Bayesian Analysis

Bayesian Analysis with Priors, Posteriors and MCMC in Health Technology Assessment

In health technology assessment, a Bayesian analysis treats each unknown input, such as a relative treatment effect, a baseline risk, a cost or a utility, as a quantity with a probability distribution and revises that distribution as trial and other evidence arrive. Its outputs answer decision questions directly, such as the probability that a technology is cost-effective at a given threshold. It matters because NICE prefers the network meta-analysis methods of its Decision Support Unit, whose core document is Bayesian, and because posterior samples can feed a probabilistic cost-effectiveness model. Bayesian network meta-analysis follows NICE DSU TSD 2, and the same posterior samples also support probabilistic sensitivity analysis and value of information calculations. Bayes' theorem and frequentist analysis have their own pages.

Prior, likelihood and posterior

Every Bayesian analysis has three ingredients. The FDA's 2026 draft guidance on Bayesian methods for drugs defines them in terms that serve HTA equally well. The prior is the pre-study probability distribution for model parameters, the likelihood describes the relationship between the parameters and the study data, and the posterior combines the two through Bayes' theorem, as follows.

$$p(\theta \mid y) \propto p(y \mid \theta)\, p(\theta)$$

where $\theta$ is the unknown parameter, $y$ is the observed data, $p(\theta)$ is the prior, $p(y \mid \theta)$ is the likelihood and $p(\theta \mid y)$ is the posterior. Inference rests on posterior summaries such as the posterior mean, a credible interval that contains the parameter with a stated probability under the posterior, and posterior probabilities such as the probability that the effect is greater than zero.

Gelman and colleagues, in Bayesian Data Analysis, describe the posterior as a compromise between prior information and data, with the data in greater control as the sample size grows. When prior and likelihood form a conjugate pair the posterior has a closed form; a beta prior for a probability with binomial data gives a beta posterior, as in the worked example below. Most HTA models have no closed form and need simulation.

Choosing and defending priors

The prior is where a Bayesian analysis is most often challenged, so its source and influence need to be shown. The FDA draft separates noninformative or minimally informative priors, often used as a reference scale for measuring the influence of another prior; sceptical priors, which express scepticism about very large treatment effects; and informative priors, such as one built from earlier trials of the same drug. Its example is the phase 3 trial of REBYOTA for recurrent Clostridioides difficile infection, whose primary analysis used a Bayesian model to incorporate data from an earlier phase 2 placebo-controlled study.

Spiegelhalter, reviewing Bayesian ideas in health-care evaluation, recommends a community of priors, with a sceptical prior centred on no effect alongside an enthusiastic prior. In his trial-monitoring example the posterior from the sceptical prior was emphasised to check whether the evidence would persuade a reasonable sceptic.

For evidence synthesis, TSD 2 recommends vague priors, such as a normal distribution with mean 0 and standard deviation 100, for trial baselines and relative treatment effects, and states that informative priors for relative effects would require special justification. The NICE manual (PMG36, sections 3.4.16 to 3.4.19) agrees that informative priors for relative effectiveness are not recommended except in very specific circumstances such as very sparse adverse event data, while in networks with few studies an informative prior for between-study heterogeneity may be preferable, with its source stated and a sensitivity analysis across candidate priors. Using historical data as a prior often assumes past and current studies are exchangeable, which Spiegelhalter calls a strong assumption; his alternatives model bias or discount past data.

Computing the posterior with Markov chain Monte Carlo

Outside conjugate cases the posterior is approximated by simulation. Markov chain Monte Carlo draws values sequentially, each depending on the last, so that the draws form a Markov chain whose distribution approaches the posterior; Gibbs sampling is one such algorithm. Gelman and colleagues call the discarded early iterations warm-up and assess convergence with the potential scale reduction factor $\hat{R}$, built from between-chain and within-chain variances, which declines to 1 as the chains run longer.

TSD 2 advises running at least three chains from widely different but sensible initial values, keeping both the burn-in and the retained sample conservatively large and reporting both, and cites a guideline that Monte Carlo error should be no more than 5% of the posterior standard deviation. The FDA draft asks sponsors to report burn-in, iterations, chains and convergence diagnostics, and warns that MCMC may not adequately sample part of the posterior. Monte Carlo error is imprecision from a finite run, larger when draws are autocorrelated; more iterations reduce it but do not narrow the posterior.

Bayesian network meta-analysis under NICE DSU TSD 2

The NICE manual (PMG36, section 3.4.11) asks for a network meta-analysis when technologies have not been compared within a single trial and states that NICE prefers the methods in the DSU evidence synthesis series. Its core document, TSD 2 by Dias and colleagues, sets out a generalised linear modelling framework for pairwise and network meta-analysis, takes a Bayesian approach and supplies WinBUGS code for MCMC, with models compared by the deviance information criterion and fit judged by the residual deviance.

TSD 2 explains why this suits decision modelling. Sampling from the posterior serves posterior inference and the Monte Carlo approach to probabilistic modelling at the same time, and it keeps the correlation between parameters estimated from the same data. Where the network has closed loops, the posterior samples should be used in the decision model or the correlations otherwise propagated, because distributions based on posterior marginal summaries are not adequate. For a frequentist analysis, TSD 2 suggests sampling from a multivariate normal distribution built from maximum likelihood estimates and their covariance matrix, which should be approximately the same as the Bayesian posterior. Bayesian meta-analysis covers heterogeneity priors in more depth.

Carrying the posterior into PSA and value of information

A probabilistic cost-effectiveness model needs a joint distribution for its inputs, and a posterior supplies one. Spiegelhalter separates a two-stage approach, in which distributions are produced and then propagated through the model by Monte Carlo simulation, the usual form of probabilistic sensitivity analysis, from an integrated approach that fits evidence synthesis and economic model together by MCMC and so propagates the dependence between unknown quantities. The NICE manual (PMG36, section 4.7.10) requires distributions for probabilistic analysis to represent the available evidence, not to be chosen arbitrarily, and section 4.7.12 states that the committee's preferred cost-effectiveness estimate should come from a probabilistic analysis when possible, unless the model is linear.

With posterior draws of net monetary benefit, the cost-effectiveness acceptability curve plots, for each threshold, the posterior probability that incremental net benefit is positive. The expected value of perfect information is the expected gain from resolving all uncertainty before choosing.

$$\text{EVPI} = E_{\theta}\big[\max(\text{INB}(\theta),\,0)\big] - \max\big(E_{\theta}[\text{INB}(\theta)],\,0\big)$$

where $\text{INB}(\theta)$ is the incremental net benefit of the new technology over its comparator at parameter values $\theta$, and both expectations are over the posterior. When adoption is optimal on current evidence this reduces to Spiegelhalter's form, $E[\max(-\text{INB}(\theta),\,0)]$. Value of information analysis builds research prioritisation on this quantity.

FDA guidance on Bayesian methods and its status on 5 October 2026

For drugs and biological products, the FDA issued Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products as draft guidance in January 2026 (docket FDA-2025-D-3217), and on 5 October 2026 its guidance page still lists it as a draft, not for implementation. It describes success criteria of the form $\Pr(d > a) > c$, where $d$ is the population treatment effect, $a$ is the minimum beneficial effect and $c$ is the minimum posterior probability that supports effectiveness. The thresholds can be calibrated to control the familywise Type I error rate, typically at 0.025 one-sided, read directly as posterior probabilities when the prior accurately summarises existing information, or set by decision-theoretic criteria that minimise expected loss.

For devices, Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials was issued on 5 February 2010 and is still listed as final. It notes that a correctly used Bayesian approach may be less burdensome than a frequentist one, and recommends that sponsors provide the type I and type II error rates of a Bayesian design. Neither document sets HTA rules; a posterior probability of benefit that satisfies a regulator says nothing by itself about cost-effectiveness.

Worked example of a response rate updated and carried into net benefit

All numbers are illustrative. A new treatment's response probability $p$ is uncertain. An earlier phase 2 study is summarised as a Beta(6, 14) prior, with mean 0.30 and standard deviation 0.10, and a new trial records 18 responders among 40 patients.

1. Update the prior. By beta-binomial conjugacy the responders and non-responders are added to the prior parameters.

$$p \mid y \sim \text{Beta}(6 + 18,\ 14 + 22) = \text{Beta}(24,\ 36)$$

where 18 is the number of responders and 22 the non-responders (40 less 18). The posterior mean, 24/60 = 0.40, weights the prior mean 0.30 by 6 + 14 = 20 and the trial proportion 18/40 = 0.45 by 40 to give (20 × 0.30 + 40 × 0.45)/60 = 0.40. The 95% equal-tailed credible interval is 0.281 to 0.526.

2. Convert to net benefit. Each responder gains 0.5 QALYs, the treatment costs an extra GBP 5,000 per patient, and the threshold is GBP 20,000 per QALY.

$$\text{INB}(p) = 20000 \times 0.5 \times p - 5000 = 10000\,p - 5000$$

where $\text{INB}(p)$ is incremental net benefit per patient in GBP, positive only when $p$ exceeds 0.5. Posterior expected INB is 10,000 × 0.40 − 5,000 = −GBP 1,000, so rejection is preferred on current evidence.

3. Read the uncertainty. The posterior probability that $p$ exceeds 0.5, the acceptability curve at GBP 20,000 per QALY, is 0.059. Because rejection is optimal, EVPI is the posterior expectation of max(10,000$p$ − 5,000, 0), which is GBP 16 per patient by numerical integration over Beta(24, 36), matched to the nearest pound by one million Monte Carlo draws.

4. Test the prior. A flat Beta(1, 1) prior gives a Beta(19, 23) posterior with mean 19/42 = 0.452 and 95% credible interval 0.307 to 0.603. Expected INB becomes 10,000 × 0.452 − 5,000 = −GBP 476, the probability of cost-effectiveness 0.266 and EVPI GBP 125 per patient.

5. Interpret. Both priors lead to rejection, but the phase 2 prior pulls the estimate from 0.45 to 0.40 and cuts the per-patient value of further research from GBP 125 to GBP 16. Whether that borrowing is justified turns on the relevance of the phase 2 data, the question the FDA draft asks of any informative prior, and here the prior mean of 0.30 sits well below the trial's 0.45.

Assumptions, boundaries and common misreadings

A posterior is conditional on its prior, likelihood and model structure, so a well-converged MCMC run can still describe the wrong model. TSD 2 notes that priors cannot be totally non-informative, which is why the choice of reference treatment in a network can affect posterior estimates although the likelihood is unchanged.

• A vague prior on one scale can be informative on another. A flat prior on a probability is not flat on the log odds scale.
• Decisions follow expected net benefit. The probability of cost-effectiveness and EVPI describe how uncertain the preferred choice is; they do not replace it.
• Independent sampling of marginal posteriors loses correlation, which TSD 2 calls inadequate when the network has closed loops.
• More iterations shrink Monte Carlo error only, leaving parameter uncertainty unchanged.

Bayesian analysis is the workflow of priors, models, computation and decision use built on Bayes' theorem. Bayesian cost-effectiveness analysis and Bayesian adaptive design apply the same machinery to trial-based economic evaluation and to trial design.

Sources

• 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. University of Sheffield; August 2011, last updated September 2016. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD2-General-meta-analysis-corrected-2Sep2016v2.pdf
• Food and Drug Administration (CDER, CBER). Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products: Draft Guidance for Industry. January 2026, docket FDA-2025-D-3217; draft status checked 5 October 2026. https://www.fda.gov/media/190505/download
• Food and Drug Administration. Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials. Final guidance, 5 February 2010; status checked 5 October 2026. https://www.fda.gov/media/71512/download
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd edition. CRC Press; 2013. Chapters 2 and 11. 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 3.4 and 4.7. 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