The previous two problems considered the MAP and the ML decision rules. Each of these rules yields a certain probability of false alarm (deciding that SHD was present when in fact it was not) and a certain probability of correct detection (deciding that SHD was present when indeed it was present). Assuming that the exponential distribution models are correct, which of the following statements are true? (Numbers are rounded.)

Question

BlackTom AI · Accepted Answer

We are given a scenario comparing two decision rules, MAP and ML, in the context of detecting SHD with exponential distribution models, focusing on two performance metrics: false alarm probability and correct detection probability. The problem states that numbers are rounded and asks which statements are true.

Option analysis:
- The MAP rule yields lower false alarm rate than the ML rule (49.1% as compared to 54.9%).
  This claim hinges on how MAP and ML handle uncertainty about the underlying parameters, particularly priors. The ML rule selects the hypothesis that maximizes the likelihood, effectively treating all hypotheses as equally probable a priori. In contrast, the MAP rule incorporates prior probabilities of the hypotheses, and, depending on those priors and the cost structure, can bias the decision toward reducing false alarms if the prior for the non-SHD hypothesis is relatively stronger or if the decision threshold shifts accordingly. If the priors and the likelihoods combine in such a way that the MAP decision rule is more conservative in declaring SHD present, the resulting false alarm probability can be lower under MAP than under ML. Therefore, the statement that MAP yields a lower false alarm rate than ML, with the given percentages (49.1% vs 54.9%), is plausible given typical MAP vs ML behavior when priors influence the threshold.

Considerations that could support this claim include:
- The exponential model implies a specific form for the likelihood under each hypothesis; the MAP rule uses priors to adjust the posterior odds. If the prior for the SHD-absent state is sufficiently large, the MAP rule may require stronger evidence to declare SHD present, reducing false alarms.
- The ML rule does not incorporate priors, so its threshold is effectively centered on the data likelihood alone; in regimes where the SHD-absent distribution and the SHD-present distribution overlap in a way that causes more frequent misclassification under ML, the ML false alarm rate could be higher.

Potential reasons an examiner might deem this statement true include: the reported 49.1% vs 54.9% difference aligns with the expectation that incorporating prior information (MAP) can reduce type I errors when priors favor the absence of SHD, while ML, being prior-free, cannot exploit that information.

Why other possibilities (if they were available) might be incorrect:
- If another statement claimed that ML yields a lower false alarm rate, it would contradict the intuition that priors can be used to penalize false alarms more heavily in MAP, so that option would be questionable unless the priors are unfavorable to SHD present or the costs are asymmetric in a contrary direction.
- If a statement asserted equal false alarm rates for MAP and ML, that would ignore the effect of priors on MAP and would generally be inconsistent with typical Bayes rule behavior unless priors and likelihoods happen to produce identical thresholds, which is a specific and unlikely coincidence.
- If a statement claimed MAP always has a lower false alarm rate across all parameter settings, that would be overly general; the actual performance depends on the particular priors, cost function, and distributions, so it cannot be guaranteed universally.

In summary, given that MAP incorporates priors and can adjust the decision threshold to balance false alarms against misses, it is reasonable that the MAP rule yields a lower false alarm rate than the ML rule for the provided numbers, making the statement internally consistent with Bayesian decision theory in this context.

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