A Compass for an Uncertain World: When Bayes Meets Kelly

Forum topic · 小凯 · 2026-03-16

Summary

This essay explains how to combine Bayesian inference with the Kelly criterion into a complete decision-making system for investing under uncertainty. Using stories about Edward Thorp beating blackjack in 1956 and John Kelly Jr.'s 1956 formula at Bell Labs, the author introduces Bayes' theorem as the 'cognitive engine' for updating probability estimates with new evidence, and the Kelly formula f* = (bp − q)/b as the 'action engine' for optimal position sizing. A worked example walks through a full loop: setting a prior (60% win probability), Bayesian updating after an earnings beat (to 75%), estimating odds (b = 2), computing a Kelly stake of 62.5%, and rebalancing as new information arrives. Practical caveats include fractional Kelly (half or one-third Kelly) to guard against estimation error, dynamic odds, correlated multi-asset portfolios, and emotional discipline. The piece frames the combination as mathematically expressed 'unity of knowledge and action' and cites foundational references including Kelly (1956), Thorp (2006), and Poundstone's Fortune's Formula.

Key points

The central thesis: Bayesian inference should serve as the *cognitive engine* (updating your probability estimates), while the Kelly criterion serves as the *action engine* (converting those estimates into optimal position sizes). Together they form a closed loop: belief → action → new information → updated belief → adjusted action.
Historical framing: In 1956, mathematician Edward Thorp used card-counting to systematically beat blackjack, raising the question of how much to bet when you know you have an edge. John Kelly Jr. at Bell Labs answered it the same year: the optimal betting fraction maximizes compound (log) growth while avoiding ruin.
The Bayes half: The essay illustrates Bayes' theorem with the classic medical-test example — a 90%-accurate test for a disease with 1% prevalence yields only ~9% true positive probability — and a stock example where a 60% prior is updated to 70% after an earnings beat (80% likelihood if the stock rises, 30% if not). Core principle: today's posterior is tomorrow's prior; extreme beliefs (0% or 100%) are dangerous because they cannot be updated.
The Kelly half: For a biased coin with p = 0.6, q = 0.4, and b = 1, the Kelly formula gives f* = (1×0.6 − 0.4)/1 = 20% of capital. Betting everything guarantees eventual ruin; betting too little wastes the edge. Kelly maximizes expected *log* returns — long-run compound growth — not single-bet expected value.
A full decision loop (worked example): prior p = 0.6 → Bayesian update on an earnings beat → p = 0.75 → odds b = 2 (target +30%, stop −15%) → Kelly stake f* = (2×0.75 − 0.25)/2 = 62.5% → later adverse news triggers re-estimation (p = 0.65, b = 1.5) and a new optimal position (~40%) with rebalancing.

Practical caveats

Fractional Kelly: Because probability estimates have error, practitioners use f_actual = f* × k with k < 1 — half Kelly (k = 0.5) is most common, one-third Kelly for high uncertainty. The punishment for over-betting (ruin) is asymmetrically worse than under-betting.
Dynamic odds: b changes as price moves toward target or stop; moving stops can adjust the odds profile over time.
Multi-asset portfolios: Kelly is designed for single independent bets; correlated holdings require multi-asset Kelly formulations or risk-parity-style allocation.
Emotion as the biggest enemy: Written rules, automated alerts, and trading-journal reviews help prevent fear (under-betting after losses) and greed (over-betting after wins). As the author puts it: *the system exists to protect you from yourself*.

Key formulas

Bayes' theorem:

$P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$

Kelly criterion:

$f^* = \frac{bp - q}{b}$ , where $$q = 1 - p$$

Notable quotes

> "Although the expected return is positive, without an optimal betting strategy the player still has a high probability of eventually going bankrupt." — paraphrasing Kelly's point that knowing *what* to bet on (positive edge) and knowing *how much* to bet (Kelly sizing) are different problems.

References

1. Kelly, J. L. (1956). "A New Interpretation of Information Rate." *Bell System Technical Journal*, 35(4), 917-926. 2. Jaynes, E. T. (2003). *Probability Theory: The Logic of Science*. Cambridge University Press. 3. Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." *Handbook of Asset and Liability Management*, 385-428. 4. Poundstone, W. (2005). *Fortune's Formula*. Hill and Wang. 5. MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (2011). *The Kelly Capital Growth Investment Criterion: Theory and Practice*. World Scientific.

The essay closes with three questions to ask before any investment decision: What is my current probability estimate (prior)? How does new information change it (Bayesian update)? And what is the optimal position given the updated estimate (Kelly)? It is a compass, not a silver bullet — but it reliably points north when you are lost.