Sequential Bayesian updating over three weeks

Question

You track whether a market-making strategy has a profitable day, modeled as a yes/no outcome with unknown daily success probability $p$ . You start from a $\text{Beta}(2, 2)$ prior. The results arrive in three weekly batches:

Week 1: 3 profitable days out of 5
Week 2: 4 profitable days out of 5
Week 3: 5 profitable days out of 5

You update after each week, feeding one week's posterior in as the next week's prior.

After all three weeks, what is the probability that the next day is profitable?

Accepted Answer

Intuition first. A powerful feature of Bayesian updating is that it does not matter whether you digest the data all at once or piece by piece: last week's posterior simply becomes this week's prior, and the final belief is identical either way. For a Beta with yes/no data, each update just adds that batch's profitable days to the first number and unprofitable days to the second, so the whole sequence collapses to adding up all the days. Update step by step. Beta prior 	ext{Beta}(a,b) plus k profitable and m unprofitable days gives 	ext{Beta}(a+k, b+m). - Start: 	ext{Beta}(2, 2). - After We…

Sequential Bayesian updating over three weeks

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