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UI STAT 4520 - Bayesian Statistics

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122S:138Bayesian StatisticsWhat is Bayesian Statistics?Lecture 1Aug. 25, 2008Kate Cowles374 SH, [email protected] Scientific Method1(But it’s not just for “science”...)1. Ask a question or pose a problem.2. Assemble and evaluate the relevant information.• (Take stock of what is already known.)3. Based on current information, design an investigationor experiment (or perhaps no experiment) to addre ssthe question posed in step 1.• Consider costs and benefits of the available experi-ments, including the value of any information theymay contain.• Recognize that step 6 is coming.4. Carry out the investigation or experiment.5. Use the evidence from step 4 to update the previouslyavailable information; draw conclusions, if only tenta-tive ones.6. Repeat steps 3 through 5 as necessary.1as stated by Don Berry3Where does statistics fit in?• Central to steps 2, 3, and 5• May help with step 1– can help show that a question is inappropriate– may show that answering the question will be dif-ficult or imp os s ible• Bayesian statististics is particularly well-suited forsteps 2 and 5.4Who started it all? Thomas BayesBorn: 1702 in London, EnglandDied: 17 April 1761 in Tunbridge Wells, Kent, England• ordained “Nonconformist” minister in England• Essay towards solving a problem in the doctrine ofchances– set out Bayes’s theory of pr obabili ty– published in the Philosophical Transactions of theRoyal Society of London in 1764– The paper was sent to the Royal Society by RichardPrice, a friend of Bayes’, who wrote:I now send you an es say which I havefound among the papers of our deceased friendMr Bayes, and which, in my opinion, hasgreat merit... In an introduction which he haswrit to this Essay, he says, that his design atfirst in thinking on the subject of it was, tofind out a method by which we might judgeconcerning the probability that an event hasto happen, in given circumstances, upon sup-posit ion that we know nothing concerning it5but that, under the same circumstances, ithas happened a certain number of times, andfailed a certain other number of times.• Bayes’s con clus ions were accepted by Laplace in a1781 memoir, redis covered by Condorcet (as Laplacementions), and remained unchallenged until Boole ques-tioned them in the Laws of Thoug ht. Since thenBayes’ techniques have been su bject to controversy.• elected a Fellow of the Royal Society in 1742 despitethe fact that at that time he had no published workson mathematics. Indeed none were published in hislifetime under his own name.6Some settings in which Bayesian statistics isused today• economics and econometrics• marketing• social science• education• health policy• medical research– more common in England than in US– but FDA has approved some new medical devicesbased on Bayesian analysis and is pushing the useof Bayesian methods in device testing• weather• the law• etc., etc.7Simple inference using Bayes’ ruleExample: Do you have a rare disease?• Your friend is diagnosed with a rare dis ea se that hasno obvious symptoms.• You wish to determine how likely it is that you, too,have the disease.That is, you a re uncertain about your true diseasestatus.• Your friend’s doctor has told her th at– The proportion of people in the general populationwho have the disease is .001.– The disease is not contagious.• A blood test exists for this disease, but it sometimesgives incorrect results.8Quantifying uncertainty using probabilitiesThe long-run frequency definition of the proba bility of aneventThe probability of an event is the proportionof the time it would occur in a long sequence ofobservations (i.e. as the number of trials tends toinfinity).• example: when we say that the probability of gettinga head on a toss of a fair coin is .5, we mean that wewould expect to get a head half the time if we flip pedthe coin a huge number of times under exactly thesame conditions• requires a sequence of repeatable experiments• no frequency interpretation possible for probabilitiesof many kinds of events– including the event that you have the rare disease9Probability as degree of beliefThe subjective definition of probability isA probability of an event is a number between0 and 1 that measures a par ticular person ’s sub-jective opinion as to how likely that event is tooccur (or to have occurred).• applies whenever th e person in question has an opin-ion about the event– if we count ignorance as an opinion, always applies!• Different people may have different subjective proba-bilities regarding the same event.• The same person’s subjective probability may changeas more information comes in.– where Bayes’ rule comes in10Properties of probabilitiesThese properties apply to probability whichever definitionis bei ng us ed.• Probabilities must not be negative. If A is any event,thenP (A) ≥ 0• All possible outcomes together must have probability1.If S is the sample space in a probability model thenP (S) = 111Back to the example• two possible events or models1. you have the disease2. you don’t have the dise ase• before taking any blood test, you think your chanceof having the disease is similar to that of a randomlyselected p er son in the pop ulation– so you assign the following prior probabilities tothe two modelsMODELPRIORHave disease .001Don’t have disease.99912Data• You decide to take the blood test.– the new information that you obtain to learn aboutthe different models is called data– the different possible data results are called obser-vations or outcomes– the data in this example is the result of the bloodtest• The two possible observations are– a “positive” blood test (+) — suggests you havethe disease– a “negative” blood test (-) — suggest you don’thave the disease13Likelihoods• The probabilities of the two poss ibl e test results aredifferent depending on whether you have the diseaseor not.• these probabilities are called likelihoods — the proba-bilities of the different data outcomes conditional oneach p os si ble model.LIKELIHOODSMODELPRIOR P(+ | MODEL) P(- | MODEL)Have disease .001 .95 .05Don’t have disease.999 .05 .9514Using Bayes’ rule to update probabilities• Bayes’ rule is the formula for upda ting your probabil-ities ab out t he models given the data.• enables you to compute posterior probabilities giventhe observed data– posterior means afterBayes’ rule


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