6.2 Bayesian inference and MCMC methods

Bayesian inference revolutionizes statistical analysis in biology by updating beliefs with new evidence. It uses prior knowledge, likelihood functions, and posterior distributions to estimate parameters and compare models, making it invaluable for understanding complex biological systems.

Markov Chain Monte Carlo methods are powerful tools for sampling complex distributions in Bayesian analysis. These techniques, including Metropolis-Hastings and Gibbs sampling, allow researchers to estimate parameters, quantify uncertainty, and make predictions in various biological contexts.

Fundamentals of Bayesian Inference

Fundamentals of Bayesian inference

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• Bayesian inference applies probabilistic approach to statistical inference based on Bayes' theorem updating beliefs with new evidence
• Key components include prior distribution representing initial beliefs, likelihood function relating data to parameters, posterior distribution combining prior and likelihood
• Parameter estimation incorporates prior knowledge and observed data providing probability distributions for parameters (population growth rates, infection rates)
• Model fitting compares different models using Bayesian model selection accounting for model complexity and fit to data (predator-prey interactions, disease transmission)

Application of Bayes' theorem

• Bayes' theorem formula $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ updates prior beliefs with new evidence
• Components in Bayesian inference:
  • $P(A)$ Prior probability initial belief about parameter
  • $P(B|A)$ Likelihood probability of data given parameter
  • $P(A|B)$ Posterior probability updated belief about parameter
  • $P(B)$ Evidence or marginal likelihood normalizing constant
• Updating process starts with prior beliefs incorporates new data through likelihood obtains posterior distribution (species abundance, genetic mutation rates)

Markov Chain Monte Carlo Methods

Principles of MCMC methods

• MCMC principles combine Monte Carlo integration with Markov chain properties to sample complex distributions
• Key concepts:
  • Stationary distribution target posterior distribution
  • Ergodicity chain explores entire parameter space
  • Detailed balance ensures correct stationary distribution
• Implementation steps:
  1. Define target distribution (posterior)
  2. Choose proposal distribution for generating new samples
  3. Set up acceptance/rejection criteria based on Metropolis-Hastings ratio
  4. Run chain for sufficient iterations to achieve convergence
• Convergence diagnostics assess chain mixing and stability:
  • Trace plots visualize parameter values over iterations
  • Autocorrelation measures dependence between samples
  • Gelman-Rubin statistic compares multiple chains

Utilization of MCMC techniques

• Metropolis-Hastings algorithm:
  • General MCMC method for sampling from complex distributions
  • Acceptance probability determines whether to accept or reject proposed samples
  • Symmetric proposals (random walk) vs. asymmetric proposals (independence sampler)
• Gibbs sampling:
  • Special case of Metropolis-Hastings for multivariate distributions
  • Samples from conditional distributions of each parameter
  • Useful for high-dimensional problems (gene regulatory networks, ecological community dynamics)
• Parameter estimation provides:
  • Point estimates mean, median, or mode of posterior distribution
  • Credible intervals quantify uncertainty in parameter estimates
• Uncertainty quantification:
  • Posterior predictive distributions simulate future observations
  • Bayesian model averaging combines predictions from multiple models