Concept Architecture
Concept
Theoretically, Posterior Odds are the odds that a hypothesis or condition is true after incorporating new evidence. They represent the updated measure of belief obtained by combining prior information with the evidential value of observed data through Bayes' Theorem. The concept is founded on Bayesian probability theory and exists to support probabilistic reasoning and decision-making under uncertainty.
Mathematically, Posterior Odds are represented as the product of Prior Odds and the Likelihood Ratio. This formulation provides a direct method for updating beliefs following the observation of new evidence. Posterior Odds may subsequently be converted into Posterior Probability, facilitating interpretation in clinical, epidemiological and health economic applications.
In practice, Posterior Odds are estimated by first calculating Prior Odds from the pre-test probability of disease and then multiplying these odds by the appropriate positive or negative Likelihood Ratio. They are widely applied in diagnostic testing, Bayesian evidence synthesis, health technology assessment and decision-analytic modelling to estimate the probability of disease or treatment effectiveness after new evidence becomes available.
Purpose
Used to update the odds of a hypothesis after incorporating new evidence, support Bayesian inference, estimate post-test disease likelihood and improve decision-making in clinical practice and health economic evaluation.
Mathematical Formulae
Primary Formula
Posterior Odds = Prior Odds ? Likelihood Ratio
Supporting Formulae
Prior Odds = Prior Probability / (1 ? Prior Probability)
Posterior Probability = Posterior Odds / (1 + Posterior Odds)
Likelihood Ratio = P(E�H?) / P(E�H?)
where:
- E = observed evidence
- H? = hypothesis of interest
- H? = alternative hypothesis
Related Mathematical Methods
- Bayes' Theorem
- Prior Odds
- Posterior Probability
- Likelihood Ratio
- Positive Likelihood Ratio
- Negative Likelihood Ratio
- Bayesian Inference
- Diagnostic Testing
Example
A patient has a pre-test probability of disease of 20%.
Prior Odds:
Prior Odds = 0.20 / (1 ? 0.20) = 0.25
A diagnostic test produces a positive result with a Positive Likelihood Ratio of 6.
Posterior Odds:
Posterior Odds = 0.25 ? 6 = 1.50
Posterior Probability:
Posterior Probability = 1.50 / (1 + 1.50) = 0.60
Following the positive test result, the estimated probability of disease increases from 20% to 60%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Division | =B2/(1-B2) | Converts prior probability into prior odds. |
| Multiplication | =C2*D2 | Calculates Posterior Odds using the Likelihood Ratio. |
| Division | =E2/(1+E2) | Converts Posterior Odds into Posterior Probability. |
| IF | =IF(F2>0.80,"Treat","Consider further investigation") | Supports treatment decisions using updated probability estimates. |
VBA (Optional)
A VBA macro can automatically update Posterior Odds and Posterior Probabilities for multiple diagnostic scenarios using user-defined prior probabilities and likelihood ratios.
Sources
- Bayes T. An Essay towards Solving a Problem in the Doctrine of Chances. 1763.
- Gelman A, Carlin JB, Stern HS, et al. Bayesian Data Analysis. 4th ed.
- Spiegelhalter DJ, Abrams KR, Myles JP. Bayesian Approaches to Clinical Trials and Health-Care Evaluation.
- Deeks JJ, Altman DG. Diagnostic tests 4: likelihood ratios. BMJ. 2004;329:168?169.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What are posterior odds?
The updated odds that a hypothesis is true after incorporating new evidence, found by multiplying prior odds by the likelihood ratio.
Source: Bayes 1763
What do posterior odds represent after evidence is taken into account?
Posterior odds represent the odds that a hypothesis is true once new evidence has been folded in, obtained by multiplying the prior odds by the likelihood ratio of that evidence. In diagnosis they are the revised odds a patient has a disease after a test result, having started from the odds before the test. Expressing the updated belief as odds makes the Bayesian calculation simple, and the odds can then be converted back into a probability. The belief after the evidence is what they capture. Sackett and colleagues (1991) describe this.
Source: Sackett et al. 1991
How are posterior odds calculated?
Posterior odds are calculated by multiplying the prior odds of the hypothesis by the likelihood ratio of the observed evidence, a form of Bayes' theorem on the odds scale. The prior odds represent belief before the evidence, and the likelihood ratio captures how much the evidence favours the hypothesis over its alternative. So posterior odds are calculated as prior odds times the likelihood ratio, which provides a simple way to update belief when evidence can be summarised by a likelihood ratio, as with diagnostic tests, and the resulting posterior odds can then be converted into a posterior probability if desired.
Source: Bayes 1763
How do posterior odds relate to prior odds?
Posterior odds relate to prior odds as the updated version after evidence: the prior odds are the odds of the hypothesis before the evidence, and multiplying them by the likelihood ratio gives the posterior odds afterward. The prior odds are the starting point and the posterior odds the result of incorporating the evidence. So posterior odds and prior odds are the before and after of Bayesian updating on the odds scale, connected by the likelihood ratio, which quantifies how strongly the evidence shifts the odds, so that stronger evidence moves the posterior odds further from the prior odds.
Source: Bayes 1763
How are posterior odds used in diagnosis?
In diagnosis, posterior odds are used to obtain a patient's probability of disease after a test result: the pre-test, or prior, odds of disease are multiplied by the test's likelihood ratio for the observed result to give the post-test, or posterior, odds, which are then converted to a post-test probability. So posterior odds are used in diagnostic reasoning as the mechanism for updating disease probability with test results, since working on the odds scale lets the likelihood ratio be applied by simple multiplication, providing a practical way to combine a patient's starting probability with the evidence a test provides.
Source: Sackett et al. 1991
How are posterior odds converted to a probability?
Posterior odds are converted to a probability by dividing the odds by one plus the odds, since odds are the ratio of the probability of an event to the probability of its non-occurrence. For example, posterior odds of three to one correspond to a probability of three-quarters. So posterior odds are converted to a posterior probability through this simple relationship between odds and probability, which allows the result of Bayesian updating, often carried out on the odds scale for convenience, to be expressed as a probability that is more readily interpreted, such as a patient's post-test probability of disease.
Source: Bayes 1763
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 10 Dec 2025
Content version: 1.0.0
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