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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?
notion image
  • 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

 

Example : Drug efficacy (Lambert p237)

 
 

Reference

(TB) — The best source to get the gist. thinkbayes.pdf (amazonaws.com)
ETC2420 - Statistical thinkingETC2420 - Tutorial solution (W1-3) CDF; Hypo Testing; Violin Plot