Bayes decision rule with misclassification costs
The problem, from first principles
Choosing the most probable class is only optimal when every mistake costs the same. If wrongly clearing a fraudulent payment is far worse than wrongly flagging a good one, the best decision is the one with the smallest expected cost, which can differ from the most probable class.
From theory to code
Implement minimum_risk_class(posteriors, cost_matrix). For each possible prediction, compute its expected cost under the posterior, and return the prediction with the lowest cost.
Constraints
posteriorsis a one-dimensional array of class probabilities summing to 1.cost_matrix[i][j]is the cost of predicting classjwhen the true class isi; it is square.- Return a Python
int. On ties, return the smallest class index.
Hints
Open one at a time. Each gives away a little more than the last.
Hint 1
The expected cost of predicting j is sum_i posteriors[i] * cost_matrix[i][j].
Hint 2
That is a vector-matrix product: posteriors @ cost_matrix gives every prediction's risk at once.
Keywords
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