In Likelihood Weighting, evidence nodes E1 = 1 and E2 = 0 are observed. Given parents, P(E1 = 1 | parents) = 0.60 and P(E2 = 0 | parents) = 0.75. What is the sample weight w assigned to this generated sample?
$Sample weight w = prod P(e_{i} | parents) = 0.60 * 0.75 = 0.45.$
A Bayesian Network has 5 binary variables X1, X2, X3, X4, X5 arranged in a simple chain X1 -> X2 -> X3 -> X4 -> X5. How many TOTAL CPT parameters are needed to fully specify this network?
X1 (no parents): 1 parameter. X2, X3, X4, X5 (1 parent each): 2 parameters each -> 4 * 2 = 8. Total = 1 + 8 = 9 parameters.
Given prior P(Disease) = 0.01, sensitivity P(Positive | Disease) = 0.90, and false positive rate P(Positive | ~Disease) = 0.05. Calculate the total probability P(Positive).
$P(Pos) = P(Pos|D)P(D) + P(Pos|~D)P(~D) = 0.90*0.01 + 0.05*0.99 = 0.009 + 0.0495 = 0.0585.$
If a fair 4-sided die is rolled, what is the Shannon Entropy H(X) in bits?
$H(X) = - 4 * (0.25 * log2 0.25) = - 4 * (0.25 * -2) = 2.0 bits.$
How many independent parameters are needed to specify a full joint distribution for $n = 4 binary variables$?
$Parameters = 2^4 - 1 = 16 - 1 = 15 parameters.$