Two posterior bars show P(A|x)=0.7 and P(B|x)=0.3, with A favored under equal costs. A loss table is introduced where misclassifying B as A costs 5 and misclassifying A as B costs 1. Expected losses are computed as 1.5 for choosing A and 0.7 for choosing B, flipping the decision despite B's lower probability. The comparison is generalized algebraically to 5p > 1-p, yielding a 1/6 threshold. Useful for first-year AI students learning Bayes decision theory.
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# The most probable class is not always the best decision Two posterior probability bars show P(A given x) = 0.7 and P(B given x) = 0.3. First choose A under equal error costs. Then introduce a loss table: predicting A when the true class is B costs 5, predicting B when the true class is A costs 1, and correct decisions cost zero. Animate the expected losses: choosing A costs 5 times 0.3 = 1.5, while choosing B costs 1 times 0.7 = 0.7. The preferred decision changes to B even though B is less probable. Let p denote the posterior for B and transform the comparison into 5p > 1-p, giving a threshold of 1/6 for choosing B. Clearly distinguish probabilities from expected losses, which can exceed one. Use black backgrounds, cyan and orange probability bars, and a fixed row-and-column convention in the loss table. Useful for understanding minimum-risk decisions. Audience: 一年级人工智能专业的学生