Likelihood Ratios & Bayes Factors

Likelihood ratios and Bayes factors measure how much a piece of evidence should change your mind. Think of them as a translation tool that converts raw information into a clear "strength adjustment" for your beliefs.

These tools work by comparing how well two competing explanations predict the same evidence. If hypothesis A predicts the observed result ten times more strongly than hypothesis B, the Bayes factor is 10 in favor of A. This ratio then updates your prior beliefs to produce new posterior beliefs through Bayesian updating.

In medical testing, likelihood ratios above 10 are considered strong evidence for a diagnosis, while ratios below 0.1 strongly rule it out. However, a "positive" test result for a rare disease (prevalence 0.1%, or 1 in 1,000) with a likelihood ratio of 100 still leaves only about a 9% chance you actually have the disease—because your starting belief was so skeptical. The evidence moved you from 0.1% to 9%, which is actually a 90-fold increase, but the absolute probability remains low.

Beyond medicine, these tools apply everywhere. In legal reasoning, a forensic match might have a likelihood ratio of 1,000 to 1 favoring guilt—but if the prior probability was only 1 in 10,000 based on other evidence, the updated probability is still under 10%. In business decisions, customer survey results showing 70% satisfaction might seem positive, but if the response rate was only 5%, the evidence may be too weak to override your prior skepticism about the sample's representativeness.

The practical lesson: Always ask how strong the evidence is relative to your starting belief, not just whether it's "positive" or "negative." Bayes factors force you to be explicit about both the evidence quality and your initial assumptions.

Bayes factors quantify the relative evidence for competing hypotheses by comparing how well each predicts observed data. Formally, a Bayes factor is the ratio of marginal likelihoods—the probability of the data under each hypothesis, averaged over all possible parameter values. This differs from classical likelihood ratios, which compare maximum likelihoods at single parameter estimates. Kass & Raftery (1995) provide standard interpretation categories: Bayes factors of 1-3 represent "not worth more than a bare mention," 3-20 "positive evidence," 20-150 "strong evidence," and above 150 "very strong evidence" [1]. Jeffreys (1961) originally proposed this logarithmic evidence scale, noting that human perception of evidence strength follows a roughly logarithmic pattern [2].

• Kruschke (2014): "The Bayes factor is the degree to which the data shift belief away from one hypothesis toward another"—emphasizing that Bayes factors describe change in belief states rather than absolute truth [3].
• Morey et al. (2016): Bayes factors avoid the "p-value fallacy" by directly quantifying evidence strength for competing models rather than testing against an arbitrary null [4].
• Wagenmakers et al. (2011): In psychology research, Bayes factors allow researchers to actually support null hypotheses rather than merely failing to reject them—a critical advantage for testing theories that predict no effect [5].

Gloss: The marginal likelihood averages how well a hypothesis predicts data across all its parameter values, weighted by prior beliefs about those parameters.

Bayes factors depend on the specified prior distributions for model parameters, which introduces subjectivity—different reasonable priors can yield different Bayes factors for the same data. They also become computationally expensive for complex models. The interpretation scales (Jeffreys, Kass-Raftery) are heuristics rather than universal thresholds—context matters for what constitutes "strong" evidence in different fields. Additionally, Bayes factors assume you've correctly specified at least one of the competing models; if all models are misspecified, the ratio may be misleading.