In applying Bayes' Rule, if the likelihood ratio is .20/.75, what can you determine based on this alone?
Added by David M.
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The likelihood ratio is the ratio of the probability of the evidence under two different hypotheses. In this case, it is given as 0.20 (for one hypothesis) and 0.75 (for another hypothesis). Show more…
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2 Bayes Rule: Likelihood-Ratio - 2 points P(A|n) / P(B|n) = P(A) / P(B) x P(n|A) / P(n|B) In words, we can say that the posterior equals the prior multiplied by the signal. Recall the model introduced in the lectures. A professor is trying to decide if a student is good or bad. The prior belief that the student is good is 1/2. Then the professor receives signals (tests), where g represents a good test performance and b represents a bad test performance. Once a professor believes it more than 50% likely that a student is good (bad), she will misinterpret a bad (good) test performance as good (bad) with probability q. Also good students do good on tests and bad students do bad on tests with P(g|G) = P(b|B) = p. (a) Suppose q = 1/2 and p = 3/4. Given a naive professor observes ĝĝĝ, what is her posterior probability that the student is good? (b) Suppose the sophisticated professor's prior probability that the student is good is 50/100? What is her posterior given she observes ĝĝ? (c) Suppose the sophisticated professor's prior probability that the student is good is 51/100? What is her posterior given she observes ĝĝ? (d) Compare your answers to (b) and (c). What is counter-intuitive here? Where is it coming from in the model? (e) What is the professor's posterior if it is equally likely that good students and bad students do well on tests and she has no bias? Answer in words, appealing to the likelihood-ratio formulation of Bayes' Rule above.
Sri K.
Let $X$ have one of the following distributions: $$\begin{array}{ccc} \hline X & H_{0} & H_{A} \\ \hline x_{1} & .2 & .1 \\ x_{2} & .3 & .4 \\ x_{3} & .3 & .1 \\ x_{4} & .2 & .4 \\ \hline \end{array}$$ a. Compare the likelihood ratio, $\Lambda,$ for each possible value $X$ and order the $x_{i}$ according to $\Lambda$. b. What is the likelihood ratio test of $H_{0}$ versus $H_{A}$ at level $\alpha=.2 ?$ What is the test at level $\alpha=.5 ?$ c. If the prior probabilities are $P\left(H_{0}\right)=P\left(H_{A}\right),$ which outcomes favor $H_{0} ?$ d. What prior probabilities correspond to the decision rules with $\alpha=.2$ and $\alpha=.5 ?$
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