Beta Distribution

Beta Distribution for Uncertain Probabilities and Utilities in Decision Models

In a probabilistic sensitivity analysis, every uncertain model input is given a distribution, and the beta distribution is the usual choice for inputs that must stay between zero and one, above all the probabilities that drive decision trees and Markov models. Because its two shape parameters can be read as counts of events and non-events, it can be fitted directly from trial or registry data, or from a published mean and standard error. Both fitting routes, and the link to Bayesian updating of binomial data, can be compared on an illustrative progression probability. The beta distribution is a poor choice for utilities that can fall below zero, and a row of multinomial transitions needs the Dirichlet distribution instead of separate betas.

Shape, mean and variance on the unit interval

The beta distribution is defined for values strictly between zero and one, which matches the range of a probability. Two positive parameters control its location and spread, so the same family can be centred near zero for a rare complication or near one for a high survival probability. Its density is as follows.

$$f(p \mid \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\,\Gamma(\beta)}\, p^{\alpha - 1} (1 - p)^{\beta - 1}, \quad 0 < p < 1$$

where $p$ is the uncertain probability, $\alpha$ and $\beta$ are the two positive shape parameters, and $\Gamma(\cdot)$ is the gamma function, which makes the density integrate to one. The mean and variance are as follows.

$$E(p) = \frac{\alpha}{\alpha + \beta}, \qquad \operatorname{Var}(p) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$$

where $E(p)$ is the expected value of the probability and $\operatorname{Var}(p)$ its variance. The mean depends only on the ratio of the two parameters, while their sum, $\alpha + \beta$, acts like a sample size, and the larger it is, the narrower the distribution. Statistical texts describe the two parameters as prior sample sizes for this reason. When both parameters equal one the distribution is uniform, and when both are below one it is U-shaped, with most of its weight near zero and one.

Why decision models sample probabilities from a beta

A decision model needs a mean for each input to produce a point estimate, and a distribution for each input to show how uncertain that estimate is. The distribution describes parameter uncertainty, the imprecision in the mean, not variation between patients. The ISPOR-SMDM modelling task force on parameter estimation and uncertainty describes beta distributions as a natural match for binomial data. It pairs gamma or log-normal distributions with right-skewed quantities such as costs, and log-normal distributions with relative risks and hazard ratios.

A probability sampled from a normal distribution can fall below zero or above one when the mean is close to a boundary or the evidence is thin, which breaks the model logic. A beta draw cannot. In NICE's manual for technology appraisal (PMG36, section 4.7), the distributions used in probabilistic analysis should not be chosen arbitrarily but should represent the available evidence on each parameter, and their use should be justified. The task force report adds that a published analysis should state each distribution and its parameters and give the reasons for the choice.

Fitting from event counts and the link to Bayesian updating

When the evidence is a count of patients who experienced an event out of a known number at risk, the parameters can be set directly from the data. With $r$ events among $n$ patients, a common choice is $\alpha = r$ and $\beta = n - r$, so the mean is the observed proportion $r/n$ and the spread reflects the sample size.

This rule has a Bayesian interpretation. The beta distribution is the conjugate prior distribution for the probability in a binomial distribution. If the prior is beta and the data are binomial, the posterior distribution is also beta.

$$p \mid r \sim \text{Beta}(\alpha_0 + r,\ \beta_0 + n - r)$$

where $\alpha_0$ and $\beta_0$ are the prior parameters, $r$ is the number of events and $n$ the number of patients at risk. Setting $\alpha = r$ and $\beta = n - r$ corresponds to the limiting prior with $\alpha_0 = \beta_0 = 0$; a uniform prior, $\alpha_0 = \beta_0 = 1$, adds one to each parameter. With large counts the choice of prior hardly matters. With small or zero counts it does. A study with no deaths among 120 patients gives $\alpha = 0$, which is not a valid beta distribution, whereas a uniform prior gives Beta(1, 121), with a mean of about 0.008. The conjugate form also allows evidence to be combined in a Bayesian analysis, for example when a registry count is added to a trial count for the same population.

Fitting by the method of moments from a mean and standard error

Published studies often report a proportion with a standard error or confidence interval rather than raw counts. In that case the parameters have to be recovered from the summary statistics. The method of moments sets the beta mean and variance equal to the reported values and solves for the two parameters, as follows.

$$\alpha + \beta = \frac{\mu (1 - \mu)}{s^2} - 1, \qquad \alpha = \mu (\alpha + \beta), \qquad \beta = (1 - \mu)(\alpha + \beta)$$

where $\mu$ is the reported mean probability and $s$ is its standard error. The solution exists only when $s^2 < \mu (1 - \mu)$; otherwise the sum is negative and no beta distribution matches. When the standard error is large relative to the mean, one or both fitted parameters can fall below one, piling the distribution up against a boundary, which rarely reflects what the evidence says. The input must be the standard error of the mean, not the standard deviation across patients. Using the patient-level standard deviation describes variation between individuals and gives a distribution far too wide for parameter uncertainty.

Worked example of a one-year progression probability

The figures below are illustrative. A cohort Markov model with a one-year cycle needs the probability that a patient in a stable state progresses within a year. Two sources are available, a cohort in which 24 of 120 patients progressed, and a published estimate of 0.20 with a standard error of 0.05.

1. Fit from the counts. With $\alpha = 24$ and $\beta = 120 - 24 = 96$, the mean is $E(p) = 24/120 = 0.20$. The variance is as follows.

$$\operatorname{Var}(p) = \frac{24 \times 96}{120^2 \times 121} = \frac{2304}{1742400} = 0.00132$$

where the terms follow the variance formula above. The standard error is the square root, about 0.036. The central 95% of the distribution comes from the inverse cumulative function. In Excel, BETA.INV(0.025, 24, 96) returns about 0.134 and BETA.INV(0.975, 24, 96) about 0.276.

2. Fit from the published mean and standard error. With $\mu = 0.20$ and $s = 0.05$, the sum of the parameters is as follows.

$$\alpha + \beta = \frac{0.20 \times 0.80}{0.05^2} - 1 = \frac{0.16}{0.0025} - 1 = 63$$

where the terms are as defined for the method of moments. So $\alpha = 63 \times 0.20 = 12.6$ and $\beta = 63 \times 0.80 = 50.4$, giving Beta(12.6, 50.4). Its central 95% runs from BETA.INV(0.025, 12.6, 50.4), about 0.112, to BETA.INV(0.975, 12.6, 50.4), about 0.306. The two fits share a mean, but the second behaves like evidence from about 63 patients rather than 120, so it is wider.

3. Draw values for the simulation. In a spreadsheet, each Monte Carlo simulation iteration can draw a value with the Excel formula BETA.INV(RAND(), 24, 96), which applies the inverse cumulative beta function to a uniform random number. Excel returns an error when either shape parameter is zero or negative, which is the visible symptom of a failed method-of-moments fit.

4. Interpret. Whichever fit is used, every sampled transition probability lies between zero and one. Because the stable state here has two exits, progressing or staying, the probability of staying is one minus the sampled value, so the cohort is conserved in every iteration. States with three or more exits need the joint approach set out in the Dirichlet section below. Choosing between the fits is a judgement about which evidence better represents the decision population, and the model report should state it.

Utilities and where the beta fits

A health state utility value has an upper bound of one, but on many preference-based instruments an individual patient's value can fall below zero for states valued as worse than dead. The NICE Decision Support Unit's Technical Support Document 12 notes that such data are often skewed, bounded and include negative values, and gives a lower limit of minus 0.594 for the UK EQ-5D-3L index. In a probabilistic analysis, however, the distribution describes uncertainty in the mean utility, not the spread of patient values. The relevant test is therefore whether the plausible range of the mean reaches zero. A beta distribution on the zero to one interval suits a mean whose plausible range sits well inside that interval, even when some patients report negative values.

The same document states that, in most cases, uncertainty in a mean utility can be described adequately by a normal distribution. Where it cannot, it suggests describing utilities as decrements from full health, one minus the utility, and sampling the decrement from a log-normal or gamma distribution, which keeps every sampled utility at or below one with no lower limit. Where a lower limit is needed, it suggests a standard beta distribution rescaled to a wider interval.

For an illustrative severe state with a mean utility of 0.10 and a standard error of 0.08, the beta fit by moments is Beta(1.31, 11.76), which places all its weight above zero and has a median of about 0.08. Fitting a gamma distribution to the disutility instead, with mean 0.90 and the same standard error, gives a shape of $k = 0.90^2 / 0.08^2 = 126.6$ and a scale of $\theta = 0.08^2 / 0.90 = 0.0071$. About 11% of the sampled mean utilities then fall below zero. Which representation is right depends on whether a mean utility below zero is plausible for this state, given the evidence and the value set.

Multinomial transitions and the Dirichlet distribution

Many model states have more than two exits in a cycle. A patient in a stable state may stay stable, progress or die, and the three probabilities must add to one in every iteration. Sampling each from its own beta distribution breaks this rule. In the illustrative cohort above, suppose 90 patients stayed stable, 24 progressed and 6 died. Independent draws from Beta(90, 30), Beta(24, 96) and Beta(6, 114) add to 1.00 on average, but in an illustrative simulation of 200,000 iterations the sum fell outside the range 0.95 to 1.05 in about 38% of draws.

The Dirichlet distribution is the multivariate generalisation of the beta distribution and the conjugate prior for multinomial data. Briggs, Ades and Price showed how it gives a fully probabilistic transition matrix for a Markov model and how a Bayesian prior avoids problems with zero counts. It can be sampled by drawing independent gamma variables with unit scale and normalising them, as follows.

$$p_j = \frac{G_j}{\sum_{k=1}^{K} G_k}, \qquad G_j \sim \text{Gamma}(a_j, 1)$$

where $p_j$ is the sampled probability of moving to destination $j$, $K$ is the number of possible destinations, $G_j$ is a gamma draw and $a_j$ is the count (plus any prior) for that destination. For the example, one illustrative set of gamma draws of 86.4, 25.8 and 5.0 adds to 117.2. Dividing each draw by that total gives probabilities of 0.7372, 0.2201 and 0.0427, which add to one. Each component of a Dirichlet has a beta marginal. The progression probability from Dirichlet(90, 24, 6) is Beta(24, 96), the distribution fitted in step 1 of the worked example. For a chance node with only two branches, the Dirichlet reduces to a single beta, with the second probability taken as one minus the first.

Choices that change the result

Once the beta family is chosen, the results depend on the choices made in fitting it. The evidence source matters most. Counts from a trial in a different population, or a standard error from a small study, can shift both the mean and the width, and NICE's manual expects alternative plausible distributions to be explored in separate probabilistic analyses.

Correlation is a second choice. Probabilities sampled independently ignore any relationship between them, such as the link between baseline risk and risk on treatment. The ISPOR-SMDM task force suggests defining parameters so that independence is plausible, for example sampling a baseline probability and a relative effect separately and deriving the treated probability from them. Where multiplying a baseline probability by a relative risk can give a value outside the zero to one range, the task force suggests switching to odds ratios. PMG36 asks analysts to reflect evidence on correlation and to avoid both artificial restriction of distributions and unsupported assumptions of perfect correlation between ordered parameters.

Time period is a third. When the evidence covers a different period from the model cycle, the distribution is fitted on the observed period and each sampled value is converted to the cycle length through rate-to-probability conversion.

Where the beta distribution is misapplied

Two errors recur in applied models. The first is fitting a beta distribution to a quantity that is not bounded by zero and one, such as a cost, a relative risk or a hazard ratio, for which gamma or log-normal distributions are the usual choices. The second is using the patient-level standard deviation in the method of moments, which treats variation between patients as uncertainty in the mean.

A last misunderstanding concerns what the distribution represents. The beta describes uncertainty about a population probability, so its width should shrink as evidence accumulates. It does not describe how individual patients differ. Outputs computed across iterations, such as the cost-effectiveness acceptability curve and the expected value of perfect information, are only as credible as the distributions behind them.

Sources

• Ara R, Wailoo A. NICE DSU Technical Support Document 12: The use of health state utility values in decision models. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; July 2011. Section 4.1.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford: Oxford University Press; 2006. Chapter 4.
• Briggs AH, Ades AE, Price MJ. Probabilistic sensitivity analysis for decision trees with multiple branches: use of the Dirichlet distribution in a Bayesian framework. Medical Decision Making. 2003;23(4):341-350.
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6. Value in Health. 2012;15(6):835-842.
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. Boca Raton: CRC Press; 2013. Section 2.4 and Appendix A.
• Microsoft. BETA.INV function. Microsoft Support; accessed 1 October 2026. https://support.microsoft.com/en-gb/office/beta-inv-function-e84cb8aa-8df0-4cf6-9892-83a341d252eb
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Section 4.7.