PSYCH 240 1st Edition Lecture 20Bayesian Statistics-The goal of Bayesian statistics is to estimate the probability that a claim or a hypothesis is true-The Bayesian approach provides a method for updating the probability that a hypothesis is true when new observations are considered-Null hypothesis significance tests only make sure that evidence is specific. That is why they cannot tell you that a hypothesis is true or false. -You have to know two other things that a hypothesis test doesn’t tell you to figure that out. Bayesian statistics attempts to consider all of the factors relevant to the truth of a claim-Claims are more likely to be true if they are…oMore plausibleoSupported by more sensitive evidenceoSupported by more specific evidence Three Ingredients of the Truth-Plausibility of the Claim: probability that the claim/hypothesis would be true given what you knowbefore you get new evidenceo"Plausible" means that p(H) is high and p(~H) is lowoGiven your background knowledge, the hypothesis is likely to be true and unlikely to be false-Sensitivity of the Evidence: probability that the new evidence would be observed if the claim was trueo"Sensitive" evidence means that p(D|H) is highoThe evidence offered is something you would expect to observe if the hypothesis was really true-Specificity of the Evidence: probability that the new evidence would be observed if the claim was false o"Specific" evidence means that p(D|~H) is lowoThe evidence offered is something you would NOT expect to observe if the hypothesis was really false-These factors can override each other; for example very strong evidence can show that even implausible claims are almost certainly true-Likelihood Ratio (LR): the probability of having the evidence if the hypothesis was true divided by the probability if the hypothesis was falseooLR of 1 means that the evidence is irrelevant to the hypothesisThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.-The evidence is equally likely to be observed regardless of whether the hypothesis it true or falseoLR above 1 means the evidence favors the hypothesis, with higher values meaning stronger evidence-LR value of 2 means the evidence is twice as likely to be observes if the hypothesis is true, 10 means 10 times more likely, etc.oLR value below 1 means the evidence goes against the hypothesis, with lower values meaning stronger evidence against-LR value of 1/2 means the evidence is twice as likely to be observed if the hypothesis isFALSE, 1/10 means 10 times more likely, etc. Symbols-H - hypothesis-D - data-p() - probability-| - "given"-~ - "not" (negates something or indicates that it is false) Bayes Theorem-op(H|D) - probability that the hypothesis is true given the observed dataop(H) (prior probability) - the overall probability that the hypothesis is true given what you already know before new evidence is offered for or against the hypothesisop(D|H) - the probability of observing the new evidence given that the hypothesis is trueoLikelihood: the probability of observing data under a given hypothesisoNumerator - overall chance that the hypothesis will be true AND you will observe the dataoDenominator - the overall probability of observing the data regardless of whether or not the hypothesis is true or false (probability of the data)-p(H) x p(D|H) - the probability that the hypothesis is true and you will observe the data-p(~H) x p(D|~H) - the probability that the hypothesis is FALSE and you will observe the dataoThe probability that the hypothesis is false will always be 1 - the prior probability that the hypothesis is true -p(~H) = 1-p(H)-Problems will only give you p(H)oThe whole equation is saying, "out of all the times you observe evidence like this, how many times is the hypothesis true." Example-You’ve got an old car, and you are worried that it might need repairs. You ask a mechanic, and she says that there is a 60% chance that your type of car would need repairs at your mileage. A week later, your check engine light comes on. The probability that the check engine light would come on if you need a repair is 30%. The probability that it would come on if you don’t need a repair is 2%. Now that the check engine light is on, what is the probability that you need a repair? op(H) = .6op(~H) = .4op(D|H) = .3op(D|~H) = .02p(H|D) =