Email open rate from a flat Beta prior

Question

A marketer wants to estimate the true open probability $p$ of a new email template. With no prior opinion, they use a flat prior $\text{Beta}(1, 1)$ , which treats every value of $p$ between 0 and 1 as equally likely. They send 20 emails and 4 are opened.

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

Accepted Answer

Intuition first. A Beta prior is like pretend prior data. 	ext{Beta}(1,1) is the "know nothing" version, equivalent to almost no pretend observations, so the data will do nearly all the talking. Updating is bookkeeping: add real opens to the first number, real non-opens to the second. The rule. Beta is conjugate to open/no-open data, so the posterior stays a Beta: 	ext{Beta}(a,b)\ +\ k 	ext{ opens and } (n-k) 	ext{ non-opens}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Plug in. Here a=1, b=1, with 4 opens and 16 non-opens: 	ext{posterior} = 	ext{Beta}(1+4,\ 1+16) = 	ext{Beta}(5, 17). Poste…

Email open rate from a flat Beta prior

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