Bayesian Statistics

Bayesian statistics is a branch of statistical inference based on Bayes' theorem that treats probability as a measure of belief or uncertainty rather than as a long run frequency. It updates prior distributions, which represent existing knowledge or beliefs about parameters, with observed data to produce posterior distributions that incorporate both the prior information and the new evidence.

Bayesian methods offer a categorically different framework for statistical reasoning than the frequentist approach that dominates much of conventional research methodology. They allow researchers to formally incorporate prior knowledge into their analyses, make direct probability statements about parameters (rather than about hypothetical repeated samples), and update conclusions as new data accumulate. These properties make Bayesian statistics particularly valuable in adaptive clinical trial designs, meta analysis, health technology assessment, and any research context where prior information is available and relevant. The FDA has issued guidance on Bayesian methods for medical device trials (2010) and draft guidance for trials of drugs and biological products (2026), and their use is growing across the biomedical and social sciences.

The foundation of Bayesian statistics is Bayes' theorem, which states that the posterior probability of a parameter given the data is proportional to the likelihood of the data given the parameter multiplied by the prior probability of the parameter. In mathematical notation: P(theta | data) is proportional to P(data | theta) times P(theta).

The prior distribution, P(theta), encodes what is known or believed about the parameter before the current data are collected. It can be informative (based on previous studies, expert opinion, or biological plausibility) or non informative (designed to have minimal influence on the posterior, allowing the data to dominate). The choice of prior is one of the most important and controversial decisions in Bayesian analysis, because informative priors can substantially influence the results, particularly when the sample size is small.

The likelihood function, P(data | theta), describes how probable the observed data are for each possible value of the parameter. It is the same function used in frequentist maximum likelihood estimation and is determined by the assumed statistical model.

The posterior distribution, P(theta | data), is the output of the Bayesian analysis. It provides a complete probability distribution for the parameter, from which researchers can derive point estimates (such as the posterior mean or median), credible intervals (the Bayesian analog of confidence intervals), and posterior probabilities for specific hypotheses (such as the probability that the treatment effect exceeds a clinically meaningful threshold).

Because the posterior distribution is often mathematically intractable for complex models, computational methods such as Markov chain Monte Carlo (MCMC) simulation are used to approximate it. Software platforms such as Stan, JAGS, and WinBUGS implement these algorithms and have made Bayesian analysis accessible to applied researchers.

Bayesian statistics is appropriate when prior information is available and relevant, when the research question calls for direct probability statements about parameters, when the analysis must incorporate accumulating evidence (as in adaptive trials or living reviews), or when the data structure is too complex for standard frequentist methods. It is particularly useful in small sample studies where prior information can improve precision, in decision analyses that require probability distributions for parameters, and in meta analyses that combine evidence from multiple sources. Bayesian methods are less advantageous when no prior information is available and results are sensitive to the choice of prior, or when transparency about prior selection is difficult to achieve.

A common criticism of Bayesian analysis, and a common source of error in it, is the subjective choice of prior distributions. If the prior is too informative and does not accurately reflect the available evidence, it can bias the posterior in ways that mislead. Researchers should conduct sensitivity analyses using different priors (including non informative priors) to demonstrate that the conclusions are not driven by the prior choice.

Another mistake is interpreting Bayesian credible intervals as if they were frequentist confidence intervals, or vice versa. Although they often produce similar numerical ranges, their interpretations differ: a 95% credible interval means there is a 95% posterior probability that the parameter falls within the interval, whereas a 95% confidence interval means that 95% of such intervals constructed from repeated samples would contain the true parameter. Conflating these interpretations can lead to miscommunication. Finally, applying Bayesian methods using default software settings without understanding the underlying model, the convergence diagnostics, or the implications of the prior can produce unreliable results.

A research team conducts a Bayesian meta analysis of six small randomized trials evaluating a new physiotherapy technique for chronic shoulder pain. Based on two earlier pilot studies and expert opinion, they specify a mildly informative prior for the treatment effect (a normal distribution centered on a 1.5 point reduction in pain score with a standard deviation of 2.0). The posterior distribution, combining the prior with data from the six trials (total n = 240), yields a posterior mean treatment effect of 1.6 points (95% credible interval: 0.3 to 2.9) with a posterior probability of 99% that the true effect exceeds zero. A sensitivity analysis using a non informative prior yields almost the same result, a posterior mean of 1.6 points (95% credible interval: 0.2 to 3.0), confirming that the conclusions are not driven by the prior. The mild prior changes little because the six trials together carry far more information than a prior with a standard deviation of 2.0.