Flipping the Conditional (Bayes' Theorem)

Why this matters

Bayes' theorem is the single most important formula in applied probability. It tells you how to invert a conditional — from $P(\text{evidence} \mid \text{hypothesis})$, which you usually have, to $P(\text{hypothesis} \mid \text{evidence})$, which you usually want. Every modern ML model that does inference is using a Bayes update somewhere. The infamous failure mode is base-rate neglect: you read 'this test is 99% accurate' and conclude that a positive result means 99% chance of disease, when the actual posterior can be under 10% if the disease is rare. One worked Bayes calculation immunises you for life.

Demo

Bayes' theorem rewrites the conditional in the other direction by combining the prior $P(H)$, the likelihood $P(E \mid H)$, and the marginal evidence $P(E)$ — typically computed via the law of total probability. The posterior is proportional to prior times likelihood.

Try it yourself

1. Run it. The posterior $P(D \mid +) \approx 0.167$, not 0.99. Read this number until the base-rate intuition clicks: a 'highly accurate' test on a rare disease still produces mostly false positives.
2. Hold sensitivity at 0.99 and increase specificity to 0.99. Recompute $P(D \mid +)$ — it jumps to about 0.5. The bottleneck for a rare-disease test is specificity, not sensitivity.
3. Hold the test characteristics fixed and bump the prior to $P(D) = 0.10$ (a high-risk subgroup). The posterior should rise to about 0.69 — same test, very different conclusion. Bayes is contextual.

References

• Wikipedia — Bayes' theorem
• Blitzstein & Hwang — Stat 110 (Ch. 2)
• 3Blue1Brown — The medical test paradox