Bayesian Model Averaging tackles model uncertainty by combining predictions from multiple models. It weighs each model based on its posterior probability, considering all plausible models instead of relying on a single one.

Ensemble diversity measures how much base learners in an ensemble disagree. Higher diversity often leads to better performance. Measures like the Kappa statistic and Q-statistic quantify pairwise agreement, while correlation-based measures assess overall ensemble diversity.

Bayesian Model Averaging

Posterior Probability and Model Uncertainty

• Bayesian Model Averaging (BMA) addresses model uncertainty by combining predictions from multiple models
  • Weights each model's predictions based on its posterior probability
  • Posterior probability indicates the likelihood of a model being true given the observed data
• Model uncertainty refers to the uncertainty in selecting the best model from a set of candidate models
  • Arises when multiple models fit the data reasonably well but make different predictions
• BMA accounts for model uncertainty by considering the predictions of all plausible models
  • Avoids relying on a single model that may not capture the full complexity of the data

Marginal Likelihood and Occam's Razor

• Marginal likelihood measures how well a model fits the data while penalizing model complexity
  • Calculated by integrating the likelihood function over the prior distribution of model parameters
  • Higher marginal likelihood indicates a better balance between model fit and complexity
• Occam's razor principle favors simpler models over complex ones when both explain the data equally well
  • BMA incorporates Occam's razor by assigning higher posterior probabilities to simpler models
  • Prevents overfitting and encourages parsimony in model selection ($P(M_i|D) \propto P(D|M_i)P(M_i)$)

Ensemble Diversity Measures

Pairwise Diversity Measures

• Ensemble diversity measures quantify the degree of disagreement among base learners in an ensemble
  • Higher diversity often leads to better ensemble performance
  • Diversity ensures that base learners make different errors and complement each other
• Kappa statistic measures pairwise agreement between two classifiers while correcting for chance agreement
  • Ranges from -1 (complete disagreement) to 1 (complete agreement)
  • Lower kappa values indicate higher diversity ($\kappa = \frac{p_o - p_e}{1 - p_e}$)
• Q-statistic measures pairwise similarity between two classifiers based on their predictions
  • Ranges from -1 (complete disagreement) to 1 (complete agreement)
  • Lower Q-statistic values indicate higher diversity ($Q_{ij} = \frac{N^{11}N^{00} - N^{01}N^{10}}{N^{11}N^{00} + N^{01}N^{10}}$)

Correlation-based Diversity and Bias-Variance-Covariance Decomposition

• Correlation-based diversity measures the average pairwise correlation between base learners' predictions
  • Lower correlation indicates higher diversity
  • Can be calculated using Pearson's correlation coefficient or rank correlation measures (Spearman's rho, Kendall's tau)
• Bias-variance-covariance decomposition breaks down the ensemble's error into three components
  • Bias: the difference between the ensemble's average prediction and the true value
  • Variance: the variability of predictions across base learners
  • Covariance: the interdependence between base learners' predictions
• Ideal ensemble has low bias, high variance, and low covariance
  • High variance ensures diversity among base learners
  • Low covariance prevents base learners from making similar errors