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# Bayesian Statistics 22S 138

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This 21 page Class Notes was uploaded by Vance Bode Sr. on Friday October 23, 2015. The Class Notes belongs to 22S 138 at University of Iowa taught by Mary Cowles in Fall. Since its upload, it has received 28 views. For similar materials see /class/228077/22s-138-university-of-iowa in Natural Sciences and Mathematics at University of Iowa.

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Date Created: 10/23/15

2281138 Bayesian Statistics Calibration Experiments and Review of Probability Lecture 2 Aug 297 2007 Kate Cowles 374 SH7 33570727 kcowles statuiowaedu Calibration experiments 0 calibration ewperiment 7 a scale used to assess a persons degree of belief that a par ticular event will occur or has occurred 0 All outcomes of the calibration experiment must be equally likely in the opinion of the person whose subjective probability is being assessed 7 Example imagine that I promise to pay you 100 if the roll of a 6rsided die comes up the number you call lf you are indifferent as to which numr ber you call7 the 6 possible outcomes are equally likely for you 2 Assessing subjective probability about events 0 We may sometimes need to quantify our subr jective probability of an event in order to make a decision or take an action 0 Example You have been offered a job as a statistir cian with a marketing rm in Cincinnati 7 ln order to decide whether to accept the job and move to Cincinnati7 you wish to quantify your subjective probability of the event that you would like the job and would like Cincinnati 7 We will talk about Bayesian decision the cry later in the semester 4 0 Calibration experiments may be useful if 7 person is not knowedgeable or comfortable with probability 7 person is uncertain as to his her opinion about the event 0 principle of using a calibration experiment to assess subjective probability 7 Person is offered a choice of 2 ways of wine ning a prize gtk through a realization of the calibration experiment with known probability of success gtk through the occurrence of the event of interest 7 The calibration experiment is adjusted at successive steps Example Using a chipsrinrarbowl experiment to assess your subjective probability regarding event A that the Department of Physics at Florida State University has more than 2 female faculty meme bersi 0 Experiment is having a blindfolded person draw one chip at random from a bowl con taining chips of the same size and shape 0 Let PS A denote your subjective probability that event A has occurred 7 o The bowl contains 3 green chips and 1 red chipi You may choose Game 1 or Game 2 If you choose Game 27 then I conclude that your 025 lt PSA lt 050 0 Then I may go on to Step 3 Now the bowl contains 5 green chips and 3 red chips You may choose Game 1 or Game 2 If you choose Game 17 I conclude that 025 lt PA lt 0375 0 Step 1 The bowl contains 1 green chip and 1 red chipi Imagine that you may choose 1 of two games 1 I will be blindfolded and draw one chip at randomi I will pay you 100 if the chip drawn is red I will pay you nothing if it is green 2 I will pay you 100 if the Physics Dept at FSU has more than 2 female faculty 1 will pay you nothing if it has 2 or fewer If you choose Game 17 then I conclude that 0 lt PSA lt 05 So 1 construct Step 2 as follows 8 0 We continue until it becomes too dif cult for you to choose between the two games o How the chips are set up in the bowl at each step is determined by your answer at the pre ceding step 0 Comments i The payoffs have to be imaginary7 because we wish to use this procedure for assessing unveri able probabilities Luckily7 a high degree of accuracy in as sessing subjective probabilties usually is not needed Quick review of probability 0 event any outcome or set of outcomes of a random phenomenon the basic element to which probability can be applied usual notation is a capital letter near be ginning of alphabet example random phenomenon is that we are drawing a patient at random from a huge database of patients insured by an gtk event A is the event that the patient we draw is under 6 years of age 7 we will denote the probability of event A as PA 0 sample space S the set of all possible out comes of a random phenomenon 7M1 11 o complement of an event A is the event not A77 notated AC or A 7 A U A S o the null event 7 an event that can never happen notated Q 0 events A and B are disjoint or mutually exclusive if they cannot occur together fie if AnB 6 7 example event A is that the patient we draw is under 6 years of age7 and event B is event that the patient is 6 to 11 years of age 10 o intersection of two events A and B is the event both A and B notated A n B 7 example if event B is the event that the patient we draw weighs at least 150 pounds7 then A n B is the event that the patient we draw is under 6 years of age and weighs at least 150 pounds 0 union of two events is event either A or B or both notated A U B o A set of events A1 A2 A3 are exhaus tive if AlUAQU A3US o additive rule of probability 7 if two events A and B are mutually exclur sive7 then PAU B PA PB PAC 1 i PA Conditional Probability o PB A 7 the probability that event B will 14 Patients in the database example gt occur given that we already know that event lt 150 pounds 7 150 pounds TOtal A h d under 6 798 2 800 as 00mm 2 6 4702 4498 9200 o multiplicative rule of probability Total 5500 4500 10000 PA B PAPB A PA B PBPA B 0 so if PA 7f 0 PA n B P B A lt 4 gt PM Independence Patients in the database example 0 Two events are independent if the occurrence lt 150 pounds gt 150 pounds Total or nonroccurrence of one of them does not green eyes 7 800 affect the probability that the other one ocr not green 5060 4140 9200 curs Events A and B are independent if Total 5500 4500 10000 PA B PA P B W 133 o multiplicative rule of probability for inde pendent events PAn B PAPB Law of Total Probability o Applies when you wish to know the marginal unconditional probability of some event7 but you only know its probability under some conditions 0 Example l have asked my friend to mail an imporr tant letter fl want to calculate PA7 the probabilr ity that the letter will reach the addressee within the next 3 days i I believe that PM7 the probability that my friend will remember to mail the letter today or tomorrow7 is 60 19 Using the Law of Total Probability to nd PA o For any events A and M A AnM u AnM 0 Events AnM and AHM are disjoint7 so the addition rule says PA PltA Mgt PAn 0 And the multiplication rule applied to both terms on the right hand side says PA PA MPM PA 7P7 o For the example PA 9560 0000140 57004 i I believe that if my friend mails the letter today or tomorrow7 the probability that the postal service will deliver it to the ad dressee within the next 3 days is 95 PA M 95 fl believe there7s only 1 chance in 10000 that the letter will get there somehow if my friend forgets to mail it PAM 0001 20 General Law of Total Probability 0 Suppose there were many different conditions under which the event of interest could occur o If M1 M2 M3 are mutually exclusive and exhaustive events then PM PAM1PM1PAM2PM2PAM3PMg Bayes7 Rule discrete case 0 My prior probability that my friend would remember to mail the letter was PM o The data is that the letter actually arrived within 3 daysl o Bayes7 rule calculates my posterior probabilr ity that my friend mailed the letter given the data7 that is PM V1 when we know PA M PA I7 and Generalized Bayes7 Rule 0 corresponds to Generalized Law of Total Probe ability 0 assumes that the event A could happen cone ditional on one of a number of different other events7 M1 M2 M3 0 assumes that we know PA M17 PA M27 etc as well as PM17 PM27 etc 0 after the event A has occurred7 we want to assess the conditional probability of one of the events Mj7 PMj A o If M1 M2 M3 are mutually exclusive and exhaustive events then PM7A P PMWJHMJ A MUPfMO PA M2PM2 PA M3PMa 22 o By the de nition of conditional probabilility P M n A PA 0 Using the multiplication rule to expand the numerator7 and the law of total probability to expand the denominator gives Bayes7 rule PAMPM PMA P M A H lt PAMPM PAMPM o For the example this is i 9560 PalHA 9560 0000140 i 57 57004 99993 228138 Bayesian Statistics Fall 2007 Instructor Cowles Information on Final Exam 0 The nal will be given at 730 am Friday December 21 2007 in the classroom 0 It will be cumulative and may cover the following topics in addition to anything that was listed for midterms 1 and 2 7 Bayesian regression 7 Hierarchical normal linear models 7 Model checking and comparison Bayes factors posterior probabilities and odds posterior predictive checks model comparison using the Deviance Information Criterion 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring three 8 12 X 11 sheets of paper with your own notes on them your table of distributions and a calculator 228138 Bayesian Statistics Fall 2006 Instructor Cowles Information on Midterm 1 o Midterm 1 will be given Fri 0929 in class 0 It will cover the following topics 7 Probability addition rule multiplication rule conditional probability indepen dence law of total probability 7 Probability Bayes theorem 7 Assessing a subjective probability 7 Priors informative and noninformative conjugate and nonconjugate Jeffreys priors 7 Likelihoods 7 Posterior distributions and inference using them 7 Posterior predictive distributions 7 Robustness 7 Bayesian inference for one parameter models binomial probability Poisson rate parameter normal mean variance assumed known normal variance or precision mean assumed known possibly others 7 use of RSplus functions that have appeared in lab lecture or homework 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring one 8 12 X 11 sheet of paper with your own notes on it the table of distributions 1 gave you and a calculator 2281387 Bayesian Statistics Fall 2007 Instructor Cowles Information on Midterm 2 o Midterm 2 will be given Fri 1102 in class 0 It will cover the following topics 7 Exchangeability 7 Normal models with both In and 02 unknown 7 Hierarchical models 7 Directed graphs 7 Deriving full conditional distributions 7 Interpreting WinBUGS output graphs and statistics 0 Questions will be mostly short answer7 with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring one 8 12 X 11 sheet of paper with your own notes on it and a calculator and a photocopy of the table of distributions from your textbook 228138 Bayesian Statistics Fall 2005 Instructor Cowles Information on Final Exam 0 The nal will be given at 215 PM Monday December 12 2005 in the classroom 0 It will be cumulative and may cover the following topics in addition to anything that was listed for midterms 1 and 2 7 Bayesian regression 7 Hierarchical normal linear models 7 Model checking and comparison Bayes factors posterior probabilities and odds posterior predictive checks model comparison using the Deviance Information Criterion 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring three 8 12 X 11 sheets of paper with your own notes on them your table of distributions and a calculator 2281387 Bayesian Statistics Fall 2008 Instructor Cowles Information on Midterm 2 o Midterm 2 will be given Fri 1031 in class 0 It will cover the following topics 7 Exchangeability 7 Normal models with both In and 02 unknown 7 Hierarchical models 7 Directed graphs 7 Deriving full conditional distributions 7 Interpreting WinBUGS output graphs and statistics 0 Questions will be mostly short answer7 with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring one 8 12 X 11 sheet of paper with your own notes on it and a calculator and a photocopy of the table of distributions from your textbook 228138 Bayesian Statistics Fall 2007 Instructor Cowles Information on Final Exam 0 The nal will be given at 1200 noon Tues Dec 16 in the classroom 0 It will be cumulative and may cover the following topics in addition to anything that was listed for midterms 1 and 2 7 Bayesian regression 7 Hierarchical normal linear models 7 Generalized linear models 7 Model checking and comparison Bayes factors posterior probabilities and odds posterior predictive checks model comparison using the Deviance Information Criterion 7 Likelihood principle 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring three 8 12 X 11 sheets of paper with your own notes on them your table of distributions and a calculator 228138 Bayesian Statistics Fall 2008 Instructor Cowles Information on Midterm 1 o Midterm 1 will be given Fri 1003 in class o It will cover the following topics 7 Probability addition rule multiplication rule conditional probability indepen dence law of total probability 7 Probability Bayes7 theorem 7 Calibration experiments for assessing a subjective probability 7 Priors informative and noninformative conjugate and nonconjugate Jeffreys7 priors 7 Likelihoods 7 Posterior distributions and inference using them 7 Posterior predictive distributions 7 Robustness 7 Bayesian inference for oneparameter models binomial success probability Poisson rate parameter normal mean variance assumed known normal variance or precision mean assumed known possibly others 7 use of RSplus functions that have appeared in lab lecture or homework 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring one 8 12 X 11 sheet of paper with your own notes on it a photocopy of the table of distributions from your textbook and a calculator 228138 Bayesian Statistics Fall 2005 Instructor Cowles Information on Midterm 1 o Midterm 1 will be given Fri 0930 in class 0 It will cover the following topics 7 Probability addition rule multiplication rule conditional probability indepen dence law of total probability 7 Probability Bayes theorem 7 Assessing a subjective probability 7 Priors informative and noninformative conjugate and nonconjugate Jeffreys priors 7 Likelihoods 7 Posterior distributions and inference using them 7 Posterior predictive distributions 7 Robustness 7 Bayesian inference for one parameter models binomial probability Poisson rate parameter normal mean variance assumed known normal variance or precision mean assumed known possibly others 7 use of RSplus functions that have appeared in lab lecture or homework 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring one 8 12 X 11 sheet of paper with your own notes on it a photocopy of the table of distributions from your textbook and a calculator 225 138 Bayesian Statistics Calibration Experiments and Review of Probability Lecture 2 Aug 23 2006 Kate Cowles 374 SH7 33570727 kcowles statuiowaedu Calibration experiments 0 calibration experiment 7 a scale used to assess a persons degree of belief that a pare ticular event Will occur or has occurred 0 All outcomes of the calibration experiment must be equally likely in the opinion of the person Whose subjective probability is being assessed 7 Example imagine that I promise to pay you 100 if the roll of a 6rsided die comes up the number you call elf you are indifferent as to Which numr ber you call7 the 6 possible outcomes are equally likely for you 2 Assessing subjective probability about events 0 We may sometimes need to quantify our subr jective probability of an event in order to make a decision or take an action 0 Example 7 You have been offered a job as a statistir cian with a marketing rm in Cincinnati 7 In order to decide Whether to accept the job and move to Cincinnati7 you Wish to quantify your subjective probability of the event that you would like the job and would like Cincinnati 7 We Will talk about Bayesian decision the cry later in the semester 4 0 Calibration experiments may be useful if 7 person is not knowedgeable or comfortable with probability person is uncertain as to hisher opinion about the event 0 principle of using a calibration experiment to assess subjective probability 7 Person is offered a choice of 2 ways of Wine ning a prize gtk through a realization of the calibration experiment with known probability of success gtk through the occurrence of the event of interest 7 The calibration experiment is adjusted at successive steps Example Using a chipsrinrarbowl experiment to assess your subjective probability regarding event A that the Department of Physics at Florida State University has more than 2 female faculty meme bers 0 Experiment is having a blindfolded person draw one chip at random from a bowl con taining chips of the same size and shape 0 Let 13514 denote your subjective probability that event A has occurred 7 o The bowl contains 3 green chips and 1 red chip You may choose Game 1 or Game 2 If you choose Game 2 then I conclude that your 025 lt P5A lt 050 0 Then I may go on to Step 3 Now the bowl contains 5 green chips and 3 red chips You may choose Game 1 or Game 2 If you choose Game 17 I conclude that 025 lt PA lt 0375 0 Step 1 The bowl contains 1 green chip and 1 red chip Imagine that you may choose 1 of two games 1 I will be blindfolded and draw one chip at random 1 will pay you 100 if the chip drawn is red I will pay you nothing if it is green 2 I will pay you 100 if the Physics Dept at FSU has more than 2 female faculty 1 will pay you nothing if it has 2 or fewer If you choose Game 17 then I conclude that 0 lt P5A lt 05 So 1 construct Step 2 as follows E 0 We continue until it becomes too dif cult for you to choose between the two games o How the chips are set up in the bowl at each step is determined by your answer at the pre ceding step 0 Comments i The payoffs have to be imaginary7 because we wish to use this procedure for assessing unveri able probabilities Luckily7 a high degree of accuracy in as sessing subjective probabilties usually is not needed Quick review of probability 0 event any outcome or set of outcomes of a random phenomenon the basic element to which probability can be applied usual notation is a capital letter near be ginning of alphabet example random phenomenon is that we are drawing a patient at random from a huge database of patients insured by an HMO gtk event A is the event that the patient we draw is under 6 years of age we will denote the probability of event A as PA 0 sample space S the set of all possible out comes of a random phenomenon 7PS1 11 o complement of an event A is the event not A notated AC or A 7 A u A S o the null event 7 an event that can never happen notated Q 0 events A and B are disjoint or mutually exclusive if they cannot occur together fie ifAnB Q iexample event A is that the patient we draw is under 6 years of age7 and event B is event that the patient is 6 to 11 years of age 10 o intersection of two events A and B is the event both A and B notated A n B 7 example if event B is the event that the patient we draw weighs at least 150 pounds7 then A n B is the event that the patient we draw is under 6 years of age and weighs at least 150 pounds 0 union of two events is event either A or B or both notated A U B o A set of events A17 A27 A37 are exhaus tive if AlUAQU A3U 3 o additive rule of probability 7 if two events A and B are mutually exclur sive7 then PAUB PA 133 PAC17 PA Conditional Probability o PB A 7 the probability that event B Will 14 Patients in the database example gt occur given that we already know that event lt 150 pounds 7 150 pounds TOtal A h d under 6 798 2 800 as 000mm 2 6 4702 4498 9200 o multiplicative rule of probability Total 5500 4500 10000 PA B PAPB A PA B PBPA B 0 so if PA 7 0 PM n B P B A lt l gt PM Independence Patients in the database example 0 TWO events are independent if the occurrence lt 150 pounds gt 150 pounds Total or nonroccurrence of one of them does not green eyes 7 800 affect the probability that the other one ocr not green 5060 4140 9200 curs Events A and B are independent if Total 5500 4500 10000 PA B PM PBA 133 o multiplicative rule of probability for inde pendent events PAnB PAPB Law of Total Probability o Applies when you wish to know the marginal unconditional probability of some event but you only know its probability under some conditions 0 Example 1 have asked my friend to mail an imporr tant letter i I want to calculate PA the probabilr ity that the letter will reach the addressee within the next 3 days if believe that PM the probability that my friend will remember to mail the letter today or tomorrow is 60 Using the Law of Total Probability to nd PM o For any events A and M A AmM U AmM 0 Events AnM and AHM are disjoint so the addition rule says PA PAmM PA M 0 And the multiplication rule applied to both terms on the right hand side says PM PA MPM PA MPM o For the example PA 9560 0000140 57004 i I believe that if my friend mails the letter today or tomorrow the probability that the postal service will deliver it to the ad dressee within the next 3 days is 95 PA M 95 if believe there7s only 1 chance in 10000 that the letter will get there somehow if my friend forgets to mail it PAM 0001 20 General Law of Total Probability 0 Suppose there were many different conditions under which the event of interest could occur o If M1 M2 M3 are mutually exclusive and exhaustive events then PA PltA M1PltM1PA M2PM2PA M3PMg Bayes7 Rule discrete case 0 My prior probability that my friend would remember to mail the letter was PM o The data is that the letter actually arrived within 3 days 0 Bayes7 rule calculates my posterior probabilr ity that my friend mailed the letter given the data that is PUWA when we know PAlM PAlM and Generalized Bayes7 Rule 0 corresponds to Generalized Law of Total Probe ability 0 assumes that the event A could happen cone ditional on one of a number of different other events M1 M2 M3 0 assumes that we know PAlM1 PAlM2 etc as well as PM1 PM2 etc 0 after the event A has occurred we want to assess the conditional probability of one of the events Mj PMle o If M1 M2 M3 are mutually exclusive and exhaustive events then PltMAAgt P PMWJ My AlM1PM1 PAlM2PM2 PAlM3PM3 o By the de nition of conditional probabilility PMnA PM 0 Using the multiplication rule to expand the numerator and the law of total probability to expand the denominator gives Bayes7 rule PAlMPM P M A H l PA MPM PAlMPM o For the example this is PMA 9560 PltMW 57 57004 7 9999 228138 Bayesian Statistics Fall 2007 Instructor Cowles Information on Midterm 1 o Midterm 1 will be given Fri 1005 in class 0 It will cover the following topics 7 Probability addition rule multiplication rule conditional probability indepen dence law of total probability 7 Probability Bayes7 theorem 7 Calibration experiments for assessing a subjective probability 7 Priors informative and noninformative conjugate and nonconjugate Jeffreys7 priors 7 Likelihoods 7 Posterior distributions and inference using them 7 Posterior predictive distributions 7 Robustness 7 Bayesian inference for oneparameter models binomial probability Poisson rate parameter normal mean variance assumed known normal variance or precision mean assumed known possibly others 7 use of RSplus functions that have appeared in lab lecture or homework 0 Questions will be mostly short answer with a little calculation Some will involve interpreting computer output that I will provide 0 You may bring one 8 12 X 11 sheet of paper with your own notes on it a photocopy of the table of distributions from your textbook and a calculator

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