Chapter 10 Key Takeaways: Bayesian Thinking for Bettors
Key Concepts
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Bayes' Theorem: The mathematical rule for updating beliefs in light of new evidence. The posterior probability is proportional to the prior probability times the likelihood: $P(H \mid E) \propto P(H) \cdot P(E \mid H)$. For bettors, this formalizes the process of combining preseason knowledge with in-season results.
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Prior Distributions: Encode what you know before seeing data. In sports betting, informative priors built from preseason analysis, historical patterns, and domain expertise are a competitive advantage. The spectrum runs from uninformative (Beta(1,1) --- maximum ignorance) through weakly informative (Beta(4,4) --- league-average default) to strongly informative (Beta(14,6) --- specific team projection). The effective sample size $\alpha + \beta$ determines the prior's influence.
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Conjugate Priors: Prior-likelihood pairs where the posterior has the same distributional form as the prior. The two essential pairs for sports bettors are Beta-Binomial (for win rates and proportions) and Normal-Normal (for scoring margins and point spreads). Conjugate models allow closed-form updating without MCMC.
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Sequential Updating: Last week's posterior becomes this week's prior. This makes Bayesian reasoning naturally suited to the week-by-week rhythm of sports seasons. The order of processing independent evidence does not affect the final posterior.
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Bayesian Shrinkage / Regression to the Mean: The prior pulls extreme estimates toward the population mean, preventing overreaction to small samples. Early-season records are "shrunk" toward preseason expectations. This is the mechanism behind one of the most reliable edges in sports betting: fading teams with extreme early records.
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Hierarchical Models (Partial Pooling): Individual team parameters are modeled as draws from a league-wide distribution. This "borrows strength" across teams, producing better estimates for every franchise. Teams with less data are shrunk more; teams with more data are shrunk less. The degree of shrinkage is learned from the data.
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Posterior Predictive Distribution: The distribution of future outcomes that accounts for both parameter uncertainty and inherent randomness. This gives full probability distributions over game outcomes, enabling direct computation of win probability, cover probability, and expected value for betting.
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MCMC (Markov Chain Monte Carlo): The computational engine for complex Bayesian models. Modern samplers (NUTS/Hamiltonian Monte Carlo) in PyMC and Stan handle models with hundreds of parameters. Key diagnostics: $\hat{R} < 1.01$, sufficient effective sample size, no divergent transitions.
Key Formulas
| Formula | Expression | Application |
|---|---|---|
| Bayes' Theorem | $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$ | Update beliefs with evidence |
| Proportional Form | $\text{Posterior} \propto \text{Prior} \times \text{Likelihood}$ | Core Bayesian relationship |
| Beta-Binomial Posterior | $\text{Beta}(\alpha + w, \beta + n - w)$ | Update win rate after $w$ wins in $n$ games |
| Posterior Mean (Beta) | $\hat{\theta} = \frac{\alpha + w}{\alpha + \beta + n}$ | Weighted average of prior and data |
| Prior Weight | $\frac{\alpha + \beta}{\alpha + \beta + n}$ | Prior's share in the posterior mean |
| Data Weight | $\frac{n}{\alpha + \beta + n}$ | Data's share in the posterior mean |
| Bayes Factor | $BF = \frac{P(E \mid H_1)}{P(E \mid H_2)}$ | Strength of evidence for $H_1$ vs. $H_2$ |
| Posterior Odds | $\frac{P(H_1 \mid E)}{P(H_2 \mid E)} = BF \times \frac{P(H_1)}{P(H_2)}$ | Updated model comparison |
| Normal-Normal Posterior Mean | $\frac{\frac{\mu_0}{\sigma_0^2} + \frac{n\bar{x}}{\sigma^2}}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}}$ | Precision-weighted average for margins |
| Normal-Normal Posterior Variance | $\frac{1}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}}$ | Posterior uncertainty for margins |
The Bayesian Bettor's Workflow
Step 1 --- Set priors (preseason). For each team, translate preseason analysis into a formal prior distribution. Use Beta($\alpha$, $\beta$) for win rates or Normal($\mu_0$, $\sigma_0^2$) for scoring margins. The effective sample size of the prior should reflect your confidence: 10--20 for moderate conviction, 20+ for strong conviction based on extensive analysis.
Step 2 --- Update weekly (in-season). After each game, apply the conjugate update rule. Wins add to $\alpha$; losses add to $\beta$. For more complex models, re-run MCMC with the updated data. The posterior evolves continuously, always reflecting all available evidence.
Step 3 --- Generate predictions. For each upcoming game, compute the posterior predictive distribution. This gives you the full probability distribution over margins, not just a point estimate. From this distribution, directly compute win probability, cover probability for any spread, and over/under probabilities for any total.
Step 4 --- Compare to the market. Convert sportsbook lines to implied probabilities. When your posterior predictive probability diverges meaningfully from the market's implied probability, you have identified a potential edge.
Step 5 --- Size the bet. Use the full posterior distribution (not just the point estimate) when applying Kelly Criterion or fractional Kelly sizing. The posterior's width determines your confidence in the edge, which should scale the bet size.
Step 6 --- Monitor calibration. Track your posterior predictions against actual outcomes. Run posterior predictive checks regularly. If your model consistently over- or under-predicts, diagnose and fix the problem before it costs more money.
Quick-Reference Decision Framework
| Situation | Recommended Approach | Reason |
|---|---|---|
| NFL team ratings (17 games) | Bayesian with informative priors | Small sample; prior is critical |
| NBA team ratings (82 games) | Bayesian or frequentist | Large sample; both work by midseason |
| MLB player batting average (April) | Bayesian | Tiny sample; shrinkage prevents wild estimates |
| Championship probability | Bayesian | Rare event; base rate (prior) matters enormously |
| Play-by-play analysis (millions of rows) | Frequentist or ML | Massive data; computation speed matters |
| Comparing two betting strategies | Bayesian A/B test | Posterior probability of superiority is intuitive |
| Early-season line shopping | Bayesian shrinkage | Market may overreact; fading extremes has edge |
| Late-season team assessment | Either | Prior is washed out; data dominates |
Jeffreys' Scale for Bayes Factors
| Bayes Factor | Evidence Strength |
|---|---|
| 1 -- 3 | Barely worth mentioning |
| 3 -- 10 | Substantial |
| 10 -- 30 | Strong |
| 30 -- 100 | Very strong |
| > 100 | Decisive |
Common Pitfalls
Pitfall 1: Overconfident priors. Setting $\alpha + \beta$ too large prevents the model from learning. If you use Beta(50, 50) for an NFL team, you need about 100 games of data to reduce the prior's weight to 50%. In a 17-game season, the prior dominates. Use the effective sample size interpretation: would you genuinely bet as if you had observed that many games?
Pitfall 2: Ignoring priors when they matter most. Flat priors in the NFL waste the most valuable informational advantage you have: preseason knowledge. A Beta(1,1) prior for an NFL team means a 4-0 start gives a posterior mean of 0.833 --- far too extreme.
Pitfall 3: Failing to check the model. A Bayesian model that passes no posterior predictive checks is just a sophisticated way to be wrong with confidence. Always compare simulated data from the model to actual observations.
Pitfall 4: Confusing posterior with prediction. The posterior distribution is over the parameter (true win rate). The posterior predictive distribution is over future observations (next game outcome). Use the posterior predictive for betting decisions; it includes both parameter uncertainty and game-to-game noise.
Pitfall 5: Not updating the prior when the world changes. Bayesian updating handles gradual evidence accumulation. It does not automatically handle structural breaks: injuries, trades, rule changes, coaching firings. When the data-generating process changes, your prior needs manual revision.
Ready for Part III? Self-Assessment Checklist
Before moving on to Part III (The Betting Marketplace), confirm that you can do the following:
- [ ] Apply Bayes' theorem to compute posterior probabilities from priors and likelihoods
- [ ] Set informative Beta priors from preseason analysis and explain effective sample size
- [ ] Perform Beta-Binomial conjugate updates and compute posterior means and credible intervals
- [ ] Perform Normal-Normal conjugate updates for scoring margin estimation
- [ ] Track a team's win probability through a season using sequential Bayesian updating
- [ ] Explain Bayesian shrinkage and why it prevents overreaction to small samples
- [ ] Distinguish when Bayesian methods outperform frequentist methods (and vice versa)
- [ ] Build a simple Bayesian model in PyMC with appropriate priors
- [ ] Interpret MCMC diagnostics ($\hat{R}$, effective sample size, divergences)
- [ ] Generate posterior predictive distributions and use them for betting decisions
- [ ] Run posterior predictive checks to validate model fit
- [ ] Compute and interpret Bayes factors for hypothesis comparison
If you can check every box with confidence, you have mastered the statistical toolkit of Part II. In Part III, we enter the marketplace where these tools meet money.