Chapter 17 — Bayesian thinking for clinicians

The “posterior probability” hunch

You’re evaluating a new sepsis biomarker. The paper reports: “Posterior probability of mortality given a positive test = 78% (95% credible interval 62–89%).” You pause — what does posterior probability mean? How is it different from the p‑value you usually see?

Bayesian statistics offers an intuitive way to update beliefs with data: start with a prior belief (what you think before seeing the data), look at the likelihood (how well the data support each possibility), and end with a posterior belief (what you think after seeing the data). In the ICU, this mirrors how we think at the bedside: we start with a pretest probability, adjust with test results, and get a posttest probability.

If you’re comfortable with likelihood ratios (Chapter 14), you’ve already done a simple Bayesian update. This chapter formalises that intuition and shows when a full Bayesian analysis (with priors, likelihoods, and posteriors) adds value — especially in adaptive trials, interim analyses, or when combining multiple sources of evidence.

The three ingredients: prior, likelihood, posterior

Bayes’ theorem in words:

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• Prior – your belief before seeing the current data. Can be based on previous studies, expert opinion, or physiological reasoning.
• Likelihood – how probable the observed data are under each possible hypothesis (e.g., treatment works vs does not work).
• Posterior – your updated belief after seeing the data.

In mathematical form for two hypotheses (H₀: no effect, H₁: effect):

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The ratio on the left is the posterior odds; the first fraction on the right is the likelihood ratio (LR); the second fraction is the prior odds.

Thus, if you already know how to convert a pretest probability to odds, multiply by an LR, and convert back to probability, you are doing a Bayesian update with a binary hypothesis (disease present/absent). The same logic applies to treatment effects.

Why Bayes can be useful in the ICU

1. Interim monitoring – Adaptive RCTs (e.g., REMAP‑CAP) use Bayesian posterior probabilities to stop early for efficacy, futility, or safety. Instead of relying on a p‑value threshold, they monitor the posterior probability that the treatment effect exceeds a clinically important threshold.
2. Incorporating external evidence – When planning a new ICU trial, you can use prior data from observational studies or meta‑analyses to inform the prior distribution of the treatment effect, potentially reducing required sample size.
3. Handling small samples – With sparse data (e.g., rare adverse events), a Bayesian analysis with a sensible prior can produce more stable estimates than frequentist methods that may converge to extreme values.
4. Probabilistic statements – A Bayesian credible interval says: “There is a 95% probability that the true effect lies in this interval,” given the model and the prior. This matches how clinicians think about uncertainty.

A simple ICU example: evaluating a new vasopressor

Suppose you want to know whether a new vasopressor increases MAP by at least 5 mmHg. You run a small pilot (n = 20 per arm) and observe a mean difference of 4 mmHg with SD = 8 mmHg.

• Frequentist approach: t‑test p = 0.32 → “not significant.”
• Bayesian approach: Choose a prior for the treatment effect, say a normal distribution centred at 0 with SD = 10 mmHg (reflecting uncertainty but skepticism about large effects). Combine with the likelihood from the data (normal with mean = 4, SE ≈ 2.5). The posterior distribution ends up centred around ~2 mmHg with SD ≈ 2 mmHg. The posterior probability that the true effect exceeds 5 mmHg might be only 15% — low, supporting the decision not to adopt the vasopressor based on this pilot.

If instead the observed mean difference were 7 mmHg, the posterior probability of exceeding 5 mmHg might rise to 78%, suggesting a promising signal worth a larger trial.

Prior specification: where do priors come from?

• Weakly informative priors – Broad distributions that let the data speak (e.g., normal(0, 20) for a mean difference in mmHg). Use when you genuinely have little prior information.
• Informative priors – Based on previous trials, meta‑analyses, or physiological limits (e.g., you know a vasopressor cannot raise MAP by more than 30 mmHg without causing ischemia). Use when you have strong external evidence.
• Sceptical priors – Centre at zero with a modest spread, reflecting the belief that most new interventions have little or no effect.
• Enthusiastic priors – Centre at a clinically important effect, used when you strongly expect benefit (requires justification).

Appraisal checkpoint: Does the paper state the prior distribution, justify it, and show sensitivity analyses (e.g., varying the prior width)? If priors are not disclosed, the Bayesian results are not reproducible.

Common Bayesian models in ICU literature

How to read a Bayesian ICU paper

1. Identify the prior – Is it stated? Is it weakly informative, informative, or sceptical? Is a sensitivity analysis shown?
2. Check the likelihood – Is the model appropriate for the data type (normal for continuous, binomial for binary, etc.)?
3. Look for the posterior summary – Posterior mean, median, credible interval (usually 95% highest density interval).
4. See the probabilistic statement – e.g., “There is a 85% probability that the treatment reduces mortality by more than 5%.”
5. Assess model checking – Posterior predictive checks, trace plots (for MCMC), convergence diagnostics (R̂, effective sample size).
6. Consider clinical usefulness – Does the posterior probability map onto a decision rule you would use at the bedside (e.g., adopt if posterior probability of benefit > 80%)?

Go deeper

• StatPearls – “Bayesian Statistics” (NBK560123): free, clear introduction to prior, likelihood, posterior, and ICU‑relevant examples. https://www.ncbi.nlm.nih.gov/books/NBK560123/
• OpenIntro Statistics – Chapter 12 “Introduction to Bayesian Statistics” (free PDF): worked examples of Bayesian updating with discrete and continuous data, includes R code. https://www.openintro.org/stat/textbook.php
• BMJ Statistics Notes – “Bayesian statistics” (Altman & Bland): concise explanation of why Bayes matches clinical thinking. https://www.bmj.com/content/bmj_stats_notes
• PMC7189812 – “Bayesian adaptive platform trials in critical care” (free full text): describes REMAP‑CAP and similar designs, with prior specification and decision rules. https://doi.org/10.1186/s13054-020-03392-2 (PMC free)
• PMC6052478 – “A practical guide to reporting Bayesian analyses” (free): checklist for priors, likelihood, convergence, and sensitivity. https://doi.org/10.1186/s12874-020-01075-2 (PMC free)

Next: Chapter 18 — Meta‑analysis & systematic reviews (PRISMA)