Bayesian Odds: Combining Market Lines With Your Model

Why Bayesian?

Sports betting markets are efficient — but they're not perfect. If you have a model that produces win probabilities, you face a question: should you trust the market or your model?

The answer is both. Bayesian inference gives us a principled way to combine them.

The Beta-Binomial Framework

We model Team A's true win probability $p$ as a random variable. The market gives us a prior belief, and our model provides evidence.

Prior: The Market

Convert the market moneyline to implied probability (removing vig):

$$p_{\text{fair}} = \frac{p_{\text{implied},A}}{p_{\text{implied},A} + p_{\text{implied},B}}$$

This becomes a Beta distribution prior:

$$p \sim \text{Beta}(\alpha_0, \beta_0) \quad \text{where } \alpha_0 = p_{\text{fair}} \cdot N_{\text{market}}, \quad \beta_0 = (1 - p_{\text{fair}}) \cdot N_{\text{market}}$$

The parameter $N_{\text{market}}$ controls how much you trust the market. Higher values = tighter prior = more trust in the market.

Evidence: Your Model

Your model says Team A wins with probability $p_{\text{model}}$. We treat this as observing $N_{\text{model}}$ pseudo-trials:

$$\alpha_{\text{post}} = \alpha_0 + p_{\text{model}} \cdot N_{\text{model}}$$ $$\beta_{\text{post}} = \beta_0 + (1 - p_{\text{model}}) \cdot N_{\text{model}}$$

Posterior

The posterior win probability is:

$$p_{\text{posterior}} = \frac{\alpha_{\text{post}}}{\alpha_{\text{post}} + \beta_{\text{post}}}$$

This is a precision-weighted average. If $N_{\text{market}} = 100$ and $N_{\text{model}} = 50$, the posterior is 2/3 market + 1/3 model.

Edge Calculation

Once you have a posterior probability, compare it to the available odds:

$$\text{Edge} = p_{\text{posterior}} - p_{\text{implied}}$$ $$\text{EV} = p_{\text{posterior}} \cdot \left(\frac{1}{p_{\text{implied}}} - 1\right) - (1 - p_{\text{posterior}})$$

If both edge and EV are positive and exceed your threshold, you have a bet.

Practical Tips

• Start with $N_{\text{market}} = 100$ and $N_{\text{model}} = 20\text{-}50$
• Track your results and adjust the weights over time
• The Bayesian framework naturally handles the case where your model agrees with the market (posterior stays close to the market)
• When your model strongly disagrees, the posterior moves toward your model but is tempered by market wisdom