Sequential Bayesian Updating¶

Why It Matters¶

Naive reasoning attempts to synthesize all evidence at once, which is cognitively overwhelming and prone to errors. Sequential updating breaks this into manageable steps - each piece of evidence causes a small probability shift from the current state. This corrects the systematic error of holistic synthesis when component-wise updating is more reliable.

The Key Move¶

Start with base rates for your differential diagnosis. When the first piece of evidence arrives, update each hypothesis's probability using its likelihood ratio. These updated probabilities become your new priors. When the next evidence arrives, update again from the current state. Continue until probability concentrates on one hypothesis or until the value of additional testing is low. The discipline is in making updates explicit and sequential.

Classic Application¶

Cardiac biomarker interpretation. A patient with chest pain has three hypotheses: MI (10% prior), unstable angina (20%), non-cardiac (70%). First troponin at 1 hour is negative (LR for MI ~ 0.3): MI drops to ~3%, others increase proportionally. EKG shows ST depression (LR for acute coronary syndrome ~ 3): MI rises to ~8%, unstable angina to ~50%, non-cardiac to ~42%. Second troponin at 3 hours is elevated (LR for MI ~ 10): MI jumps to ~60%. Sequential updating accumulates evidence systematically.

Surprising Application¶

A/B test evaluation during rollout. You release a feature to 1% of users - initial metrics show 2% engagement increase (weak positive signal, LR ~ 1.3). Update from prior expectation. Roll out to 10% - now 3% increase with tighter confidence (LR ~ 2.5). Update from 1% posterior. Roll out to 50% - still 3% increase, high confidence (LR ~ 4). The sequential process accumulates evidence while allowing early stopping if strong negative signals appear, exactly like diagnostic testing.

Failure Modes¶

Treating non-independent evidence as independent. If findings are correlated, multiplying likelihood ratios double-counts evidence. The tool requires understanding when updates should be independent versus when evidence is redundant. Also fails with confirmation bias - selectively updating on confirming evidence while ignoring disconfirming evidence. The worst failure is false precision - calculating exact probabilities from rough estimates and treating them as accurate.

Go Deeper¶

Winkler, R.L. & Murphy, A.H. (1973). "Experiments in the Laboratory and the Real World," Organizational Behavior and Human Performance, 10(2), 252-270; van den Ende, J. et al. (2002). "Bayesian Decision Analysis in Individual Patients," Annals of Emergency Medicine, 40(4), 394-402.