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Nov 15, 2021 02:40 AM
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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?
- 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
(TB) — The best source to get the gist. thinkbayes.pdf (amazonaws.com)
- Author:Jason Siu
- URL:https://jason-siu.com/article/9e8a8b1f-e93e-423a-9423-175afe30e1ff
- Copyright:All articles in this blog, except for special statements, adopt BY-NC-SA agreement. Please indicate the source!
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