🎲 Beta Distribution

Beta Distribution#

A continuous distribution on [0, 1]; the conjugate prior for a binomial probability.

Important

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What it is#

The Beta distribution is a continuous distribution on the interval \([0, 1]\), which makes it the natural way to model a probability or proportion. It has two shape parameters, \(\alpha\) and \(\beta\), and density

\[f(p \mid \alpha, \beta) = \frac{1}{B(\alpha, \beta)}\, p^{\alpha - 1} (1 - p)^{\beta - 1}, \qquad 0 \le p \le 1,\]

where \(B(\alpha, \beta)\) is the Beta function, the normalising constant.

Summary statistics#

\[\mathbb{E}[p] = \frac{\alpha}{\alpha + \beta}, \qquad \operatorname{Var}(p) = \frac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)},\]

and, when \(\alpha, \beta > 1\), the mode is \((\alpha - 1)/(\alpha + \beta - 2)\). A larger \(\alpha + \beta\) means a more concentrated (lower-variance) belief.

The shape gallery#

The two parameters give the family enormous flexibility:

\(\alpha = \beta = 1\) — uniform on \([0, 1]\).
\(\alpha > \beta\) — skewed toward 1; \(\alpha < \beta\) — skewed toward 0.
\(\alpha, \beta > 1\) — unimodal (a single bump).
\(\alpha, \beta < 1\) — U-shaped (mass piling up near 0 and 1).

Why it’s central: conjugacy#

The Beta is the conjugate prior for the Binomial likelihood. With a \(\text{Beta}(\alpha, \beta)\) prior and \(k\) successes in \(n\) trials, the posterior is again Beta:

\[p \mid \text{data} \sim \text{Beta}(\alpha + k,\; \beta + n - k).\]

So Bayesian updating is just adding successes to \(\alpha\) and failures to \(\beta\) — no integration required.

Examples#

\(\text{Beta}(1,1)\) — a flat prior, no preference before data.
\(\text{Beta}(20, 20)\) — sharply peaked at 0.5, a strong “fair coin” belief.
Start from \(\text{Beta}(1,1)\), observe 7 heads and 3 tails → \(\text{Beta}(8, 4)\), mean \(8/12 \approx 0.67\).

Where it shows up#

Bayesian inference over probabilities, A/B testing (belief about conversion rates), reliability (success/failure rates), and any proportion modelling. Its multivariate generalisation, the Dirichlet distribution, plays the same conjugate role for the categorical/multinomial likelihood. In code, scipy.stats.beta provides the density, sampling and quantiles.

Theme: Probability & Statistics Foundations · All terminology

Hint

Mind map — connected ideas

Binomial Likelihood · Posterior belief · Prior Belief (or Prior Probability) · Thompson Sampling (TS) in Bandits (Multi-Armed Bandit Problem (MAB)) · Bayes’ Theorem

Hint

More in Probability & Statistics Foundations

Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion · Probability