You put a Beta(2, 8) prior on a strategy's hit rate and then observe 30 wins in 60 trades. Give the posterior distribution and its mean, express that mean as a weighted average of the prior mean and the sample rate, and say where the mean sits after 300 wins in 600 trades.

You put a Beta(2, 8) prior on a strategy's hit rate and then observe 30 wins in 60 trades. Give the posterior distribution and its mean, express that mean as a weighted average of the prior mean and the sample rate, and say where the mean sits after 300 wins in 600 trades.

Approach: Add successes and failures to the two beta parameters, then rewrite the posterior mean as a prior-count weighting against the data count.

16/35. The beta prior is conjugate to the binomial likelihood, so the posterior is Beta(2 + 30, 8 + 30) = Beta(32, 38) with mean 32/70 = 16/35 = 0.4571. The prior carries 2 + 8 = 10 pseudo-observations against 60 real trades, so the posterior mean is the weighted average (10/70)*0.2 + (60/70)*0.5 = 0.4571. The shrinkage toward the prior is exactly that weight of 10/70 = 0.143, and it pulls the estimate below the raw sample rate of 0.5 because the prior was centred at 0.2. With 600 trades the prior weight falls to 10/610 and the posterior mean rises to 0.4951, so a fixed prior is overwhelmed at a rate of 1/n and the choice of prior stops mattering once the sample is large.

Follow-up: How would you set the two beta parameters if you believed the hit rate was near 0.55 and wanted the prior to carry the weight of exactly 40 trades?

Key concepts: conjugate prior, posterior mean, beta-binomial, shrinkage.