim_data) <- c("y", "X1", "X2")
      
      # Fit models
      classical_model <- lm(y ~ X1 + X2, data = sim_data)
      
      # Fit Bayesian model
      bayesian_model <- stan_glm(y ~ X1 + X2, 
                                data = sim_data,
                                family = gaussian(),
                                prior = normal(0, 10),
                                prior_intercept = normal(0, 10),
                                prior_aux = cauchy(0, 5),
                                chains = 4,
                                iter = 2000,
                                refresh = 0,  # Suppress output
                                seed = 123 + rep)
      
      # Extract estimates
      classical_coef <- coef(classical_model)
      bayesian_coef <- bayesian_summary$coefficients[, "mean"]
      
      # Calculate error metrics
      classical_mse <- mean((beta_true - classical_coef)^2)
      bayesian_mse <- mean((beta_true - bayesian_coef)^2)
      
      # Save results
      results <- rbind(results, data.frame(
        n_samples = n,
        rep = rep,
        classical_mse = classical_mse,
        bayesian_mse = bayesian_mse
      ))
    }
  }
  
  return(results)
}

# Run simulation for different sample sizes
n_samples_list <- c(20, 50, 100, 200, 500)
n_reps <- 10  # Reduced for demonstration, use more for actual analysis
sim_results <- run_simulation(n_samples_list, n_reps, beta_true, sigma_true)

# Summarize results
summary_results <- sim_results %>%
  group_by(n_samples) %>%
  summarize(
    classical_mse_mean = mean(classical_mse),
    bayesian_mse_mean = mean(bayesian_mse),
    classical_mse_sd = sd(classical_mse),
    bayesian_mse_sd = sd(bayesian_mse)
  )

# Visualize performance comparison
ggplot(summary_results, aes(x = n_samples)) +
  geom_line(aes(y = classical_mse_mean, color = "Classical")) +
  geom_line(aes(y = bayesian_mse_mean, color = "Bayesian")) +
  geom_ribbon(aes(ymin = classical_mse_mean - classical_mse_sd,
                 ymax = classical_mse_mean + classical_mse_sd,
                 fill = "Classical"), alpha = 0.2) +
  geom_ribbon(aes(ymin = bayesian_mse_mean - bayesian_mse_sd,
                 ymax = bayesian_mse_mean + bayesian_mse_sd,
                 fill = "Bayesian"), alpha = 0.2) +
  scale_color_manual(values = c("Classical" = "blue", "Bayesian" = "red")) +
  scale_fill_manual(values = c("Classical" = "blue", "Bayesian" = "red")) +
  labs(
    title = "MSE Comparison by Sample Size",
    x = "Sample Size",
    y = "Mean Squared Error",
    color = "Method",
    fill = "Method"
  ) +
  theme_minimal()

Results

The comparative analysis of classical and Bayesian regression revealed several key findings:

With large sample sizes and well-specified models, both approaches converged to similar parameter estimates
With small sample sizes, Bayesian models with informative priors provided more stable estimates
Classical confidence intervals and Bayesian credible intervals have fundamentally different interpretations
Bayesian models offered more intuitive quantification of parameter uncertainty
In the presence of outliers, robust Bayesian models showed better performance than standard OLS

Comparison of parameter estimates by sample size:

[Parameter estimate comparison plot would appear here]

Posterior distributions for key parameters:

[Posterior distribution plot would appear here]

Conclusions

This project highlighted the complementary nature of classical and Bayesian approaches to regression:

Classical methods offer computational simplicity and well-established theoretical properties
Bayesian methods provide a natural framework for incorporating prior knowledge and quantifying uncertainty
The choice between approaches should be guided by sample size, prior knowledge, and specific inference goals
Hybrid approaches that leverage strengths of both paradigms may offer optimal solutions in many cases

The simulation study demonstrated that while classical and Bayesian methods often yield similar point estimates, the Bayesian approach offers advantages in terms of uncertainty quantification and incorporation of external knowledge. These insights are valuable for researchers deciding between methodological approaches for specific applications.