📊Bayesian Statistics Unit 11 – Bayesian Model Selection & Averaging

Key Concepts

• Bayesian model selection involves comparing and selecting models based on their posterior probabilities, which quantify the support for each model given the observed data
• Bayes factors ($BF_{ij}$) measure the relative evidence for one model over another by comparing their marginal likelihoods
  • Marginal likelihood integrates the likelihood function over the prior distribution of the model parameters
• Model selection criteria, such as Bayesian Information Criterion (BIC) and Deviance Information Criterion (DIC), balance model fit and complexity to identify the most parsimonious model
• Bayesian model averaging (BMA) accounts for model uncertainty by averaging predictions or parameter estimates across multiple models, weighted by their posterior probabilities
• Prior distributions play a crucial role in Bayesian model selection and averaging, as they encode prior knowledge or beliefs about the models and their parameters
• Bayesian model selection and averaging can be applied to various domains, including regression, classification, time series analysis, and causal inference
• Computational methods, such as Markov Chain Monte Carlo (MCMC) and Laplace approximation, are often required to estimate marginal likelihoods and posterior probabilities in complex models

Bayesian Model Comparison

• Bayesian model comparison evaluates the relative plausibility of competing models $M_1, \ldots, M_K$ given the observed data $D$
• The posterior probability of each model $M_k$ is computed using Bayes' theorem: $P(M_k|D) = \frac{P(D|M_k)P(M_k)}{\sum_{i=1}^K P(D|M_i)P(M_i)}$
  • $P(D|M_k)$ is the marginal likelihood of model $M_k$, and $P(M_k)$ is the prior probability of model $M_k$
• The marginal likelihood $P(D|M_k)$ measures the average fit of model $M_k$ to the data, integrating over the prior distribution of its parameters $\theta_k$: $P(D|M_k) = \int P(D|\theta_k, M_k)P(\theta_k|M_k)d\theta_k$
• The prior probabilities $P(M_k)$ reflect the prior beliefs about the plausibility of each model before observing the data
• Bayesian model comparison naturally penalizes more complex models through the marginal likelihood, as complex models spread their prior probability over a larger parameter space
• The posterior odds ratio $\frac{P(M_i|D)}{P(M_j|D)}$ quantifies the relative support for model $M_i$ over model $M_j$ after observing the data

Bayes Factors

• Bayes factors ($BF_{ij}$) are a key tool for comparing the relative evidence for two models $M_i$ and $M_j$
• The Bayes factor $BF_{ij}$ is the ratio of the marginal likelihoods of the two models: $BF_{ij} = \frac{P(D|M_i)}{P(D|M_j)}$
  • A $BF_{ij} > 1$ indicates support for model $M_i$ over model $M_j$, while $BF_{ij} < 1$ favors model $M_j$
• Bayes factors can be interpreted as the strength of evidence for one model over another
  • For example, a $BF_{ij} = 10$ means that the data are 10 times more likely under model $M_i$ than under model $M_j$
• Bayes factors are independent of the prior model probabilities $P(M_i)$ and $P(M_j)$, making them a useful tool for model comparison when prior information is limited or controversial
• Bayes factors can be used to compare non-nested models, which is an advantage over likelihood ratio tests
• The Jeffreys scale provides a rough guideline for interpreting Bayes factors: $BF_{ij} > 100$ is decisive evidence for $M_i$, $10 < BF_{ij} < 100$ is strong evidence, $3 < BF_{ij} < 10$ is substantial evidence, and $1 < BF_{ij} < 3$ is anecdotal evidence

Model Selection Criteria

• Model selection criteria are quantitative measures that balance model fit and complexity to identify the most parsimonious model
• Bayesian Information Criterion (BIC) is a widely used model selection criterion based on the marginal likelihood approximation: $BIC = -2\log P(D|\hat{\theta}_k, M_k) + p_k \log n$
  • $\hat{\theta}_k$ is the maximum likelihood estimate of the model parameters, $p_k$ is the number of parameters in model $M_k$, and $n$ is the sample size
  • Models with lower BIC values are preferred, as they indicate a better balance between fit and complexity
• Deviance Information Criterion (DIC) is another popular Bayesian model selection criterion that extends the Akaike Information Criterion (AIC) to hierarchical models
  • DIC is computed as $DIC = \bar{D} + p_D$, where $\bar{D}$ is the posterior mean deviance and $p_D$ is the effective number of parameters
  • Models with lower DIC values are preferred, as they indicate a better fit to the data while penalizing model complexity
• Watanabe-Akaike Information Criterion (WAIC) and Leave-One-Out Cross-Validation (LOO-CV) are more recent model selection criteria that are particularly useful for Bayesian models with non-normal posteriors or hierarchical structures
• Model selection criteria should be used with caution, as they rely on asymptotic approximations and may not always select the "true" model, especially when the models are misspecified or the sample size is small

Bayesian Model Averaging

• Bayesian model averaging (BMA) is a principled approach to account for model uncertainty by combining predictions or parameter estimates from multiple models
• In BMA, the posterior distribution of a quantity of interest $\Delta$ (e.g., a future observation or a parameter) is obtained by averaging its posterior distributions under each model, weighted by the models' posterior probabilities: $P(\Delta|D) = \sum_{k=1}^K P(\Delta|D, M_k)P(M_k|D)$
• BMA provides a coherent framework for making predictions or inferences that incorporate uncertainty about the model structure
• BMA can improve predictive performance by leveraging the strengths of different models and reducing the risk of overfitting
• The weights used in BMA are the posterior model probabilities $P(M_k|D)$, which are derived from the marginal likelihoods and prior model probabilities using Bayes' theorem
• BMA can be used for model selection by identifying the models that contribute the most to the averaged posterior distribution
• Implementing BMA can be computationally challenging, especially when the number of candidate models is large or the models are complex, requiring efficient sampling or approximation techniques

Practical Applications

• Bayesian model selection and averaging have been applied to a wide range of domains, including:
  • Regression and classification problems in machine learning and statistics
  • Time series analysis and forecasting in economics and finance
  • Causal inference and policy evaluation in social sciences and epidemiology
  • Model-based clustering and mixture modeling in biology and psychology
• In variable selection problems, Bayesian model averaging can be used to identify the most important predictors while accounting for uncertainty in the model structure
  • For example, in linear regression, BMA can be used to average over all possible subsets of predictors, weighted by their posterior probabilities
• In ensemble learning, Bayesian model averaging can be used to combine predictions from multiple models, such as decision trees, neural networks, or support vector machines
• In causal inference, Bayesian model averaging can be used to estimate treatment effects or policy impacts while accounting for uncertainty in the confounding variables or the functional form of the relationship
• Bayesian model selection and averaging can help researchers and practitioners make more robust and reliable inferences by incorporating model uncertainty and avoiding the pitfalls of model selection based on a single criterion or dataset

Computational Methods

• Bayesian model selection and averaging often involve computationally intensive tasks, such as estimating marginal likelihoods, sampling from posterior distributions, or exploring large model spaces
• Markov Chain Monte Carlo (MCMC) methods, such as the Metropolis-Hastings algorithm or the Gibbs sampler, are widely used to sample from the posterior distribution of model parameters and estimate marginal likelihoods
  • MCMC methods construct a Markov chain whose stationary distribution is the desired posterior distribution, allowing for efficient sampling and estimation
• Laplace approximation is another popular method for approximating marginal likelihoods and posterior probabilities, especially when the posterior distribution is approximately normal
  • Laplace approximation uses a second-order Taylor expansion around the posterior mode to approximate the posterior distribution as a multivariate normal distribution
• Variational inference is an alternative to MCMC that approximates the posterior distribution with a simpler, tractable distribution by minimizing the Kullback-Leibler divergence between the two distributions
  • Variational inference can be faster and more scalable than MCMC, but it may provide less accurate approximations, especially for complex or multi-modal posteriors
• Bayesian optimization and adaptive sampling techniques can be used to efficiently explore large model spaces and identify the most promising models for further analysis
• Parallel and distributed computing techniques, such as MapReduce or GPU acceleration, can be employed to speed up computations and handle large datasets or model spaces

Limitations and Challenges

• Bayesian model selection and averaging rely on the specification of prior distributions for the model parameters and the model space, which can be subjective and influential on the results
  • Sensitivity analysis and robust priors can be used to assess the impact of prior choices and mitigate their influence
• The interpretation of Bayes factors and posterior model probabilities can be challenging, especially when the models are not well-defined or the prior distributions are improper
  • Careful elicitation of prior distributions and the use of default or reference priors can help ensure the validity and interpretability of the results
• Bayesian model selection and averaging can be computationally demanding, particularly when dealing with complex models, large datasets, or high-dimensional parameter spaces
  • Efficient sampling techniques, approximations, and parallel computing can help alleviate the computational burden, but they may introduce additional errors or biases
• Model misspecification, where all the candidate models are far from the true data-generating process, can lead to misleading inferences and poor predictive performance
  • Model checking, validation, and expansion can help detect and mitigate model misspecification, but they may not always be feasible or conclusive
• The choice of the model space and the prior model probabilities can have a significant impact on the results of Bayesian model selection and averaging
  • Careful consideration of the scientific context, expert knowledge, and data-driven methods can help guide the specification of the model space and priors
• Communicating the results of Bayesian model selection and averaging to non-technical audiences can be challenging, as the concepts and methods may be unfamiliar or counterintuitive
  • Clear and accessible explanations, visualizations, and case studies can help convey the key insights and limitations of the analysis