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| kb:hypothesis_testing [2021-05-10 14:04] – created jaeyoung | kb:hypothesis_testing [2024-04-30 04:03] (current) – external edit 127.0.0.1 | ||
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| Hypothesis testing involves deducing the quantity of a hypothesis $H$, which takes on one of the values $H_0, H_1, \dots$ from a measurement $R=r$. | Hypothesis testing involves deducing the quantity of a hypothesis $H$, which takes on one of the values $H_0, H_1, \dots$ from a measurement $R=r$. | ||
| - | ===== Maximum | + | ===== Maximum a posteriori rule ===== |
| 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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| where $f_{R|H}$ is the conditional PDF of the random variable $R$ given a certain $H$, and $f_R$ is the PDF of $R$. | where $f_{R|H}$ is the conditional PDF of the random variable $R$ given a certain $H$, and $f_R$ is the PDF of $R$. | ||
| - | 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 $P(H_0) f_{R|H}(r|H_0)} < P(H_1) f_{R|H}(r|H_1)}$, then announce | + | * If $P(H_0) f_{R|H}(r|H_0) < P(H_1) f_{R|H}(r|H_1)$, |
| + | ===== Likelihood ratio test ===== | ||
| + | The likelihood ratio $\Lambda(r)$ is defined as: | ||
| + | |||
| + | $$ \Lambda(r) = \frac{f_{R|H}(r|H_1)}{f_{R|H}(r|H_0)} $$ | ||
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| + | We can compare this likelihood ratio to the threshold $\eta$, which is the ratio between the a priori probabilities: | ||
| + | |||
| + | $$ \eta = \frac{P(H_1)}{P(H_0)} $$ | ||
| + | |||
| + | If $ \Lambda(r) > \eta $, then announce $' | ||
| + | |||
| + | ===== Terminology for different probabilities ===== | ||
| + | |||
| + | Probability of miss (probability we announce $H = H_0$, when in reality $H = H_1$): | ||
| + | |||
| + | $$ P_M = P(' | ||
| + | |||
| + | Probability of false alarm (probability we announce $H = H_1$, when in reality $H = H_0$): | ||
| + | |||
| + | $$ P_{FA} = P(' | ||
| + | |||
| + | Probability of detection (probability we announce $H = H_1$ given that $H = H_1$): | ||
| + | |||
| + | $$ P_D = P(' | ||
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| + | True negative rate/ | ||
| + | |||
| + | $$ 1 - P_{FA} = P(' | ||
| + | |||
| + | Positive predictive value (probability that $H = H_1$ given that we announce $H = H_1$): | ||
| + | |||
| + | $$ P(H_1| ' | ||
| + | |||
| + | Negative predictive value (probability that $H = H_0$ given that we announce $H = H_0$): | ||
| + | |||
| + | $$ P(H_0| ' | ||