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Bayesian biostatistics uses Bayes' theorem to update prior beliefs about parameters with observed data to produce posterior distributions: P(theta|data) proportional to P(data|theta) × P(theta). Unlike frequentist methods, which report what the data tell you about the probability of the data under a fixed parameter, Bayesian methods report the probability of the parameter given the data — the quantity clinicians actually want. The posterior distribution provides a complete summary of uncertainty: point estimates (posterior mean or median), interval estimates (credible intervals that have a direct probabilistic interpretation — "there is a 95% probability that the treatment effect lies in this interval"), and direct probability statements about hypotheses ("the probability that the treatment is effective is 0.93"). The choice of prior distribution is both the method's greatest strength (incorporating existing knowledge) and its primary source of controversy (potential subjectivity).
Frequentist statistics — the framework underlying p-values, confidence intervals, and hypothesis tests — dominates biostatistics training and practice. But it answers questions in a circuitous way. A p-value of 0.03 tells you: if the null hypothesis were true, there would be only a 3% chance of observing data this extreme. It does not tell you the probability that the null hypothesis is true or that the treatment works. Bayesian statistics, by contrast, directly computes the probability of hypotheses and parameter values given the observed data. This directness comes at a cost: you must specify a prior distribution reflecting what was known or believed before the data were collected.
Bayes' theorem provides the machinery: posterior ∝ likelihood × prior. The likelihood is the same function used in frequentist analysis — it captures what the data say about the parameter. The prior encodes pre-existing knowledge: previous studies, biological plausibility, or deliberate skepticism. The posterior combines both, representing updated knowledge after seeing the data. With large samples, the data dominate and the posterior is insensitive to the prior. With small samples, the prior matters more — which is either a feature (incorporating legitimate knowledge) or a concern (importing unjustified assumptions), depending on the quality of the prior information.
The practical outputs of Bayesian analysis are more intuitive than their frequentist counterparts. A credible interval has a direct probabilistic interpretation: "there is a 95% probability that the treatment effect lies between 3 and 12 units." A posterior probability answers clinical questions directly: "the probability that this treatment reduces mortality by at least 5% is 0.87." These statements are what clinicians naturally want but what frequentist inference cannot provide without additional assumptions.
Bayesian methods are increasingly used in clinical trials, particularly adaptive designs where the trial is modified based on accumulating data. Because the posterior updates continuously, Bayesian monitoring does not suffer from the multiple testing penalties that plague frequentist interim analyses. The FDA has endorsed Bayesian methods for medical device trials, and Bayesian adaptive platforms are becoming standard in oncology. Computational advances — particularly Markov Chain Monte Carlo (MCMC) methods implemented in software like Stan, BUGS, and JAGS — have made complex Bayesian models practical, overcoming the analytical intractability that historically limited Bayesian applications to simple problems.