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| kb:hypothesis_testing [2021-05-10 14:25] – jaeyoung | kb:hypothesis_testing [2024-04-30 04:03] (current) – external edit 127.0.0.1 | ||
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| We can do this by making the decision that minimizes the probability of error *conditional* on the measurement $R = r$. | We can do this by making the decision that minimizes the probability of error *conditional* on the measurement $R = r$. | ||
| - | * If $P(H_1|R = r) > P(H_0|R = r)$, that is, if it is more likely that $H = H_1$ than $H = H_0$ given that $R = r$, we decide | + | * If $P(H_1|R = r) > P(H_0|R = r)$, that is, if it is more likely that $H = H_1$ than $H = H_0$ given that $R = r$, we decide $'H_1'$. |
| - | * Otherwise, if $P(H_1|R = r) < P(H_0|R = r)$, that is, if it is more likely that $H = H_1$ than $H = H_0$ given that $R = r$, we decide | + | * Otherwise, if $P(H_1|R = r) < P(H_0|R = r)$, that is, if it is more likely that $H = H_1$ than $H = H_0$ given that $R = r$, we decide $'H_0'$. |
| The resulting conditional probability of error is: | The resulting conditional probability of error is: | ||
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| Since we are just comparing $P(H_0|R = r)$ and $P(H_1|R = r)$, we can cancel out the $f_R(r)$ on both sides, so it is equivalent to comparing $P(H_0) f_{R|H}(r|H_0)$ and $P(H_1) f_{R|H}(r|H_1)$: | Since we are just comparing $P(H_0|R = r)$ and $P(H_1|R = r)$, we can cancel out the $f_R(r)$ on both sides, so it is equivalent to comparing $P(H_0) f_{R|H}(r|H_0)$ and $P(H_1) f_{R|H}(r|H_1)$: | ||
| - | * If $P(H_0) f_{R|H}(r|H_0) > P(H_1) f_{R|H}(r|H_1)$, then announce | + | * If $P(H_0) f_{R|H}(r|H_0) > P(H_0) f_{R|H}(r|H_0)$, then announce $'H_0'$. |
| - | * If $P(H_0) f_{R|H}(r|H_0) < P(H_1) f_{R|H}(r|H_1)$, | + | * If $P(H_0) f_{R|H}(r|H_0) < P(H_1) f_{R|H}(r|H_1)$, |
| ===== Likelihood ratio test ===== | ===== Likelihood ratio test ===== | ||
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| The likelihood ratio $\Lambda(r)$ is defined as: | The likelihood ratio $\Lambda(r)$ is defined as: | ||
| - | $$ \Lambda(r) = \frac{P(H_1) | + | $$ \Lambda(r) = \frac{f_{R|H}(r|H_1)}{f_{R|H}(r|H_0)} $$ |
| We can compare this likelihood ratio to the threshold $\eta$, which is the ratio between the a priori probabilities: | We can compare this likelihood ratio to the threshold $\eta$, which is the ratio between the a priori probabilities: | ||
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| $$ \eta = \frac{P(H_1)}{P(H_0)} $$ | $$ \eta = \frac{P(H_1)}{P(H_0)} $$ | ||
| - | If $ \Lambda{r} > \eta $, then announce $`H_1'$. Otherwise, announce $`H_0'$. | + | If $ \Lambda(r) > \eta $, then announce $'H_1'$. Otherwise, announce $'H_0'$. |
| ===== Terminology for different probabilities ===== | ===== Terminology for different probabilities ===== | ||