Concept Architecture
Concept
Theoretically, Prior Distribution is a probability distribution that represents existing knowledge or beliefs about an unknown parameter before current data are observed. It is a fundamental component of Bayesian inference, where uncertainty regarding model parameters is expressed probabilistically. Prior distributions may be derived from previous studies, expert elicitation, historical evidence, or objective statistical principles, and they provide the starting point for Bayesian updating through Bayes' theorem.
Mathematically, a prior distribution assigns probabilities or probability densities to all plausible values of a parameter before analysing the observed data. The prior is combined with the likelihood function to produce the posterior distribution. Depending on the analytical context, priors may be informative, weakly informative, non-informative, or conjugate, with conjugate priors allowing closed-form posterior solutions for particular likelihood functions.
In practice, prior distributions are specified before model estimation and are incorporated into Bayesian statistical models using analytical or computational methods such as Markov Chain Monte Carlo (MCMC). In health economics they are widely applied in Bayesian evidence synthesis, network meta-analysis, probabilistic sensitivity analysis, value of information analysis, disease modelling, and health technology assessment to integrate previous evidence with new clinical and economic data.
Purpose
Used to represent existing uncertainty about unknown parameters before analysing current evidence, enabling Bayesian updating, evidence synthesis, probabilistic modelling, and decision-making under uncertainty in health economics.
Mathematical Formulae
Primary Formula
p(?)
where:
- ? = unknown model parameter
Supporting Formulae
Bayes' theorem:
p(? | y) = [L(y | ?) ? p(?)] � p(y)
Posterior ? Likelihood ? Prior
Examples of common prior distributions:
? ~ N(?, ��)
? ~ Beta(�, ?)
? ~ Gamma(�, ?)
? ~ Uniform(a, b)
Related Mathematical Methods
- Bayesian Inference
- Bayes' Theorem
- Posterior Distribution
- Likelihood Function
- Conjugate Prior
- Markov Chain Monte Carlo
- Gibbs Sampling
- Metropolis?Hastings Algorithm
- Bayesian Network Meta-Analysis
- Expert Elicitation
Example
A health economist estimates the probability that a new intervention prevents hospital admission.
Previous studies suggest moderate effectiveness, leading to the prior:
? ~ Beta(20, 30)
The prior mean is:
E(?) = � � (� + ?)
= 20 � (20 + 30)
= 0.40
After new trial data become available, this prior is combined with the likelihood to produce the posterior distribution, reflecting both previous evidence and current observations.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| BETA.DIST | =BETA.DIST(A2,20,30,FALSE) | Evaluate a Beta prior for a probability parameter. |
| NORM.DIST | =NORM.DIST(A2,0,1,FALSE) | Evaluate a Normal prior for continuous parameters. |
| GAMMA.DIST | =GAMMA.DIST(A2,3,2,FALSE) | Evaluate Gamma priors for rate or variance parameters. |
| SUMPRODUCT | =SUMPRODUCT(PriorRange,LikelihoodRange) | Combine discrete priors with likelihood values before normalisation. |
| AVERAGE | =AVERAGE(SimulationRange) | Summarise sampled values from simulated prior distributions. |
VBA (Optional)
Automate specification of prior distributions and generate Bayesian parameter inputs for probabilistic health economic models.
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
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
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 prior distribution?
In Bayesian statistics, the probability distribution representing existing beliefs about a parameter before observing new data.
Source: Bayes 1763
What does a prior distribution express before data are seen?
A prior distribution expresses what is believed about a parameter before the new data are seen, in Bayesian analysis, stated as a probability distribution over its possible values. It may be informative, encoding real prior knowledge from earlier studies, or vague, deliberately spread out to let the data dominate. Combined with the likelihood of the observed data, it yields the posterior, so the prior's influence fades as the data grow more plentiful. Belief about the parameter before the evidence is what it captures. Sutton and Abrams (2001) describe this.
Source: Sutton & Abrams 2001
What are informative and vague priors?
An informative prior reflects substantial prior knowledge about a parameter, concentrating probability on particular values and influencing the posterior appreciably, while a vague, or weakly informative, prior reflects little prior knowledge, spreading probability broadly so that the data dominate the posterior. So priors range from informative to vague according to how much prior knowledge they encode, with informative priors strengthening inference when justified but potentially biasing it if inappropriate, and vague priors letting the data speak, which is why the type of prior is chosen deliberately and its influence on the posterior considered, especially when data are limited.
Source: Bayes 1763
How does the choice of prior affect Bayesian analysis?
The choice of prior affects Bayesian analysis because the posterior combines the prior with the data, so when data are strong the prior has little effect, but when data are limited the prior can substantially shape the posterior and hence the conclusions. So the prior's influence depends on the amount of data, being important with sparse data and negligible with abundant data, which is why the prior is chosen carefully and its effect often examined through sensitivity analysis, since an informative prior can strengthen or bias inference, and reporting how the prior affects the results supports transparency in Bayesian analysis.
Source: Bayes 1763
How is a prior distribution chosen?
A prior distribution is chosen based on existing knowledge, previous studies, expert opinion, or the desire to remain neutral, with informative priors used when credible prior information exists and vague priors when it does not or when letting the data dominate is preferred. So a prior distribution is chosen according to the available prior knowledge and the goals of the analysis, and the choice is made carefully because it can influence the posterior, which is why priors are justified and their sensitivity examined, and why weakly informative priors are often used to incorporate mild, defensible assumptions without imposing strong beliefs that the data cannot override.
Source: Bayes 1763
How does a prior distribution relate to the posterior distribution?
A prior distribution relates to the posterior distribution as the starting belief that the data update: the posterior is obtained by combining the prior with the likelihood of the data through Bayes' theorem. The prior represents belief before the data, and the posterior belief afterward. So the prior and posterior are the before and after of Bayesian updating, connected by the likelihood, with the posterior reflecting both the prior and the data, which is why the prior matters most when the data are weak, and why the movement from prior to posterior expresses how much the data have changed belief about the parameter.
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
- Persistent URI
- https://healtheconomics.wiki/concept/prior-distribution
- Term code
- HE-ES-SA-167
Stable URI · Machine-readable · Resolvable · CC BY 4.0