Statistical Models Interactive

Bayesian Updating

Combine prior beliefs with new evidence. Start with what you know, update as data arrives. The foundation of rational inference under uncertainty.

📊 Bayes' Theorem

P(θ | data) ∝ P(data | θ) × P(θ)

Posterior

Updated belief after data

Likelihood

How likely is data given θ

Prior

Belief before data

Prior Belief

30 70

2 20

Observed Data

30 70

1 50

📊 Posterior

Posterior Mean 57.3

Posterior σ 3.02

95% Credible Int. [51.4, 63.2]

Prior Weight 9%

Data Weight 91%

Distribution Evolution

Balanced update: both prior and data contribute to posterior.

Key Properties

Sequential

Today's posterior is tomorrow's prior

Shrinkage

Small samples → closer to prior

Uncertainty

Posterior width shows remaining uncertainty

🏈 Betting Applications

Player Projections

Prior: historical/positional avg. Update with current season.

Win Probability

Prior: preseason. Update in-game as plays unfold.

Sharp Detection

Prior: new user is square. Update with betting patterns.

Odds Setting

Prior: opening line. Update with market action.

R Code Equivalent

# Bayesian update (conjugate normal-normal)
bayesian_update <- function(prior_mean, prior_var, obs_mean, obs_var, n) { 
  # Posterior variance
  post_var <- 1 / (1/prior_var + n/obs_var)
  
  # Posterior mean
  post_mean <- post_var * (prior_mean/prior_var + n*obs_mean/obs_var)
  
  list(
    mean = post_mean,
    sd = sqrt(post_var),
    ci_lower = post_mean - 1.96 * sqrt(post_var),
    ci_upper = post_mean + 1.96 * sqrt(post_var)
  )
}

# Example
prior <- list(mean = 50, var = 10^2)
data <- list(mean = 58, var = 100, n = 10)

posterior <- bayesian_update(prior$mean, prior$var, data$mean, data$var, data$n)
cat(sprintf("Posterior: %.1f (95%% CI: %.1f - %.1f)\n",
    posterior$mean, posterior$ci_lower, posterior$ci_upper))

✅ Key Takeaways

• Prior × Likelihood → Posterior
• More data → posterior approaches MLE
• Less data → posterior stays near prior

• Credible intervals have direct probability interpretation
• Perfect for sequential updating
• Regularizes naturally via prior