Sequential Bayes: Likelihood Ratios and Log-Odds

The rare-disease question taught one-shot Bayes. Real inference — and real trading — is sequential: evidence arrives piece by piece, and you update as it comes. The right representation makes sequential updating literally additive, and that representation is log-odds.

Bayes in odds form

Write beliefs as odds, $O = \frac{P(H)}{P(\neg H)}$. Then Bayes' theorem becomes a single multiplication:

$$O_{\text{posterior}} = O_{\text{prior}} \times \underbrace{\frac{P(E \mid H)}{P(E \mid \neg H)}}_{\text{likelihood ratio}}$$

The likelihood ratio (LR) is evidence strength in its purest form: how much more probable is this observation if the hypothesis is true than if it's false? LR = 1 is no evidence at all — no matter how "consistent with H" the observation feels. The disease question in this form: prior odds 1:99, LR = 99 (sensitivity/false-positive rate), posterior odds 1:1 — the whole calculation in one line, and the base-rate lesson built into the prior where it belongs.

Log-odds: evidence adds

Take logs and independent evidence stacks additively:

$$\log O_{\text{post}} = \log O_{\text{prior}} + \sum_i \log \text{LR}_i$$

This is the natural currency of accumulating information (Turing and Good ran wartime cryptanalysis in these units — the "ban"). Three consequences worth internalizing:

• Weak evidence compounds. Ten independent signals each with LR 1.2 give combined LR $1.2^{10} \approx 6$ — a nothing signal, stacked, becomes decisive. This is the statistical structure of quant trading: many tiny edges, added in log-odds, sized by the total (the Kelly connection: optimal bet size is a function of exactly these odds).
• Correlated evidence double-counts. Adding log-LRs requires independence; three analysts who all read the same report are one signal wearing three hats. Over-updating on correlated evidence is the sequential version of the portfolio lesson's correlation trap.
• The sign of drift. If $H$ is true, log-odds drift upward in expectation on each observation (the expected log-LR is a KL divergence, always ≥ 0) — but individual updates go both ways. Sequential testing (Wald's SPRT: stop when log-odds cross a threshold) formalizes "trade when conviction is sufficient," and connects to the optional-stopping lesson.

Updating as a practiced skill

The logistic-regression lesson called log-odds the natural scale for classification; here it's the natural scale for you. Habits that interviews (and desks) reward:

• Pre-commit to the LR. Before the evidence arrives, ask: what would I expect to see if I'm right, and if I'm wrong? A trader who can't state what would change their mind has an LR of 1 for everything and is updating on vibes.
• Small updates, many times beats rare dramatic conversions. Markets reprice by increments; so should beliefs.
• The market-maker's fill-updating from the microstructure course is exactly this: each trade against your quote carries an LR (informed flow buys more often than uninformed when you're cheap), and the fair-value shift is the posterior.

The interview version

"You think a coin is 60/40 biased to heads with prior probability 50%. It lands heads. New belief?" — Odds 1:1 × LR (0.6/0.5 = 1.2) → 1.2:1 → ~55%. "How many heads in a row to be 95% sure?" — Need posterior odds 19:1, so $1.2^n \ge 19$: $n \approx 16$. The arithmetic is trivial; the demonstrated skill is representing beliefs as numbers that move by rules — which is, compressed to a sentence, the job description.