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Bayesian inference gets its name fromBayes’s theorem, expressing posterior probabilities for hypotheses about a data generating process as the (normalized) product of prior probabilities and a likelihood function. But Bayesian inference uses all of probability theory, not just Bayes’s theorem. Many hypotheses of scientific interest arecomposite hypotheses, with the strength of evidence for the hypothesis dependent on knowledge about auxiliary factors, such as the values of nuisance parameters (e.g., uncertain background rates or calibration factors). Many important capabilities of Bayesian methods arise from use of the law of total probability, which instructs analysts to compute probabilities for composite hypotheses bymarginalizationover auxiliary factors. This tutorial targets relative newcomers to Bayesian inference, aiming to complement tutorials that focus on Bayes’s theorem and how priors modulate likelihoods. The emphasis here is on marginalization over parameter spaces—both how it is the foundation for important capabilities, and how it may motivate caution when parameter spaces are large. Topics covered include the difference between likelihood and probability, understanding the impact of priors beyond merely shifting the maximum likelihood estimate, and the role of marginalization in accounting for uncertainty in nuisance parameters, systematic error, and model misspecification.
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When pleas precede evidence: Using Bayesian analyses to establish the importance of a reasonable standard for evidence prior to plea offers
In most U.S. jurisdictions, prosecutors are not required to clearly establish a reasonable basis for guilt prior to offering defendants plea deals. We apply Bayesian analyses, which are uniquely suited to illuminate the impact of prior probability of guilt on the informativeness of a particular outcome (i.e., a guilty plea), to demonstrate the risks of plea offers that precede evidence. Our primary prediction was that lower prior probabilities of guilt would coincide with a significantly higher risk for false guilty pleas. We incorporated data from Wilford, Sutherland into a Bayesian analysis allowing us to model the expected diagnosticity of plea acceptance across the full range of prior probability of guilt. Our analysis indicated that, as predicted, when plea offers are accepted at lower prior probabilities of guilt, the probability that a plea is actually false is significantly higher than when prior probabilities of guilt are higher. In other words, there is a trade-off between prior probability of guilt and information gain. For instance, in our analysis, when prior probability of guilt was 50%, posterior probability of guilt (after a plea) was 77.8%; when prior probability of guilt was 80%, posterior probability of guilt was 93.3%. Our results clearly indicate the importance of ensuring that there is a reasonable basis for guilt before a plea deal is extended. In the absence of shared discovery, no such reasonable basis can be established. Further, these results illustrate the additional insights gained from applying a Bayesian approach to plea-decision contexts.
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- Award ID(s):
- 1844585
- PAR ID:
- 10601195
- Editor(s):
- McLellan, Myles F
- Publisher / Repository:
- The University of Alberta Library
- Date Published:
- Journal Name:
- The wrongful conviction law review
- ISSN:
- 2563-2574
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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