Bayesian statistics provides a way to combine existing knowledge with new data to estimate probabilities and account for uncertainty. It is used across medical research, forecasting, and statistical modelling, particularly where previous information can help interpret new observations.
Bayesian statistics also has applications in Bayesian clinical trial design, where prior information and accumulating data can help inform decisions throughout a study.
Bayesian statistics is an approach to statistical analysis that uses probability to represent uncertainty about unknown quantities. Bayesian theory draws on Bayes’ theorem to describe how probabilities can be updated when new evidence becomes available.
This differs from frequentist statistics, which typically treats unknown parameters as fixed values and uses the sampling behaviour of data to quantify uncertainty. Frequentist methods commonly use p-values and confidence intervals, while Bayesian methods use probability distributions to describe uncertainty about the parameters themselves.
Bayesian analysis uses Bayes' theorem to calculate how new evidence should change an existing probability estimate. The calculation involves three main components:
Prior probability
The probability assigned to an event or unknown parameter before considering the new data.
Likelihood
How probable the observed data would be under different possible values of that parameter.
Posterior probability
The updated probability after combining the prior information with the observed data.
In simplified form, Bayes' theorem can be expressed as:
Posterior ∝ Likelihood × Prior
The symbol ∝ means 'proportional to'. The calculation must also account for the overall probability of the observed evidence to produce a valid posterior distribution.
In statistical analysis, the prior and posterior are often expressed as probability distributions rather than single values. A probability distribution describes the range of possible values for an unknown parameter and how probability is assigned across that range.
Bayesian analysis can also be repeated as additional evidence becomes available. The posterior from one analysis can become the prior for the next, provided the model and use of evidence are appropriate.
Medical diagnostic testing provides a practical example of Bayesian reasoning.
A diagnostic test may be highly accurate at identifying a disease, but a positive result does not necessarily mean that a patient has a correspondingly high probability of having that disease. The underlying prevalence of the condition also matters.
Consider a hypothetical screening test for a rare disease that affects one person in every 10,000. Assume the test has a sensitivity of 99%, meaning it correctly identifies 99% of people who have the disease. It also has a false-positive rate of 1%, meaning that 1% of people without the disease receive a positive result. This corresponds to a specificity of 99%.
If 100,000 people are screened under these assumptions:
There would therefore be approximately 1,010 positive results, but only around 10 would represent people who actually have the disease.
Using Bayes' theorem, the probability of having the disease after a positive result is approximately 1%. This happens because the disease is rare. Even with a relatively accurate test, the number of false positives can considerably exceed the number of true positives when prevalence is low.
Note that these figures are illustrative rather than measurements from a real screening programme. They demonstrate why sensitivity and specificity alone are insufficient to interpret a positive diagnostic result.
A posterior distribution describes which parameter values are more or less plausible given the observed data, the prior assumptions, and the statistical model.
Researchers can summarise this distribution using an estimated central value, such as the posterior mean or median, alongside a measure of uncertainty.
A commonly used measure is the credible interval. A 95% Bayesian credible interval identifies a range containing 95% of the posterior probability for a parameter. Under the specified model and assumptions, this can be interpreted as a 95% probability that the parameter lies within that interval.
A frequentist 95% confidence interval has a different interpretation. It is constructed using a procedure that, over repeated samples under the assumed model, would produce intervals containing the fixed parameter value 95% of the time.
Bayesian analysis can also produce predictive distributions. These account for uncertainty in the estimated parameters when describing possible future observations, rather than producing only a single predicted value.
Another application is comparing competing hypotheses or statistical models. One method uses Bayes factors, which measure how strongly the observed data favour one hypothesis over another. A Bayes factor describes the relative support provided by the data and can be combined with prior odds to calculate posterior odds.
Bayesian methods can be particularly useful when reliable prior information is available or when an analysis needs to express uncertainty through probabilities about unknown parameters or future outcomes. The approach can also accommodate more complex models, although its suitability depends on the research question, available data, and assumptions that can be justified.
In medical research, Bayesian methods can help estimate disease prevalence, interpret diagnostic evidence, and investigate relationships between biological or clinical variables.
In forecasting, Bayesian models can combine historical observations with incoming data to estimate possible future outcomes. Rather than providing only a single forecast, they can describe a range of outcomes and the uncertainty associated with them.
Machine learning also uses Bayesian methods for probabilistic classification and prediction. For example, the Naive Bayes classifier applies probability principles to tasks such as identifying whether an email is likely to be spam.
Bayesian methods are also used in finance, economics, and biodiversity research, where analysts may need to account for uncertain relationships or complex data structures.
A Bayesian model is a statistical representation of how observed data relates to unknown parameters, with uncertainty about those parameters expressed through probability distributions.
For example, a regression model might examine the relationship between two variables. A Bayesian version assigns prior distributions to the model parameters and updates them using the observed data.
More complex approaches, such as hierarchical Bayesian models, can analyse related groups while accounting for differences between them. Information can be shared across groups according to the relationships and assumptions specified in the model.
Selecting an appropriate prior distribution is an important part of Bayesian analysis because it determines how existing information influences the estimated parameters.
Priors can be informed by historical clinical data, established scientific knowledge, or reasonable assumptions about the possible values of a parameter.
An informative prior places greater probability on values supported by existing evidence. For example, a researcher with reliable previous estimates of a population parameter may use those estimates to inform the prior distribution.
When reliable information is limited, a weakly informative prior may be more appropriate. This can place broad limits on plausible parameter values without concentrating probability too strongly on a particular estimate.
Diffuse or relatively non-informative priors can also be used, although they are not necessarily free from assumptions or influence.
The choice matters because different priors can produce different posterior distributions, especially when the observed data is limited or provides relatively little information.
As more informative data becomes available, its influence will often increase relative to the prior. However, the extent of this change depends on the statistical model, the strength of the prior, and the information contained in the observations.
Researchers can assess this influence through prior sensitivity analysis, which involves repeating an analysis using alternative plausible prior distributions and examining how much the conclusions change. If the results differ substantially under reasonable alternatives, the conclusions may depend heavily on assumptions that require further justification.
Bayesian statistics has several practical limitations that need to be considered when planning an analysis. These include computational requirements, the suitability of model assumptions, and whether a Bayesian approach is appropriate for the research question.
For relatively simple models, posterior distributions can sometimes be calculated directly. More complex models may require numerical approximation because the necessary calculations cannot be solved analytically.
Markov Chain Monte Carlo (MCMC) is one commonly used technique. It uses simulation to approximate posterior distributions, allowing researchers to analyse models that would otherwise be difficult to calculate. However, these methods can require substantial computing resources, and researchers need to check whether the simulation has adequately explored the distribution.
Bayesian results also depend on the assumptions used in the model, including the prior distribution and the likelihood, which describes how the data is assumed to arise. If these assumptions are inappropriate, the resulting posterior distribution may give a misleading account of uncertainty.
Model checking is therefore an important part of the analysis. Researchers may use prior predictive checks to assess whether their assumptions imply plausible data before fitting the model, and posterior predictive checks to compare model-generated observations with the data actually collected.
A Bayesian model can produce mathematically precise probability statements without necessarily describing the underlying data-generating process adequately.
Frequentist approaches remain appropriate in many settings and may offer simpler analyses when prior information is unavailable or when the research question is well addressed using conventional methods.
Choosing between the approaches requires considering the objective of the analysis, the evidence available, and the assumptions that can reasonably be supported. In clinical trials, the selected methods and their underlying assumptions should be documented in the statistical analysis plan (SAP).
Bayesian statistics offers a practical way to assess evidence and quantify uncertainty, particularly when previous information can inform an analysis. Its usefulness depends on selecting appropriate prior assumptions, checking the statistical model, and interpreting the results in the context of the research question. These considerations help determine when Bayesian methods are suitable and how much confidence can be placed in their findings.
Quanticate’s statistical consultancy team supports sponsors with our statistical planning, modelling, analysis, and interpretation throughout clinical development. Our statisticians can help assess analytical approaches the assumptions behind them, ensuring the chosen methods are appropriate for the research objectives. Request a consultation to discuss your statistical requirements.