Find the posterior mean for α and π by using the formula for the total probability P(yi = 1).

Suppose a binary response has true prevalence Pr(Y = 1) = π but that observed responses are subject to misclassification with probabilities α₀ = Pr(y = 1|Y = 0), and α₁ = Pr(y = 0|Y = 1). Assuming α₀ = α₁ = α, state the total probability P(y_i = 1) in terms of the true prevalence probabilities P(Y = 1) and P(Y = 0) and the conditional probabilities P(y = 1|Y = 1) and p(y = 1|Y = 0). Winkler and Gaba (1990, p. 307) note that high values of α are unlikely and provide observed data on a juvenile survey question ‘have you beaten up on someone’, with r = 21 and n = 104. They assume π ∼ Beta(2, 8) and α ∼ Beta(2, 18) consistent with a prior misclassification rate of 10%. Find the posterior mean for α and π by using the formula for the total probability P(y_i = 1).