Posterior Distribution

Posterior Distribution in Evidence Synthesis and Decision Models

In health economics the posterior distribution is what a Bayesian analysis hands on to the decision. It describes how plausible each value of an uncertain input is once the evidence has been absorbed, whether that input is a hazard ratio, a baseline event rate, a mean cost or a utility. Its width and shape hold the parameter uncertainty that a probabilistic cost-effectiveness model propagates, so a poorly formed or poorly summarised posterior distorts expected net benefit and the value of further research. Under a normal approximation for a log hazard ratio and a normal prior, the posterior has a simple closed form that shows how far the data pull away from the prior, and the same logic applies to the simulated posteriors of larger models.

How the posterior is formed from prior and likelihood

A Bayesian analysis starts with a prior distribution for each unknown parameter and a likelihood that says how probable the observed data are for each possible parameter value. Bayes' theorem combines the two into the posterior.

$$p(\theta \mid y) = \frac{p(y \mid \theta)\, p(\theta)}{\int p(y \mid \theta)\, p(\theta)\, d\theta}$$

where $\theta$ is the unknown parameter or vector of parameters, $y$ is the observed data, $p(\theta)$ is the prior density, $p(y \mid \theta)$ is the likelihood, and the integral in the denominator is the marginal probability of the data, a constant that makes the posterior integrate to one. Because the denominator does not depend on $\theta$, the shape of the posterior is fixed by the product of prior and likelihood alone, and most computation works with that unnormalised product.

When a model has several parameters, the posterior is a joint distribution over all of them. A network meta-analysis, for example, yields a joint posterior for the baseline log odds and every relative effect in the network, and the parameters are correlated because they were estimated from the same trials. The marginal posterior of one parameter averages the joint posterior over the others, a distinction that matters when posterior values enter a decision model.

Precision weighting in the normal case

The normal model shows the mechanics most clearly. Gelman and colleagues, in Bayesian Data Analysis, derive the posterior for a normal mean with known variance and a normal prior.

$$\mu_1 = \frac{\mu_0 / \tau_0^2 + y / \sigma^2}{1/\tau_0^2 + 1/\sigma^2}, \qquad \frac{1}{\tau_1^2} = \frac{1}{\tau_0^2} + \frac{1}{\sigma^2}$$

where $\mu_0$ and $\tau_0^2$ are the prior mean and variance, $y$ is the observation, $\sigma^2$ is its known sampling variance, and $\mu_1$ and $\tau_1^2$ are the posterior mean and variance. The inverse of a variance is the precision, and the second equation states that posterior precision equals prior precision plus data precision. The posterior mean is a precision-weighted average of the prior mean and the data, which Gelman and colleagues also write as the data shrunk towards the prior mean, each form showing the posterior as a compromise between the two.

Spiegelhalter, reviewing Bayesian ideas in health-care evaluation, puts the same result in terms that suit trials. A classical estimate of a log odds ratio or log hazard ratio with its standard error is read as a normal likelihood, and the prior is expressed as if it came from a trial of a given size.

$$\theta \mid y \sim N\left( \frac{n_0 \mu + m y}{n_0 + m},\ \frac{\sigma^2}{n_0 + m} \right)$$

where $\theta$ is the true log hazard ratio, $y$ is the estimated log hazard ratio from the trial, $m$ is the equivalent number of events in the trial, $\mu$ is the prior mean, $n_0$ is the number of events the prior is worth, and $\sigma$ is a fixed constant that Spiegelhalter sets at 2 for a log hazard ratio from a trial balanced in recruitment and follow-up. The prior and the data are then weighted by their event counts. In the CHART lung cancer trial, the elicited clinical prior worked out at about 74 deaths' worth of evidence, which puts a number on how much the prior could move the result. The beta distribution page covers the conjugate case for a probability.

When the data dominate and when the prior still matters

Updating can proceed one study at a time. Gelman and colleagues point out that the previous posterior serves as the new prior when more data arrive, so the whole calculation need not be redone. When the studies are independent given the parameter, the result matches a single analysis of all the data, as step 5 of the worked example shows. This is how a decision model can be refreshed when a new trial reports.

As the amount of data grows, the posterior concentrates around the true value and becomes approximately normal, with the likelihood dominating the prior. Gelman and colleagues draw the practical consequence that with large samples little effort is needed to make the prior reflect all available information, while with small samples the prior is a critical part of the model. Health technology assessment often meets the second situation, for example with rare adverse events or networks with few studies.

The large-sample results also fail in identifiable ways. A prior that gives zero probability to a region keeps the posterior out of that region whatever the data show, so Gelman and colleagues advise giving positive prior density to every value that is even remotely plausible. A parameter whose true value lies on the edge of its range, such as a heterogeneity variance of zero, need not acquire a normal posterior even in the limit. A normal approximation can describe almost all of the posterior mass yet still be wrong in the tails, and for a parameter constrained to be positive it admits impossible negative values.

Worked example with a sceptical prior and two trials (illustrative)

All numbers in this example are illustrative. A new treatment is expected to reduce mortality, and its effect is modelled as a log hazard ratio $\theta$ using Spiegelhalter's normal approximation with $\sigma$ equal to 2.

1. Set a sceptical prior. The prior is centred on no effect, $\mu = 0$, and is worth $n_0 = 100$ events. Its standard deviation is $\sigma/\sqrt{n_0} = 2/\sqrt{100} = 0.20$, so the 95% prior range for the hazard ratio is $e^{-0.392} = 0.68$ to $e^{0.392} = 1.48$.

2. Read the first trial as a likelihood. The trial reports a hazard ratio of 0.75 with 200 events. The estimate is $y = \ln 0.75 = -0.2877$ with standard error $\sigma/\sqrt{m} = 2/\sqrt{200} = 0.1414$, and the conventional 95% interval for the hazard ratio is 0.57 to 0.99, which excludes 1.

3. Form the posterior. The posterior mean is $(100 \times 0 + 200 \times (-0.2877))/300 = -57.54/300 = -0.1918$, and the posterior standard deviation is $\sigma/\sqrt{n_0 + m} = 2/\sqrt{300} = 0.1155$. In precision terms, $\tfrac{1}{0.20^2} + \tfrac{1}{0.1414^2} = 25 + 50 = 75$, which equals $\tfrac{1}{0.1155^2}$. The posterior median hazard ratio is $e^{-0.1918} = 0.825$, and the 95% credible interval on the log scale is $-0.1918 \pm 1.96 \times 0.1155$, or -0.418 to 0.035, giving a hazard ratio of 0.66 to 1.04.

4. Read probabilities off the posterior. The posterior probability of benefit is $\Pr(\text{HR} < 1) = \Phi(0.1918/0.1155) = \Phi(1.661) = 0.952$. The probability that the hazard ratio is below 0.8 is $\Phi((-0.2231 + 0.1918)/0.1155) = \Phi(-0.271) = 0.393$. Under a flat reference prior the posterior equals the likelihood, and the probability of benefit would be $\Phi(0.2877/0.1414) = \Phi(2.034) = 0.979$.

5. Update with a second trial. A later trial reports a hazard ratio of 0.80 with 100 events, so $y = \ln 0.80 = -0.2231$. Using the step 3 posterior as the prior, worth 300 events, the new posterior mean is $(300 \times (-0.1918) + 100 \times (-0.2231))/400 = -79.85/400 = -0.1996$, with standard deviation $\sigma/\sqrt{400} = 0.10$. Pooling the original prior with both trials at once gives the same mean. The posterior median hazard ratio is now 0.819, the 95% credible interval 0.67 to 0.996, and the probability of benefit $\Phi(1.996) = 0.977$.

6. Pass the posterior into a model. Suppose a simple model applies the hazard ratio to a one-year death risk of 0.20, so deaths averted per 1,000 patients are $\text{D} = 200\,(1-\text{HR})$, a simplification used here only for illustration. Since the posterior for $\theta$ is normal, the posterior mean hazard ratio is $e^{-0.1996 + 0.10^2/2} = e^{-0.1946} = 0.823$, which gives 200 × 0.177 = 35.4 expected deaths averted. Plugging in the median of 0.819 instead gives 36.2. In a model that is not linear in its inputs, the gap between the output at a point estimate and the expected output can be much larger.

7. Interpret. The first trial alone looks conclusive, yet a reasonable sceptic would still put about a 5% probability on no benefit, and the second trial is needed before the 95% credible interval excludes 1. The posterior, not a single summary of it, is what the decision model should receive.

Using the posterior in cost-effectiveness decisions

A posterior is usually reported through its mean or median, a credible interval and tail probabilities, but a decision model needs the distribution itself. The NICE manual (PMG36, section 4.7.10) asks that distributions for probabilistic analysis represent the available evidence on each parameter and be justified, not 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. Posterior draws are the natural input to probabilistic sensitivity analysis, and the expected net benefit, acceptability curves and value of information calculations that follow are all averages over that posterior.

The joint structure has to survive the hand-over. NICE DSU Technical Support Document 2 requires the posterior samples to be used in the decision model, or the correlations they carry to be propagated through it, when the evidence network has closed loops, because distributions based on posterior marginal summaries are not adequate. When the posterior has no closed form, as in most network and survival models, the draws come from Markov chain Monte Carlo, and Bayesian meta-analysis covers the related predictive distribution for the effect in a new setting.

Checking the posterior before relying on it

A posterior is conditional on the whole model, so its accuracy is no better than the prior and likelihood behind it. Gelman and colleagues argue that a good Bayesian analysis should include at least some check of how well the model fits the data and how plausible it is for its intended use, noting that the likelihood is typically as suspect as the prior. Their framing of sensitivity analysis is how much posterior inferences change when other reasonable models are used. PMG36 (section 4.7.10) takes a similar line for decision models, asking for separate probabilistic analyses when alternative plausible distributions could represent a parameter's uncertainty. In the worked example, the change in the probability of benefit after the first trial, 0.952 under the sceptical prior against 0.979 under the reference prior, is the kind of result such an analysis reports.

Posterior, prior, likelihood and predictive distributions

Four distributions are easily confused. The prior describes belief before the current data, and the likelihood is a function of the parameter for fixed data that is not itself a probability distribution over the parameter. The posterior combines the two. The posterior predictive distribution describes a future observation, not the parameter, and Gelman and colleagues show for the normal model that its variance is the sampling variance plus the posterior variance, so it is always wider than the posterior. A maximum likelihood estimate with its standard error often approximates the posterior mode and spread when the prior is vague and the sample large, which is why frequentist analysis and Bayesian analysis frequently agree in large trials and diverge in small ones.

Common misreadings

Most errors come from forgetting what the posterior is conditioned on or how it was summarised. Each of the following can distort what a decision model receives.

1. Reading a vague prior as no prior. A flat prior on one scale is informative on another, and the posterior inherits that choice.
2. Passing marginal summaries into a model. Independent sampling from each parameter's marginal posterior drops the correlations that TSD 2 says must be kept when the network has closed loops.
3. Exponentiating a mean and calling it a mean. When the posterior for a log hazard ratio is normal, its exponentiated mean is the posterior median hazard ratio, which is smaller than the posterior mean hazard ratio.
4. Treating the posterior as a description of patients. It describes uncertainty about a population quantity, and variation between patients is a separate matter for the model structure.
5. Ignoring what sits outside the model. Structural uncertainty, such as the choice of survival curve, is absent from a posterior unless it is built into the model.

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. Decision Support Unit, 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
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd edition. CRC Press; 2013. Chapters 1, 2, 4 and 6. 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 4.7.10 and 4.7.12. 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