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 by the , 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)