9.2 Posterior odds

Posterior odds are a key concept in Bayesian statistics, quantifying how likely one hypothesis is compared to another after seeing data. They combine prior beliefs with new evidence, allowing us to update our understanding as we gather more information.

Calculating posterior odds involves multiplying prior odds by the Bayes factor, which measures how well the data supports each hypothesis. This approach offers a more nuanced alternative to traditional hypothesis testing, enabling direct probability interpretations and incorporation of existing knowledge.

Definition of posterior odds

• Posterior odds quantify the relative plausibility of competing hypotheses after observing data in Bayesian statistics
• Fundamental concept in Bayesian inference used to update beliefs based on new evidence
• Bridges prior knowledge with observed data to form updated probabilistic conclusions

Relationship to prior odds

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• Posterior odds result from updating prior odds with new evidence
• Incorporates the initial belief ratio between hypotheses before observing data
• Multiplies prior odds by the likelihood ratio to obtain posterior odds
• Reflects how initial beliefs change in light of new information

Comparison with likelihood ratio

• Likelihood ratio measures the relative support of data for competing hypotheses
• Posterior odds combine likelihood ratio with prior odds
• Likelihood ratio alone does not account for prior probabilities of hypotheses
• Posterior odds provide a more comprehensive measure of hypothesis plausibility

Components of posterior odds

Prior odds

• Ratio of prior probabilities for competing hypotheses before observing data
• Represent initial beliefs or existing knowledge about the hypotheses
• Derived from previous studies, expert opinions, or theoretical considerations
• Influence the posterior odds more strongly when sample size is small
• Can be uninformative (equal prior probabilities) or informative (favoring one hypothesis)

Bayes factor

• Quantifies the relative evidence provided by the data for one hypothesis over another
• Calculated as the ratio of marginal likelihoods under competing hypotheses
• Independent of prior probabilities, focusing solely on the data's contribution
• Ranges from 0 to infinity, with values greater than 1 supporting the alternative hypothesis
• Interpreted using guidelines (weak, moderate, strong evidence) proposed by statisticians (Kass and Raftery)

Calculation of posterior odds

Mathematical formula

• Posterior odds = Prior odds × Bayes factor
• Expressed mathematically as: $\frac{P(H_1|D)}{P(H_0|D)} = \frac{P(H_1)}{P(H_0)} \times \frac{P(D|H_1)}{P(D|H_0)}$
• $P(H_1|D)$ and $P(H_0|D)$ represent posterior probabilities of hypotheses given data D
• $P(H_1)$ and $P(H_0)$ denote prior probabilities of hypotheses

Step-by-step process

• Specify prior probabilities for each hypothesis based on existing knowledge
• Calculate prior odds by dividing prior probability of H1 by prior probability of H0
• Compute likelihood of observed data under each hypothesis
• Determine Bayes factor by dividing likelihood under H1 by likelihood under H0
• Multiply prior odds by Bayes factor to obtain posterior odds

Interpretation of posterior odds

Strength of evidence

• Posterior odds greater than 1 indicate support for the alternative hypothesis
• Values between 0 and 1 suggest evidence favoring the null hypothesis
• Magnitude of posterior odds reflects the strength of evidence for one hypothesis over another
• Logarithmic scale often used for easier interpretation (log posterior odds)
• Guidelines proposed by statisticians (Jeffreys) for categorizing evidence strength

Decision-making threshold

• No universally agreed-upon threshold for accepting or rejecting hypotheses
• Researchers often use cutoff values based on specific contexts or requirements
• Common thresholds include 3:1, 10:1, or 100:1 for strong evidence
• Decision thresholds may vary depending on the consequences of false positives or negatives
• Consideration of practical significance alongside statistical evidence

Applications in hypothesis testing

Model selection

• Posterior odds used to compare and select between competing statistical models
• Allows for simultaneous comparison of multiple models, not just pairwise
• Incorporates model complexity through prior probabilities (favoring simpler models)
• Provides a natural implementation of Occam's razor in model selection
• Enables calculation of model-averaged estimates and predictions

Bayesian vs frequentist approach

• Posterior odds offer a direct probability interpretation, unlike p-values
• Allow incorporation of prior knowledge, which is not possible in frequentist hypothesis testing
• Provide evidence in favor of both null and alternative hypotheses
• Do not rely on sampling distributions or assumptions about repeated sampling
• Enable more nuanced conclusions beyond binary reject/fail to reject decisions

Posterior odds vs posterior probability

Conversion between odds and probability

• Posterior odds can be converted to posterior probabilities using the formula: $P(H_1|D) = \frac{\text{Posterior odds}}{1 + \text{Posterior odds}}$
• Posterior probability ranges from 0 to 1, while posterior odds range from 0 to infinity
• Odds of 1:1 correspond to a probability of 0.5
• Logarithmic odds (log-odds) provide a symmetric scale around 0

Advantages of each representation

• Posterior odds facilitate easy updating with new data through simple multiplication
• Posterior probabilities offer more intuitive interpretation for non-specialists
• Odds representation avoids issues with probabilities very close to 0 or 1
• Probability scale useful for decision-making based on expected utility theory
• Odds often preferred in fields like epidemiology for reporting relative risks

Limitations and considerations

Sensitivity to prior specification

• Posterior odds can be strongly influenced by choice of prior probabilities
• Sensitivity analyses recommended to assess impact of different prior specifications
• Uninformative priors may lead to counterintuitive results in some cases
• Elicitation of informative priors from experts can be challenging and subjective
• Robustness to prior choice increases with larger sample sizes

Computational challenges

• Calculation of marginal likelihoods for Bayes factors can be computationally intensive
• Numerical integration or Monte Carlo methods often required for complex models
• Approximation methods (BIC) sometimes used but may have limitations
• High-dimensional models pose particular challenges for accurate computation
• Specialized software and algorithms developed to address these computational issues

Examples and case studies

Medical diagnosis

• Posterior odds used to update disease probabilities based on test results
• Incorporates prevalence (prior probability) and test accuracy (likelihood ratio)
• Helps clinicians make informed decisions about further testing or treatment
• Accounts for false positive and false negative rates in diagnostic tests
• Allows for sequential updating as multiple test results become available

Forensic evidence evaluation

• Posterior odds assess the strength of forensic evidence in legal proceedings
• Used in DNA profiling to quantify match probabilities between samples
• Incorporates population genetics data and laboratory error rates
• Helps courts interpret complex scientific evidence in a probabilistic framework
• Addresses issues of evidence combination and interdependence

Posterior odds in model comparison

Bayesian model averaging

• Posterior odds used to weight predictions from multiple competing models
• Accounts for model uncertainty in making inferences and predictions
• Combines strengths of different models based on their posterior probabilities
• Improves predictive performance by not relying on a single "best" model
• Particularly useful when no single model clearly outperforms others

Occam's razor principle

• Posterior odds naturally implement Occam's razor by favoring simpler models
• Complex models penalized through lower prior probabilities
• Balances model fit with parsimony to avoid overfitting
• More complex models require stronger evidence to overcome simplicity preference
• Helps researchers select parsimonious models that generalize well to new data

Reporting and communicating results

Graphical representations

• Posterior odds often visualized using forest plots or tornado diagrams
• Log scales frequently employed for easier comparison across wide ranges
• Density plots of posterior distributions complement odds ratios
• Bayes factor robustness plots show sensitivity to prior specifications
• Interactive visualizations allow exploration of posterior odds under different scenarios

Verbal interpretation guidelines

• Standardized language proposed for describing strength of evidence
• Phrases like "strong evidence," "moderate evidence," or "weak evidence" linked to specific ranges
• Emphasis on describing uncertainty and avoiding overly definitive statements
• Encouragement to report full posterior distributions alongside summary measures
• Guidelines for communicating Bayesian results to non-technical audiences developed by statisticians