Prior Distribution

Prior Distribution in Bayesian Health Technology Assessment

In a Bayesian health technology assessment, the prior is the analyst's formal statement of what is already known about an uncertain input, such as an adverse event rate or the spread of treatment effects between trials, before the current data are used. As a probability distribution it carries a best guess and its uncertainty, and Bayes' theorem combines it with the likelihood of the data to give the posterior distribution. With plentiful data the prior has little effect, but HTA often works with rare events and networks of few trials, where the prior can move the estimate that enters a cost-effectiveness model. NICE methods set specific expectations for priors in evidence synthesis.

Where a prior comes from

Gelman and colleagues, in Bayesian Data Analysis, give two readings of a prior. In the population reading it describes a population of possible parameter values from which the current one was drawn, as when earlier trials of similar drugs supply a range of plausible effects. In the state of knowledge reading it expresses what is known, and how uncertain that knowledge is.

Spiegelhalter, reviewing Bayesian ideas in health-care evaluation, holds that no prior is correct and adopts the idea of a community of priors, so that an analysis maps a range of stated prior beliefs to the posterior beliefs they imply. His community includes clinical priors elicited from experts, evidence-based priors that summarise earlier studies, reference priors meant as a baseline, sceptical priors centred on no effect and enthusiastic priors centred on the effect a trial was designed to detect. Bayesian analysis sets these choices in the wider workflow.

Matching the distribution family to the parameter

The family of a prior should respect the range and shape of the parameter. The ISPOR-SMDM task force report on parameter estimation states that beta distributions are a natural match for binomial data, gamma or log normal distributions suit right-skewed parameters, and log normal distributions suit relative risks and hazard ratios. It also questions whether convenient but implausible uniform or triangular distributions should have any role in probabilistic analysis. A probability therefore takes a beta distribution and a cost a gamma distribution or log-normal distribution.

Gelman and colleagues call it conjugacy when the posterior stays in the family of the prior, as a beta prior with a binomial likelihood gives a beta posterior. They note that the two beta parameters can be fixed by two features such as the mean and variance, which is the method of moments applied to a published estimate.

$$\alpha = \bar{x}\, k, \qquad \beta = (1 - \bar{x})\, k, \qquad k = \frac{\bar{x}(1 - \bar{x})}{s^2} - 1$$

where $\bar{x}$ is the published mean probability, $s$ is its standard error, $k$ equals $\alpha + \beta$, and $\alpha$ and $\beta$ are the parameters of the beta prior. After $y$ events among $n$ new patients the posterior is $\text{Beta}(\alpha + y,\ \beta + n - y)$, whose mean is a weighted average.

$$E(p \mid y) = \frac{\alpha + y}{\alpha + \beta + n} = \frac{\alpha + \beta}{\alpha + \beta + n} \cdot \frac{\alpha}{\alpha + \beta} + \frac{n}{\alpha + \beta + n} \cdot \frac{y}{n}$$

where $p$ is the event probability, $\alpha / (\alpha + \beta)$ is the prior mean and $y / n$ is the observed proportion. The prior mean is weighted by $\alpha + \beta$ and the data by $n$, so $\alpha + \beta$ reads as the number of patients the prior is worth. The FDA's 2026 draft guidance on Bayesian methods for drugs calls this kind of measure the prior effective sample size, the information in a distribution expressed as an equivalent number of patients in the target population.

Vague, weakly informative and reference priors

A vague prior is meant to let the data speak for themselves, but no prior is neutral on every scale. Gelman and colleagues note that a density flat in one parameterisation is not flat in another, and that the improper Beta(0, 0) prior, flat on the log odds scale, gives an improper posterior if every patient or no patient has the event. They call a prior weakly informative if it is proper but intentionally weaker than the knowledge available, and warn that a uniform prior on the probability of a rare disease can grossly overestimate it when data are weak. Spiegelhalter makes the same point, that apparently innocuous prior assumptions can have a strong impact when events are rare, so a vague prior on a rare adverse event needs as much justification as an informative one.

Priors in evidence synthesis under NICE methods

Network meta-analysis submitted to NICE usually follows the Decision Support Unit's Technical Support Document 2, which recommends vague priors such as $N(0, 100^2)$ for trial baselines and relative treatment effects and says that informative priors for relative effects would require special justification. For the between-trial standard deviation on the log odds scale, TSD 2 gives a uniform prior with an upper limit of 2 as usual practice. With a median odds ratio of 1.5, a standard deviation of 2 lets the true odds ratios of 95% of trials lie between 1.5 × exp(−3.92) = 0.03 and 1.5 × exp(3.92) = 76.

TSD 2 also warns that vague heterogeneity priors behave poorly when trials are few, a problem that Bayesian meta-analysis covers in depth. Where data are thin TSD 2 suggests a prior from a larger meta-analysis of the same outcome or one elicited from a clinician who knows the field. It also notes that priors cannot be totally non-informative, which is why the choice of reference treatment can change posterior estimates although the likelihood is unchanged.

The NICE manual (PMG36, sections 3.4.16 to 3.4.19) follows this line. In networks with few studies an informative prior for the between-study heterogeneity parameter may be preferable, distributions tailored to particular outcomes and disease areas are recommended, the source must be stated and justified, and a sensitivity analysis across candidate priors is expected. Informative priors for relative effectiveness are not recommended except in very specific circumstances, such as very sparse adverse event data.

Borrowing historical evidence and expert judgement

A prior built from earlier studies depends on how far those studies speak to the current question. Spiegelhalter sets out a ladder of assumptions with irrelevance of the historical data at one end, complete pooling of exchangeable patients at the other, and bias models and discounting in between. In discounting, the historical likelihood is raised to a power between 0 and 1, and he reports that Greenhouse and Wasserman downweighted a previous trial of 176 subjects to be equivalent to only 10. He also records the criticism by Eddy and colleagues that this power has no operational interpretation and so no clear way of choosing its value. A hierarchical model instead draws the studies' parameters from a common distribution, so the amount borrowed follows their similarity, and historical controls raise the same question for comparator arms.

The FDA draft guidance on Bayesian methodology for drugs and biological products, issued in January 2026 and still listed as a draft on 28 October 2026, describes a power prior as static, with a discounting factor where 1 weights external and trial patients equally and 0 borrows nothing. Mixture priors, which combine an informative and a noninformative component with weights read as the probability that the external source applies, borrow dynamically. The draft defines prior-data conflict as a prior that places its mass mainly where the observed data would be surprising.

Where there are no data, the prior comes from experts. The NICE manual (PMG36, section 3.3.21) prefers structured elicitation, which should follow an existing protocol such as the Medical Research Council protocol and typically assesses probability distributions after training experts in cognitive biases. Section 4.7.10 points to formal elicitation when data are lacking. Spiegelhalter reports Chaloner's view that fairly simple methods are adequate, including elicitation of 2.5% and 97.5% percentiles from as many experts as possible. Calibration of experts and anchoring bias in elicitation deal with two known weaknesses of such judgements.

Worked example of an adverse event prior built from published data

All numbers in this example are illustrative. A cost-effectiveness model needs the probability $p$ of a serious adverse event on a new drug. A published study in a related population reports a mean probability of 0.10 with a standard error of 0.03, and a small study in the target population has 6 events among 30 patients, a proportion of 0.20.

1. Build the prior. With a mean of 0.10 and a standard error of 0.03, $k = (0.10 \times 0.90)/0.0009 - 1 = 99$, so $\alpha = 0.10 \times 99 = 9.9$ and $\beta = 0.90 \times 99 = 89.1$. The Beta(9.9, 89.1) prior has a 95% central range of 0.049 to 0.166 and is worth 99 patients against 30 in the new study.

2. Check for conflict. The prior predictive probability of 6 or more events among 30 patients, from the beta-binomial distribution, is 0.099, so the new result is unusual under the prior but not extreme.

3. Update with full borrowing. The posterior is $\text{Beta}(9.9 + 6,\ 89.1 + 24) = \text{Beta}(15.9,\ 113.1)$, with mean 15.9/129 = 0.123. The weighted form agrees, since (99 × 0.10 + 30 × 0.20)/129 = 15.9/129. The 95% credible interval is 0.073 to 0.185.

4. Discount the published evidence. Reading Beta(9.9, 89.1) as 9.9 events among 99 patients from a negligible starting prior, a power prior with discounting factor 0.3 multiplies both parameters by 0.3. The result, Beta(2.97, 26.73), keeps the mean of 0.10 but is worth 29.7 patients. The posterior is Beta(8.97, 50.73), with mean (29.7 × 0.10 + 30 × 0.20)/59.7 = 8.97/59.7 = 0.150.

5. Compare a uniform prior. A Beta(1, 1) prior gives a Beta(7, 25) posterior with mean 7/32 = 0.219.

6. Carry it into the model. At GBP 2,000 per event, the expected adverse event cost per patient is 2,000 × 0.123 = GBP 247 with full borrowing, 2,000 × 0.150 = GBP 301 with discounting and 2,000 × 0.219 = GBP 438 with the uniform prior.

7. Interpret. The data are the same in each analysis, yet the input cost rises by about three quarters between full borrowing, where the prior is worth 99 patients, and the uniform prior, worth 2. Which result is defensible turns on how relevant the related population is to the target one, a question about the evidence that a sensitivity analysis can display but not settle.

How the prior reaches the cost-effectiveness decision

The NICE manual (PMG36, section 4.7.10) asks that distributions for probabilistic analysis represent the available evidence, be justified and not be chosen arbitrarily, with separate analyses where alternative plausible distributions exist. In a fully Bayesian analysis the distribution passed to probabilistic sensitivity analysis is a posterior, and the task force report names this as the exception to basing the distribution on the assumption used for a 95% confidence interval. Only when no data exist and the distribution is elicited is the PSA input a prior in the plain sense.

The prior therefore reaches the decision through the posterior means that drive expected net benefit and the posterior spread that drives the value of information. A prior that is too tight understates parameter uncertainty, while one that is too loose on a sparse parameter lets a small study set the mean almost on its own, as step 5 shows. The FDA draft notes that, alongside an informative prior, an analysis under a noninformative prior often helps show how sensitive the results are.

Common misreadings of the prior

Most errors with priors come from treating a modelling choice as neutral or fixed. Five matter most for HTA models.

1. Reading prior as earlier in time. Spiegelhalter quotes Cox that prior refers to what the evidence would have been without the current data, not to time.
2. Treating a flat prior as no prior. A prior flat on the probability scale is not flat on the log odds scale, and with rare events it can pull the estimate upwards.
3. Assuming vague heterogeneity priors are harmless. TSD 2 reports upward bias in the between-trial variance when trials are few.
4. Using an informative prior on relative effects without justification. PMG36 and TSD 2 both treat this as an exception that needs a specific reason.
5. Treating the discounting factor as obvious. Its value is a judgement about relevance, and results should be shown across a range.

Sources

• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 2: A generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Decision Support Unit; August 2011, last updated September 2016. Sections 6.1, 6.2 and 9. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD2-General-meta-analysis-corrected-2Sep2016v2.pdf
• Food and Drug Administration. Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products. Draft guidance for industry, January 2026; draft status checked 28 October 2026. https://www.fda.gov/media/190505/download
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. CRC Press; 2013. Sections 2.4, 2.8 and 2.9. http://www.stat.columbia.edu/~gelman/book/
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 3.3.21, 3.4.16 to 3.4.19 and 4.7.10. https://www.nice.org.uk/process/pmg36
• Spiegelhalter DJ. Incorporating Bayesian ideas into health-care evaluation. Statistical Science. 2004;19(1):156-174. https://doi.org/10.1214/088342304000000080