Decision Rules: a mapping from data to hypothesis
Type I Error: When H_0 negative result is true, we delcare H_1 positive result. (false positive: you tested positive, but is false - false alarm)
Type II Error: When H_1 positive result is true, we delcare H_0 negative result. (false negative: you tested negative, but is false - miss)
If you decrease one type of error, it usually increase another type of error.
We view H_0 or H_1 as latent variable.
Likelihood Ratio Test: if \land(k) \geq \tau, we declear H_0, otherwise H_1.
Example: X = x are COVID-19 detected level (the higher level is, usually the more likely you have COVID-19), H_0 is you actually don't have COVID-19.
Pr\{X=k \| H_*\} | X=0 | X=1 | X=2 | X=3 |
---|---|---|---|---|
H_0 | 0.6 | 0.3 | 0.1 | 0.0 |
H_1 | 0.1 | 0.2 | 0.3 | 0.4 |
Maximim Likelihood (ML): special case of Likelihood Ratio Tests (LRT) when \tau = 1.
If Pr\{H_0 | X = k\} \geq^? Pr\{H_1 | X = k\}, then declear H_0, otherwise H_1. This is essentially equal to:
Example: given Pr\{H_0\} = 0.8, Pr\{H_1\} = 0.2, we transform the table to joint distribution.
Pr\{X=k\cap H_*\} | X=0 | X=1 | X=2 | X=3 |
---|---|---|---|---|
H_0 | 0.48 | 0.24 | 0.08 | 0 |
H_1 | 0.02 | 0.04 | 0.06 | 0.08 |
MAP \to LRT: This is equivalent to set \tau = \frac{Pr\{H_1\}}{Pr\{H_0\}}. Also you see MAP is skewed by the prior probability, which is not good. (People in non-COVID cluster might have lower chance to be detected.)
It is essentially the sum of two joint probability in case we are wrong.
Theorem: MAP minimizes AEP.
Proof: AFSOC (\exists \sigma)(\sigma \text{ is not MAP } \land \sigma \text{ achieves lowest AEP}). Since \sigma is not MAP, then there exists X = k' such that it is either:
WLOG, we are in case 1.
But then there exists another policy \sigma' that has AEP of following
This is equivalent to set
\tau = \frac{Pr\{H_1\} E[\text{Cost} | \text{Declare }H_1 \land H_0]}{Pr\{H_0\} E[\text{Cost} | \text{Declare }H_0 \land H_1]} = \frac{Pr\{H_1\} E[\text{T1 Cost}]}{Pr\{H_0\} E[\text{T2 Cost}]}
We just change likelihood: f_{X | H_0}(x), f_{X | H_1}(x). However, the prior is still discrete because they are prior for hypothesis.
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