Let X= (X₁,X₂,X₃) be the results of exams a student takes. Let each X_i be binary {0,1}. 0 indicating fail and 1 indicating pass. Let the label Y indicate overall grade and Y is a binary variable which takes a value 0 for fail and 1 for passing grade. The prior on Y are P(Y=0) =0.9 and P(Y=1)=0.1. The likelihood is as follows:

P(X=(X₁,X₂,X₃)|Y=y) = Product from i=1 to 3 { (0.5y + 0.25)^(X_i) * (0.75 -0.5y)^(1-X_i) }. If h is the predicted outcome and y is the ground truth then the loss function L(h,y) is as follows : L(0,0)= 0, L(0,1)=1000, L(1,0) =100, L(1,1)=-500.

Q1. Find Bayes optimal hyposthesis h_{Bayes-optimal}

_{Q2. What is Bayes optimal risk (minimum expected loss) ?}

If you can not solve this please explain what it means by Bayes optimal hypothesis and bayes risk.