Concept Architecture
Concept
Theoretically, Posterior Probability is the probability assigned to an uncertain event or parameter after observed evidence has been incorporated. It is the central quantity in Bayesian inference, combining prior knowledge with information from observed data to produce an updated probability distribution. In health economics, posterior probabilities quantify uncertainty in model parameters such as treatment effects, transition probabilities, costs and utilities, and form the basis of probabilistic decision making.
Mathematically, posterior probability is obtained by applying Bayes' theorem, which updates a prior probability according to the likelihood of the observed data. The posterior distribution represents all available information about an unknown parameter after evidence has been incorporated. For complex models, posterior probabilities are typically estimated numerically using Markov Chain Monte Carlo methods rather than calculated analytically.
In practice, posterior probabilities are estimated using Bayesian statistical software such as JAGS, WinBUGS, OpenBUGS or Stan. Posterior distributions are used to populate probabilistic health economic models, estimate treatment effectiveness through Bayesian network meta-analysis, quantify parameter uncertainty, and calculate decision uncertainty measures including cost-effectiveness acceptability curves and expected value of information.
Purpose
Used to quantify updated uncertainty after incorporating observed evidence, providing probability distributions for Bayesian estimation, probabilistic sensitivity analysis and health economic decision making.
Mathematical Formulae
Primary Formula
P(? | y) = (P(y | ?) P(?)) / P(y)
where:
- P(? | y) = posterior probability
- P(y | ?) = likelihood
- P(?) = prior probability
- P(y) = marginal probability of the observed data
Supporting Formulae
Marginal probability:
P(y) = ? P(y | ?) P(?) d?
Posterior proportionality:
P(? | y) ? P(y | ?) P(?)
Related Mathematical Methods
- Bayesian inference
- Bayes' theorem
- Markov Chain Monte Carlo
- Metropolis-Hastings algorithm
- Gibbs sampling
- Bayesian network meta-analysis
- Probabilistic sensitivity analysis
Example
A Bayesian meta-analysis evaluates the effectiveness of a new anticoagulant.
The prior distribution reflects evidence from previous studies. Data from a new clinical trial are incorporated through the likelihood function.
Following Bayesian updating, the posterior probability that the treatment reduces stroke risk is estimated as 0.96. Posterior samples generated using Markov Chain Monte Carlo are subsequently used to estimate incremental costs, QALYs and the probability that the intervention is cost-effective at a willingness-to-pay threshold of �20,000 per QALY.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| PRODUCT | =B2*C2 | Calculate the unnormalised posterior numerator (likelihood ? prior) |
| SUM | =SUM(D2:D101) | Calculate the normalising constant |
| DIVIDE | =D2/$D$102 | Normalise posterior probabilities after Bayesian updating |
| SUMPRODUCT | =SUMPRODUCT(B2:B101,C2:C101) | Calculate weighted posterior expectations |
| RAND | =RAND() | Generate random values for posterior simulation methods |
VBA (Optional)
Automate Bayesian updating, posterior probability calculation and export of posterior distributions for probabilistic health economic analyses.
Sources
- Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. CRC Press; 2013.
- Bernardo JM, Smith AFM. Bayesian Theory. Wiley; 1994.
- Brooks S, Gelman A, Jones GL, Meng XL, eds. Handbook of Markov Chain Monte Carlo. CRC Press; 2011.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
2
Parameter Estimation and Uncertainty: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6 — Briggs, Weinstein, Fenwick, Karnon, Sculpher & Paltiel, Task Force Report 6 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on parameter estimation and the characterisation of uncertainty in decision models, covering probabilistic sensitivity analysis, distributional choices, and correlation between parameters.
Journal ArticleView source →Bayesian Theory — José M. Bernardo & Adrian F. M. Smith, 1st Edition ed., 1994 (John Wiley & Sons)
A comprehensive theoretical account of Bayesian inference, prior and posterior distributions, probability, information and statistical decision theory.
BookView source →
Frequently Asked Questions (6)
What is posterior probability?
The updated probability of a hypothesis after accounting for observed evidence, combining a prior probability with the likelihood of that evidence.
Source: Bayes 1763
How does evidence change a prior into a posterior?
A posterior probability is what a prior belief becomes once new data have been taken into account. Bayes's theorem combines the prior, the belief held before the evidence, with the likelihood, how probable the observed data are under the hypothesis, to yield the revised probability. Strong evidence pulls the posterior far from the prior, while weak evidence leaves it close to where it started. The posterior is thus the updated state of belief after learning. Spiegelhalter and colleagues (2004) describe this updating.
Source: Spiegelhalter et al. 2004
How is posterior probability calculated?
Posterior probability is calculated using Bayes's theorem, which combines the prior probability of a hypothesis with the likelihood of the observed evidence given that hypothesis. The posterior is proportional to the prior times the likelihood, normalised so the probabilities sum to one across hypotheses. In effect, the prior belief is weighted by how well each hypothesis explains the data. The result is the posterior probability, expressing revised belief about the hypothesis after the evidence has been taken into account.
Source: Bayes 1763
How does posterior probability relate to prior probability?
Posterior probability relates to prior probability as its update in light of evidence: the prior is belief before the data, and the posterior is belief after, obtained by revising the prior using the likelihood of the observed evidence. Strong evidence moves the posterior far from the prior, while with little data the posterior stays close to the prior. The two are connected by Bayes's theorem, so the posterior combines what was believed beforehand with what the data indicate.
Source: Bayes 1763
Why is posterior probability important in Bayesian analysis?
Posterior probability is important because it is the output of Bayesian inference, expressing everything known about a hypothesis or parameter after combining prior information with the data. From the posterior distribution, estimates, credible intervals, and probabilities of interest are derived. Because it integrates prior knowledge and evidence coherently, the posterior provides a full representation of uncertainty for decision making. In health economics, posterior distributions from Bayesian analysis inform parameters and their uncertainty in models.
Source: O'Hagan & Stevens 2001
How is posterior probability used in health economic modelling?
In health economic modelling, posterior probabilities and distributions from Bayesian analysis provide parameter estimates and their uncertainty, combining prior evidence with new data such as trial results. This is useful for synthesising evidence and for representing uncertainty coherently, and posterior distributions can be sampled directly in probabilistic sensitivity analysis. Bayesian methods, yielding posteriors, also connect to value-of-information analysis. Using posterior probability lets a model reflect all available evidence about its parameters and propagate the resulting uncertainty to the results.
Source: O'Hagan & Stevens 2001
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 10 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-MP-032
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