6.5 Bayesian inference with MCMC

Bayesian inference with MCMC is a powerful method for updating beliefs based on evidence. It combines prior knowledge with observed data to create posterior distributions, allowing us to make informed decisions in the face of uncertainty.

MCMC methods help us sample from complex posterior distributions when analytical solutions aren't possible. By using tools like R packages, we can run simulations, assess convergence, and improve performance to get reliable results for our Bayesian analyses.

Bayesian Inference Principles

Updating Beliefs Based on Evidence

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• Bayesian inference updates prior beliefs about parameters based on observed data to obtain posterior distributions
• Incorporates prior knowledge through the prior distribution, which represents initial beliefs or knowledge about parameters before observing data
• Combines prior distribution and likelihood function using Bayes' theorem to obtain posterior distribution
  • Posterior distribution represents updated beliefs about parameters after observing data
  • Proportional to product of prior distribution and likelihood function
• Allows for incorporation of prior knowledge and provides framework for updating beliefs based on evidence (scientific experiments, real-world observations)

Components of Bayesian Inference

• Prior distribution: Initial beliefs or knowledge about parameters before observing data
  • Can be based on previous studies, expert opinion, or theoretical considerations
  • Examples: Uniform distribution (equal probability for all parameter values), normal distribution (prior centered around a specific value)
• Likelihood function: Quantifies probability of observing data given parameter values
  • Describes how likely the observed data is for different parameter values
  • Depends on the assumed statistical model and its assumptions
• Posterior distribution: Updated beliefs about parameters after observing data
  • Combines prior distribution and likelihood function using Bayes' theorem
  • Represents the uncertainty and beliefs about parameters based on both prior knowledge and observed data
  • Can be used for inference, prediction, and decision-making

MCMC Methods in R

Sampling from Complex Posterior Distributions

• Markov Chain Monte Carlo (MCMC) methods are computational techniques used to sample from complex posterior distributions in Bayesian inference
• Generate a Markov chain of parameter samples from the posterior distribution
  • Markov chain: Sequence of random variables where future states depend only on current state
  • Converges to target posterior distribution under certain conditions
• Commonly used MCMC methods:
  • Metropolis-Hastings algorithm: Proposes new parameter values based on proposal distribution and accepts or rejects proposals based on acceptance probability
  • Gibbs sampling: Samples from full conditional distributions of parameters iteratively
• R packages for MCMC implementations:

  [rjags](https://www.fiveableKeyTerm:rjags)

  ,

  [rstan](https://www.fiveableKeyTerm:rstan)

  ,

  [MCMCpack](https://www.fiveableKeyTerm:mcmcpack)

  • Provide functions and frameworks for specifying models and running MCMC simulations
  • Example:

    jags()

    function from

    rjags

    package for running MCMC using JAGS (Just Another Gibbs Sampler)

Assessing Convergence and Improving MCMC Performance

• Convergence diagnostics: Techniques to assess whether MCMC chains have converged to the posterior distribution
  • Trace plots: Visualize parameter values across MCMC iterations to check for stability and mixing
  • Gelman-Rubin statistic: Compares variance within and between multiple MCMC chains to assess convergence
• Techniques to improve MCMC performance:
  • Thinning: Retaining only every nth sample from MCMC chain to reduce autocorrelation
  • Burn-in period: Discarding initial portion of MCMC chain to remove transient behavior and ensure convergence
  • Choosing appropriate proposal distributions and tuning parameters to improve mixing and convergence
• Importance of running multiple MCMC chains with different initial values to assess convergence and robustness of results

Posterior Distributions and Credible Intervals

Summarizing Posterior Distributions

• Posterior distribution summarizes uncertainty and beliefs about parameters after observing data
• Posterior summary statistics provide point estimates of parameters
  • Mean: Average value of parameter samples from posterior distribution
  • Median: Middle value of parameter samples when sorted
  • Mode: Most frequent or highest probability value of parameter samples
• Posterior distribution shape and spread provide insights into uncertainty and range of plausible parameter values
  • Narrow and peaked distribution indicates high certainty and precision
  • Wide and flat distribution indicates high uncertainty and low precision
• Visualizing posterior distributions using density plots, histograms, or box plots
  • Helps communicate uncertainty and distribution of parameter estimates
  • Example: Plotting posterior distribution of regression coefficients to assess their significance and uncertainty

Credible Intervals and Posterior Predictive Checks

• Credible intervals quantify uncertainty associated with parameter estimates
  • Typically reported as 95% credible intervals
  • Indicates 95% probability that true parameter value lies within the interval
  • Interpreted differently from frequentist confidence intervals
    • Credible intervals have direct probability interpretation
    • Confidence intervals have long-run frequency interpretation
• Posterior predictive checks assess model fit by comparing observed data to simulated data generated from posterior distribution
  • Generate replicated datasets from posterior predictive distribution
  • Compare observed data to replicated datasets using summary statistics or graphical checks
  • Helps identify model misspecification or lack of fit
  • Example: Comparing observed and replicated data distributions to assess goodness-of-fit

Bayesian vs Frequentist Inference

Philosophical and Conceptual Differences

• Bayesian inference treats parameters as random variables and assigns probability distributions to them
  • Allows for incorporation of prior knowledge and beliefs
  • Provides posterior distributions that quantify uncertainty about parameters
• Frequentist inference treats parameters as fixed unknown quantities
  • Relies solely on observed data and sampling distributions
  • Provides point estimates and confidence intervals based on long-run frequency interpretation
• Bayesian inference allows for computation of posterior probabilities and hypothesis testing based on posterior distribution
  • Can directly calculate probability of hypothesis being true given the data
  • Example: Calculating posterior probability of a parameter being positive
• Frequentist inference relies on p-values and significance testing
  • Assesses evidence against null hypothesis based on sampling distribution
  • Provides indirect measure of evidence and does not quantify probability of hypothesis being true

Strengths and Limitations

• Bayesian inference is well-suited for small sample sizes and complex models
  • Incorporates prior knowledge to compensate for limited data
  • Can handle hierarchical and nonlinear models more naturally
• Frequentist inference may have limitations in small sample sizes and complex models
  • Relies on asymptotic properties and large-sample approximations
  • May have issues with model identifiability and convergence
• Bayesian inference allows for direct probability statements and decision-making based on posterior distribution
  • Can calculate probabilities of parameters falling within specific ranges
  • Facilitates decision-making under uncertainty (clinical trials, business decisions)
• Frequentist inference focuses on long-run performance and error rates
  • Provides methods for controlling Type I and Type II errors
  • Emphasizes reproducibility and frequentist properties of estimators and tests
• Choice between Bayesian and frequentist approaches depends on research question, available prior knowledge, computational resources, and philosophical preferences
  • Bayesian approach is gaining popularity due to advancements in computational methods and software
  • Frequentist approach remains widely used and is the foundation of many classical statistical methods