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Bayesian statistics for clinicians

You already reason in priors and posteriors. The formalism just makes it honest.

The bedside intuition

Before the troponin comes back, you already know something. A 78-year-old with crushing chest pain and a smoking history does not walk into the department with the same probability of infarction as a 23-year-old with a sore rib, and no clinician pretends otherwise. That number you carry before the test — the pretest probability — is a prior.

The test result does not replace it; it updates it. Bayesian statistics is nothing more exotic than that update, written down honestly: what you believed before, what the data were worth, and what you are entitled to believe now. Clinicians have been doing this at the bedside for as long as there have been bedsides. The formalism's only demand is that the prior be stated out loud instead of smuggled in.

What a posterior answers that a p-value does not

A p-value answers a question nobody at the bedside is asking: how surprising would data like these be, if the treatment did nothing? The question the clinician actually has is the inverse: given these data, how likely is it that the treatment helps? Those two are not the same question, and no amount of squinting turns one into the other. A posterior probability answers the second question directly — and it answers it on a scale a human can act on. "p = 0.04" and "there is a 97% probability the treatment improves outcomes" feel similar and are profoundly different objects; only one of them is a statement about the treatment. The cost of the honest answer is that you must declare what you believed beforehand. That is not a weakness of the method. It is the method.

A trial every reader trusts: DAWN

If posterior probabilities sound like a novelty, consider that one of the most practice-changing trials in stroke was Bayesian from the ground up. DAWN (Nogueira RG, et al. N Engl J Med 2018; doi:10.1056/NEJMoa1706442; NCT02142283) tested thrombectomy 6 to 24 hours from stroke onset using a Bayesian adaptive-enrichment design: prespecified priors, interim looks that could adapt enrolment, and a stopping rule written in posterior terms. It was stopped early on a prespecified interim analysis, and its primary results are stated in posterior terms — a posterior probability of superiority above 0.999 — with 95% credible intervals, the Bayesian object that means what most readers already think a confidence interval means. Every physician who takes a patient to the angiography suite in the late window is acting on Bayesian evidence, and has been since 2018, whether or not anyone called it that on rounds.

What a reanalysis can add: EOLIA

The other direction matters more. EOLIA, the randomised trial of early ECMO in very severe ARDS (Combes A, et al. N Engl J Med 2018; doi:10.1056/NEJMoa1800385), "failed" by the conventional reading: 60-day mortality 35% against 46%, relative risk 0.76 (95% CI 0.55–1.04), p = 0.09 — not significant. Goligher and colleagues reanalysed it within a Bayesian framework (JAMA 2018; PMID 30347031), asking the question the original analysis could not: given these data, what is the probability that ECMO reduces mortality? The answer — the posterior probability of a relative risk below 1 — ran from 88% under a strongly sceptical prior to 96% under a minimally informative one and 99% under enthusiastic or evidence-informed priors. The data had not changed by one patient. What changed was the honesty of the question. A "negative" trial is very often not a demonstration of no effect; it is a quantifiable probability of effect that the significance test was never built to express.

Bayesian statistics for clinicians

Same trial, three priors, one honest answer.

One trial, one dataset, three defensible starting positions. The p-value returned not significant; the reanalysis asks instead how much of the posterior for the relative risk lies below one, and answers it under each prior.

EOLIA (ECMO in severe ARDS): 60-day mortality 35% against 46%, relative risk 0.76 (95% CI 0.55–1.04), p = 0.09 — not significant.

Bayesian reanalysis: Goligher and colleagues, JAMA 2018.

Probability that ECMO reduces mortality: 96 %

the posterior probability of a relative risk below 1, under a minimally informative prior

  • Prior
  • Likelihood — the trial data, fixed
  • Posterior
Prior, likelihood and posterior for EOLIA under a minimally informative prior.A relative-risk axis from 0.4 to 1.4 with a dashed line at 1. Three schematic curves: a minimally informative prior, the fixed likelihood centred on the trial estimate of 0.76, and the posterior that results. Below the axis, the EOLIA estimate of 0.76 with its 95% confidence interval from 0.55 to 1.04. The reanalysis reports a posterior probability of 96 % that the relative risk is below 1 under this prior.SCHEMATIC CURVES — SHAPE ONLY, NOT THE PUBLISHED DENSITIESRR 1 · no difference0.40.60.81.01.21.4Relative risk of death with ECMOEOLIA 0.76 (95% CI 0.55–1.04)
Prior, likelihood and posterior for EOLIA under a minimally informative prior.A relative-risk axis from 0.4 to 1.4 with a dashed line at 1. Three schematic curves: a minimally informative prior, the fixed likelihood centred on the trial estimate of 0.76, and the posterior that results. Below the axis, the EOLIA estimate of 0.76 with its 95% confidence interval from 0.55 to 1.04. The reanalysis reports a posterior probability of 96 % that the relative risk is below 1 under this prior.SCHEMATIC — NOT THE PUBLISHED DENSITIESRR 10.40.60.81.01.21.4Relative risk of death with ECMOEOLIA 0.76 (95% CI 0.55–1.04)
Curves are schematic illustrations of the shape of the update, not the published densities; the three probabilities are those reported in the reanalysis.

The data had not changed by one patient. What changed was the honesty of the question.

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The video curriculum

A curated ladder, shortest first. Every link was fetched and verified in this build; each line says what the entry teaches.

  1. 3Blue1Brown — "Bayes theorem, the geometry of changing beliefs" — the single best visual explanation of what updating a belief actually does. Watch
  2. 3Blue1Brown — "The medical test paradox, and redesigning Bayes' rule" — why a positive test does not mean what it seems to; the clinician's daily Bayes problem. Watch
  3. StatQuest with Josh Starmer — "Bayes' Theorem, Clearly Explained!!!!" — the rule itself, slowly, with nothing assumed. Watch
  4. 3Blue1Brown — "Binomial distributions | Probabilities of probabilities, part 1" — the bridge from counting successes to reasoning about a probability itself; the doorway to the Beta distribution. Watch
  5. Veritasium — "The Bayesian Trap" — what happens when the prior never updates; the failure mode of the whole method, honestly told. Watch
  6. Richard McElreath — "Statistical Rethinking 2023 - 01 - The Golem of Prague" — the first lecture of the best full course in existence: why models are golems and what that demands of their operators. Watch
  7. Richard McElreath — "Statistical Rethinking 2023" (full lecture playlist) — the whole course, when the first lecture lands. Playlist
  8. Ben Lambert — "A Student's Guide to Bayesian Statistics" (full course playlist) — a complete, patient university course; the reference ladder for anyone who wants the mathematics under the intuition. Playlist

Where this foundation leads

Stroke trials keep answering yes or no at the 0.05 gate while clinicians need graded probability. What that asks of us — transparent priors, reproducible engines, pre-registration before results — is the subject of its own page: Bayesian synthesis in stroke medicine.

Where it has already led is the Bayesian Centre: one Reading a month, taking the numbers a published stroke paper printed and producing one probability against a decision line fixed before the arithmetic ran, with the counts, the code and the execution record published beside it and open to anyone with the credentials to check.