Bayesian Updating: Learn a Win Probability Game by Game

The bootstrap and the permutation test both ask a frequentist question: how much would this number wobble if we reran the season? Bayesian updating asks something different — given what we believed before the data, what should we believe after it? For a yes/no outcome like "did the home team win?", the Beta distribution is the natural container for that belief. We'll start from a flat prior, feed in 1,231 real games one at a time, and watch a vague guess sharpen into a tight estimate of the true home-win probability — with a credible interval read straight off the curve.

This pairs with the bootstrap (the frequentist take on the same home-edge question) and builds on Summary Statistics and Distributions. The data is the bundled nba_home_results.csv (real 2023-24 games), so it runs offline.

Related chapter: Probability: The Foundation of Inference (DataField.dev).

1. The prior: what we believe before any data

   A Beta distribution lives on 0 to 1, which is exactly the range of a probability. Its two parameters, alpha and beta, act like a running tally of wins and losses. Beta(1, 1) is perfectly flat — it says every home-win rate from 0 to 1 is equally plausible, the honest starting point when we claim to know nothing.

   import numpy as np, pandas as pd, math
   
   df = pd.read_csv("nba_home_results.csv")
   home_win = (df.home_pts > df.away_pts).to_numpy(int)   # 1 = home won
   
   def beta_pdf(x, a, b):                                  # density, in log-space
       logB = math.lgamma(a) + math.lgamma(b) - math.lgamma(a + b)
       return np.exp((a - 1)*np.log(x) + (b - 1)*np.log(1 - x) - logB)

   We compute the density in log-space with math.lgamma so that large counts — alpha and beta in the hundreds by season's end — don't overflow.

2. The update rule: one game, one increment

   The Beta is the conjugate prior for a yes/no outcome, which is a technical way of saying the update is trivially simple: after n games with k home wins, the posterior is Beta(1 + k, 1 + (n - k)). Every home win adds one to alpha, every loss adds one to beta. No integrals, no simulation — just counting.

   def posterior_after(n):
       wins = int(home_win[:n].sum())
       a, b = 1 + wins, 1 + (n - wins)
       return a, b, a / (a + b)          # params and the posterior mean
   
   for n in (10, 50, 200, 1231):
       print(n, posterior_after(n))

   The belief sharpening game by game

   Prior: Beta(1, 1) - a flat line, every home-win rate equally plausible.
   after   10 games: Beta(   7,   5)  mean 0.583  95% CI [0.308, 0.833]
   after   50 games: Beta(  27,  25)  mean 0.519  95% CI [0.385, 0.652]
   after  200 games: Beta( 107,  95)  mean 0.530  95% CI [0.461, 0.598]
   after 1231 games: Beta( 670, 563)  mean 0.543  95% CI [0.516, 0.571]
   Each game nudges the belief; the interval tightens as evidence piles up.

   After just 10 games the mean is a jumpy 0.583 with a 95% interval spanning 0.31 to 0.83 — almost useless. By the full 1,231 games the estimate has settled to 0.543 with an interval of just 0.516 to 0.571. The belief didn't jump to the answer; it was dragged there, one game at a time.

3. Plot the posteriors on top of each other

   The whole story is visible at a glance if you draw the posterior density at a few checkpoints on one axis. Early curves are wide and low; later curves are tall and narrow, concentrating on the true rate.

   import matplotlib.pyplot as plt
   grid = np.linspace(0.30, 0.75, 500)
   fig, ax = plt.subplots()
   for n in (10, 50, 200, 1231):
       a, b, mean = posterior_after(n)
       ax.plot(grid, beta_pdf(grid, a, b), label=f"after {n} games")
   ax.axvline(0.5, ls=":"); ax.legend()
   fig.savefig("posterior_evolution.png", dpi=144, bbox_inches="tight")

   

   The credible interval — the 2.5th to 97.5th percentile of the posterior — is the Bayesian cousin of a confidence interval, and here you can say the plain-English thing a confidence interval technically can't: given a flat prior and this data, there's a 95% probability the true home-win rate is between 0.516 and 0.571. Every part of that interval sits above 0.50, so the home edge is real — the same conclusion the bootstrap reached, arrived at from the other direction.

Troubleshooting