Beta: probability of a probability (conjugate to Bernoulli)
A probability of a probability. Conjugate prior to Bernoulli β Beta$(a,b)$ updated by $s$ successes and $f$ failures becomes Beta$(a+s, b+f)$.
Method · Beta
Intro
Horas (built on work by Krishnavedala) β public domain · Public Domain · Wikimedia Commons
Beta is the conjugate prior to Bernoulli/Binomial β the Bayesian counterpart that closes the loop on those discrete distributions before we move continuous.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
We want to estimate the bias p of a coin. Our prior on p is Uniform(0,1), i.e. Beta(Ξ±=1, Ξ²=1). We observe 7 heads and 3 tails in 10 flips. (a) Identify the posterior distribution of p. (b) Compute the posterior mean. (c) Compute the posterior probability that p > 0.5.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
Where does this distribution sit in the story chain β what question does it answer that the previous distribution couldn't? Try to recall the key moment formulas (mean, variance) without looking them up.
