Concept Architecture
Concept
Theoretically, Posterior Distribution is the probability distribution of an unknown parameter after observed data have been incorporated through Bayes' theorem. It represents the updated state of knowledge about a parameter by combining prior beliefs with empirical evidence. The posterior distribution is the fundamental inferential quantity in Bayesian statistics and provides a complete probabilistic description of parameter uncertainty.
Mathematically, the posterior distribution is obtained by multiplying the prior distribution by the likelihood function and normalising the result to ensure that the distribution integrates to one. The posterior distribution may take a closed-form solution for conjugate prior?likelihood combinations or require numerical approximation using methods such as Markov Chain Monte Carlo (MCMC). Posterior summaries, including the mean, median, variance, and credible intervals, are derived directly from the posterior distribution.
In practice, posterior distributions are estimated using Bayesian statistical software employing analytical or simulation-based methods. In health economics they are widely used in Bayesian evidence synthesis, network meta-analysis, probabilistic sensitivity analysis, value of information analysis, disease modelling, and health technology assessment, where uncertainty surrounding model parameters is propagated through economic evaluations.
Purpose
Used to update uncertainty about unknown parameters by combining prior information with observed evidence, providing probabilistic estimates that support Bayesian inference, decision modelling, and uncertainty analysis in health economics.
Mathematical Formulae
Primary Formula
p(? | y) = [L(y | ?) ? p(?)] � p(y)
where:
- ? = unknown parameter
- y = observed data
- L(y | ?) = likelihood
- p(?) = prior distribution
- p(y) = marginal likelihood (normalising constant)
Supporting Formulae
p(y) = ? L(y | ?) ? p(?) d?
Posterior ? Likelihood ? Prior
Posterior Mean:
E(? | y) = ? ? ? p(? | y) d?
Posterior Variance:
Var(? | y) = ? [? ? E(? | y)]� ? p(? | y) d?
Related Mathematical Methods
- Bayes' Theorem
- Bayesian Inference
- Prior Distribution
- Likelihood Function
- Marginal Likelihood
- Markov Chain Monte Carlo
- Gibbs Sampling
- Metropolis?Hastings Algorithm
- Bayesian Network Meta-Analysis
- Bayesian Decision Analysis
Example
A health economist estimates the probability that a new intervention prevents hospital admission.
Prior belief:
? ~ Beta(20, 30)
Clinical trial results:
40 patients avoid admission and 10 do not.
Using the conjugate Beta?Binomial model:
Posterior = Beta(20 + 40, 30 + 10)
Posterior = Beta(60, 40)
Posterior mean:
E(? | y) = 60 � (60 + 40)
= 0.60
The updated evidence indicates an estimated 60% probability that the intervention prevents hospital admission, while fully accounting for prior information and observed trial data.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| BETA.DIST | =BETA.DIST(A2,60,40,FALSE) | Evaluate the posterior Beta distribution for a probability parameter. |
| BETA.INV | =BETA.INV(0.975,60,40) | Calculate the upper limit of a 95% Bayesian credible interval. |
| GAMMA | =GAMMA(A2) | Support calculations involving Beta and Gamma distributions. |
| SUMPRODUCT | =SUMPRODUCT(PriorRange,LikelihoodRange) | Combine discrete prior probabilities with likelihood values before normalisation. |
| AVERAGE | =AVERAGE(SimulationRange) | Estimate posterior means from Monte Carlo or MCMC samples. |
VBA (Optional)
Automate Bayesian updating, posterior summary statistics, and credible interval calculations from simulated posterior samples.
Sources
- Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed.
- Bernardo JM, Smith AFM. Bayesian Theory.
- Robert CP. The Bayesian Choice.
- Spiegelhalter DJ, Abrams KR, Myles JP. Bayesian Approaches to Clinical Trials and Health-Care Evaluation.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE Health Technology Evaluation Manual.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
3
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 →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 →The BUGS Book: A Practical Introduction to Bayesian Analysis — David Lunn, Chris Jackson, Nicky Best, Andrew Thomas & David Spiegelhalter, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
A practical guide to Bayesian modeling with BUGS, covering prior specification, posterior simulation, model checking and Markov chain Monte Carlo computation.
BookView source →
Frequently Asked Questions (6)
What is a posterior distribution?
In Bayesian statistics, the updated probability distribution for a parameter after combining a prior distribution with the likelihood of observed data.
Source: Bayes 1763
What does a posterior distribution represent in Bayesian analysis?
A posterior distribution represents everything known about a parameter after the data have been seen, in Bayesian analysis, combining the prior distribution of belief with the likelihood of the observed data. It is a full probability distribution over the possible values, so it directly says how probable each value is given the evidence, and from it point estimates and credible intervals are read. The stronger the data, the more the posterior is shaped by them rather than the prior. Belief about the parameter after the data is what it captures. Sutton and Abrams (2001) describe this.
Source: Sutton & Abrams 2001
How is a posterior distribution obtained?
A posterior distribution is obtained by combining the prior distribution with the likelihood of the data using Bayes' theorem, so that the posterior is proportional to the prior multiplied by the likelihood, then normalised. When this cannot be computed in closed form, methods such as Markov chain Monte Carlo generate samples from the posterior. So a posterior distribution is obtained by updating the prior with the data through Bayes' theorem, which yields the distribution of the parameter given the data, and because exact computation is often infeasible for complex models, computational sampling methods are commonly used to approximate the posterior and summarise it.
Source: Bayes 1763
What does a posterior distribution represent?
A posterior distribution represents the updated state of knowledge about a parameter after observing the data, combining the prior belief with the evidence, so that it captures both what was believed beforehand and what the data indicate. Its spread conveys the remaining uncertainty. So a posterior distribution represents the full probabilistic belief about a parameter given the data and prior, which is why summaries such as the posterior mean or median give point estimates and credible intervals give ranges of plausible values, with the shape and spread of the posterior conveying the uncertainty that remains after the data have been incorporated.
Source: Bayes 1763
How is a posterior distribution used?
A posterior distribution is used to draw Bayesian inferences: point estimates such as the posterior mean or median summarise the parameter, credible intervals give ranges containing it with stated probability, and the posterior can be used for prediction and decision-making. So a posterior distribution is used as the basis for all Bayesian conclusions about a parameter, since it contains the complete updated information, from which estimates, intervals, and probabilities of hypotheses are derived, and it can be propagated into predictions or combined with utilities in decision analysis, making it the central object of Bayesian inference.
Source: O'Hagan & Stevens 2001
How does the prior influence the posterior distribution?
The prior influences the posterior distribution because the posterior combines the prior with the likelihood, so when the data are strong the likelihood dominates and the prior has little effect, but when the data are limited the prior can substantially shape the posterior. So the prior's influence on the posterior depends on the amount of data, being negligible with abundant data and important with sparse data, which is why the choice of prior matters especially when information is limited, and why its effect is often examined through sensitivity analysis, since the posterior reflects both the prior and the data, and an informative prior can shift or narrow it.
Source: Bayes 1763
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-159
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