Concept Architecture
Concept
Theoretically, Prior Probability is the probability assigned to an uncertain event or parameter before the current evidence is observed. Within Bayesian inference, the prior probability represents existing knowledge, previous evidence or expert belief regarding an unknown quantity and serves as the starting point for Bayesian updating. In health economics, prior probabilities are used to incorporate information from previous studies, clinical expertise or historical data before combining this information with new evidence.
Mathematically, the prior probability forms one component of Bayes' theorem, where it is multiplied by the likelihood of the observed data to obtain the posterior probability. Priors may be informative, weakly informative or non-informative, depending on the amount of existing knowledge they are intended to represent. The choice of prior influences posterior estimates, particularly when available data are limited.
In practice, prior probabilities are specified using recognised probability distributions before Bayesian model estimation begins. They are commonly derived from systematic reviews, previous clinical trials, epidemiological studies, registries or expert elicitation. Prior distributions are subsequently updated using observed data through Bayesian estimation methods such as Markov Chain Monte Carlo to obtain posterior distributions used in health economic models.
Purpose
Used to represent existing knowledge or beliefs before analysing new evidence, providing the starting probability distribution for Bayesian estimation and probabilistic health economic modelling.
Mathematical Formulae
Primary Formula
P(?)
where:
- P(?) = prior probability assigned to parameter ?
Supporting Formulae
Bayes' theorem:
P(? | y) = (P(y | ?) P(?)) / P(y)
Posterior proportionality:
P(? | y) ? P(y | ?) P(?)
Related Mathematical Methods
- Bayesian inference
- Bayes' theorem
- Markov Chain Monte Carlo
- Metropolis-Hastings algorithm
- Gibbs sampling
- Expert elicitation
- Bayesian network meta-analysis
- Probabilistic sensitivity analysis
Example
A Bayesian evidence synthesis estimates the treatment effect of a new biologic therapy.
Before analysing the latest clinical trial, previous studies suggest that the log-relative risk is centred near zero. This information is represented by a normal prior distribution with mean 0 and variance 0.25. After combining this prior with the likelihood from the new trial, the posterior distribution is used to estimate treatment effectiveness and populate a probabilistic cost-effectiveness model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| NORM.DIST | =NORM.DIST(A2,0,0.5,FALSE) | Evaluate a normal prior density for a model parameter |
| BETA.DIST | =BETA.DIST(A2,2,8,FALSE) | Evaluate a beta prior for probabilities or utilities |
| GAMMA.DIST | =GAMMA.DIST(A2,3,0.2,FALSE) | Evaluate a gamma prior for positive cost or rate parameters |
| LOGNORM.DIST | =LOGNORM.DIST(A2,1.5,0.4,FALSE) | Evaluate a log-normal prior for relative risks or hazard ratios |
| RAND | =RAND() | Generate random values for Bayesian simulation methods |
VBA (Optional)
Automate assignment of prior distributions to model parameters before Bayesian estimation and probabilistic sensitivity analysis.
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.
- Spiegelhalter DJ, Abrams KR, Myles JP. Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Wiley; 2004.
- 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
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation 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 →
Frequently Asked Questions (6)
What is prior probability?
The probability assigned to a hypothesis before accounting for new evidence, representing existing belief later updated through Bayesian analysis.
Source: Bayes 1763
How much does a prior influence the final result?
How much a prior shapes the conclusion depends on how much data are available to update it. When evidence is plentiful, the data dominate and the choice of prior barely affects the result, but when data are scarce, the prior carries much of the weight and can drive the conclusion. This is why the prior matters most in small studies and why its choice is scrutinised there. With ample data, reasonable priors converge on the same answer. Spiegelhalter and colleagues (2004) discuss this influence.
Source: Spiegelhalter et al. 2004
Where do prior probabilities come from?
Prior probabilities come from information available before the current data: previous studies and evidence, expert judgement, or general knowledge about the plausible values of a quantity. A prior may be informative, expressing substantial prior belief, or vague, expressing little, so that the data dominate. The choice of prior reflects the state of knowledge before the analysis. Because priors can influence the results, especially with limited data, their source and justification are stated so their effect can be examined.
Source: Bayes 1763
How does the prior influence Bayesian results?
The prior influences Bayesian results by contributing, alongside the data, to the posterior: the posterior is the prior updated by the likelihood, so a strong prior pulls the posterior toward itself, while with abundant data the likelihood dominates and the prior matters less. When data are limited, the prior can substantially affect the conclusion. This influence is why the choice of prior is made explicit and its effect tested, for instance by comparing results under different priors, to show how far conclusions depend on it.
Source: O'Hagan & Stevens 2001
What is the difference between informative and vague priors?
An informative prior expresses substantial prior belief about a quantity, concentrating probability on particular values based on previous evidence or expert judgement, and it influences the posterior notably when data are limited. A vague, or diffuse, prior expresses little prior information, spreading probability broadly so that the data largely determine the posterior. The choice affects how much the prior shapes the result, so an informative prior is used when good prior evidence exists, and a vague prior when the analysis should rest mainly on the data.
Source: O'Hagan & Stevens 2001
Why is the choice of prior sometimes controversial?
The choice of prior is sometimes controversial because it introduces information beyond the current data, which can influence the results, and different analysts might choose different priors, raising concerns about subjectivity, especially in analyses informing public decisions. Critics argue that priors can bias conclusions, while proponents note that priors make assumptions explicit and can incorporate genuine prior evidence. To address this, priors are stated clearly and their influence examined through sensitivity to alternative priors, so that the effect of the prior on the conclusion is transparent.
Source: Bayes 1763
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
- Persistent URI
- https://healtheconomics.wiki/concept/prior-probability
- Term code
- HE-EM-MP-033
Stable URI · Machine-readable · Resolvable · CC BY 4.0