What & Why

What is a conjugate prior?

• When the posterior is in the same family as the prior distribution they are called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.

• That means that if the prior distribution for x is a beta distribution, the posterior is also a beta distribution. (Tb)

 

What is The beta distribution?

How does that look like?

• frequently used in Bayesian statistics (Wiki)

• The shape of it depends on two parameters, written α and β, or alpha and beta.
• is uniform from 0 to 1 when alpha=1 and beta=1. (TB p39)
• If the prior is a beta distribution with parameters and , and we see data with h heads and t tails, the posterior is a beta distribution with parameters and .
• Or Dan Simpson called it :
• Under Uniform(0, 1) prior, if y = 8 Heads from n = 10 tosses, posterior is Beta(9, 3)
• In other words, we can do an update with two additions. (TB p39)

• It is great that we can leverage this advantage :
• ∵ many realistic priors there is a beta distribution that is at least a good approximation, and for a uniform prior there is a perfect match. (TB p39)

How

图解教材:概率机器学习（Murphy)_哔哩哔哩_bilibili 第69 教你如何adjust prior

Example : Drug efficacy (Lambert p237)

Reference