Bayesian Concepts

Bayesian inference combines a prior belief with the likelihood (what the data say) to give a posterior probability — it directly answers “how probable is the hypothesis, given the data?”, which a frequentist p-value does not. A Bayes factor quantifies how strongly the data favour one hypothesis over another; a credible interval is the Bayesian analogue of a confidence interval (the range the parameter lies in with, say, 95% probability). Everyday diagnosis — pre-test probability updated by a likelihood ratio into a post-test probability — is Bayesian updating.

Belief does not appear from nowhere — it is a starting point (the prior) moved by evidence. Here a 20% prior, updated by data carrying a likelihood ratio of 10, becomes a 70% posterior. Change the prior and the same data land somewhere else; that dependence on the prior is the heart of the method — and its main controversy.

Fix the prior on the left, pivot the line through the strength of the evidence (the likelihood) in the middle, and read the posterior on the right. The posterior is a genuine probability that the hypothesis is true. A Bayes factor is just the ratio that does the pivoting; a credible interval is the posterior range. Strong data (LR far from 1) swing the line a long way; weak data (LR near 1) barely move it.

It is the way clinicians already think — gestalt is the prior, the test result is the data, the updated suspicion is the posterior. It also rescues a “negative” trial: a frequentist p > 0.05 says only “not statistically significant”, whereas a Bayesian posterior can report there is, say, a 95% probability the treatment helps — a far more useful statement for a decision at the bedside.

• posterior ∝ prior × likelihood
• Bayes factor = strength of evidence for one hypothesis vs another
• Credible interval = Bayesian analogue of a confidence interval

EOLIA (NEJM 2018) — early ECMO vs conventional management in very severe ARDS. 60-day mortality 35% vs 46% (RR 0.76), but the difference was not statistically significant (p = 0.09) and the trial was stopped for futility. A pre-planned Bayesian re-analysis (Goligher et al., JAMA 2018) found a 88–99% posterior probability that ECMO reduces mortality across a range of priors — a far more decision-useful answer than “non-significant”. The posterior depended on the prior: a strongly sceptical prior pulled the probability of benefit down. Same data, different prior, different conclusion — pre-specify the prior.

• Prior sensitivity — forgetting the posterior depends on the (subjective) prior chosen.
• Reading a p-value as a posterior — p is not the probability the hypothesis is true.
• Confusing a credible interval (Bayesian, a probability statement) with a confidence interval (frequentist).