Defect rate from a skeptical Beta prior

Question

A factory estimates the true defect probability $p$ of parts from a new machine. Engineers expect defects to be rare, so the prior is $\text{Beta}(1, 9)$ : centered at $0.1$ and leaning low, like having already seen 1 defect and 9 good parts. In a batch of 30 parts, 2 are defective.

Using Bayesian updating, what is the posterior mean estimate of the defect probability $p$ ?

Accepted Answer

Intuition first. A Beta prior encodes pretend past outcomes. 	ext{Beta}(1,9) is like 1 pretend defect and 9 pretend good parts, an opinion that defects are uncommon. Updating is bookkeeping: add real defects to the first number, real good parts to the second. The rule. Beta is conjugate to defect/good data, so the posterior stays a Beta: 	ext{Beta}(a,b)\ +\ k 	ext{ defects and } (n-k) 	ext{ good}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Plug in. Here a=1, b=9, with 2 defects and 28 good parts: 	ext{posterior} = 	ext{Beta}(1+2,\ 9+28) = 	ext{Beta}(3, 37). Posterior mean = first number ov…

Defect rate from a skeptical Beta prior

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