Decision-making and rationality

Transcription

1 Reslence Informatcs for Innovaton Classcal Decson Theory RRC/TMI Kazuo URUTA Decson-makng and ratonalty What s decson-makng? Methodology for makng a choce The qualty of decson-makng determnes success or falure of nnovaton and busness. Ratonal decson-makng No means exst that certanly lead us to a correct decson due to uncertantes n the world. Anyhow, we need to decde ratonally n some sense, wth no regret. Max. use of nformaton 1

2 Classcal decson-makng Normatve model of decson-makng Choosng one opton from many Utlty s what one wants to make maxmum n the choce. Collect nformaton Assess optons Opton 1 Opton 2 Opton 3 redefned optons Utlty of optons u 1, u 2, u 3,... Choce Opton Choce and preference reference An act to chose one opton from many Basc unt of classcal decson Order of preference n a dyad a b a f b a f b a s preferred to b. a s equvalent to b. a s preferred or equvalent to b. 2

3 Utlty and utlty functon Utlty What one wants to make maxmum n the choce Subectve value of each opton evaluated by the decson-maker Utlty functon ux Mappng from the set of optons to real numbers a f b u a u b Utlty functon s a tool to deal wth the utlty mathematcally Condton for ratonal decson In order that the preference s mathematcally consstent, t must be a weak order relaton. Completeness a f b or a pb Transtvty a f b, bf c for any par of a and b af c 3

4 xpected utlty Defnton of expected utlty s p us n 1 u s p : possble state : occurrence probablty of s : utlty of s xpected utlty hypothess The utlty of opton a under uncertanty s represented by ts expected utlty a. xample of expected utlty Wn.prob. rze Consolaton Lottery $15,000 none Lottery $8,000 none Lottery $10,000 $50 L x $15,000 $150 L x $8,000 $160 L x $10, x $50 $

5 Choce under uncertanty Uncertan stuaton s : ossble state p : Occurrence probablty of s a u : Opton of acton : Utlty when s obtans after a has been taken Varous decson crtera 1 xpected utlty crteron n a* a s.t. U a u 1 If the occurrence probablty dstrbuton s unknown, suppose p 1/n Laplace crteron Max-mn crteron a* a s.t. U a mn u Maxmze the utlty for the worst case p max max 5

6 Varous decson crtera 2 urwtz crteron a* a s.t. U a α max u + 1 α mn u α 0 and α 1 correspond to complete pessmsm max-mn crteron and complete optmsm. Regret crteron Mnmum opportunty loss a* a s.t. L a max r mn r max u u Opportunty loss r s the degree of regret compared wth the maxmum utlty obtanable wth perfect foresght. max xample of choce under uncertanty 1 You are a street food stall owner. Whch sells well depends on the weather: ce cream or hot dog. Whch wll you stock more for tomorrow s busness. Weather Sunny Cloudy Rany robablty Ice cream 1, alf & half ot dog

7 xample of choce under uncertanty 2 Crteron xpected Laplace Max-mn Ice cream alf & half ot dog Weather urwtz* Regret Ice cream alf & half ot dog * α 0.5 Condtonal probablty U : Sample space U, : Subset of U X : Sze of X Condtonal probablty U U 7

8 xample of condtonal probablty Two nvsble bns A and B A : 3 slver & 1 gold cons B : 2 slver & 4 gold cons start 1/2 1/2 3/4 S A 1/4 G 2/6 S B 4/6 G 3/8 1/8 1/6 2/6 You drew a slver con from ether of the two bns by chance. Whch bn dd you choose? A S 3/8 9 A S S 3/ 8+ 1/ 6 13 Independent events When event and satsfy the followng condtons, they are ndependent. The followng s the necessary and suffcent condton for that and are ndependent. 8

9 9 Bayes theorem General form of Bayes theorem 1, 2, n are exclusve each other and U n L n n + + L Cons-n-bns problem revsted Two nvsble bns A and B A : 3 slver & 1 gold cons B : 2 slver & 4 gold cons You drew a slver con from ether of the two bns by chance. Whch bn dd you choose? / 3 1/ 2 1/ 4 3/ 2 1/ 4 3/ + + B B S A A S A A S S A A start B 1/2 G G S 1/2 3/4 S 1/4 2/6 4/6 3/8 1/8 1/6 2/6

10 10 Bayesan nference 1 : ypothess : vdence, testmony, or symptom : ror probablty wth no evdence : osteror probablty after evdence has been obtaned Bayesan nference 2 After has been obtaned, how we should modfy the probablty of? Modfcaton of odds n terms of O O λ ror odds osteror odds

11 11 Bayesan nference 3 After another evdence has been obtaned, O O λ λ Modfcaton factor for each evdence asty doctor example You receved a cancer test. The doctor sad that 1 out of 1,000 s the average rate of cancer at your age. The test s accurate and t gves a correct result for 99% of cancer patents and 97% of non cancer patents. Your test result was postve, and the doctor recommended you hosptalzaton ASA C C C C C O

12 Three prsoners problem Monty all problem Out of three prsoners, two wll be executed and one wll be freed tomorrow. rsoner A heard from the alor that B wll be executed. A was delghted that hs alve probablty has ncreased from 1/3 to 1/2 wth ths nformaton, snce ether A or C wll be freed. A b A A b A b A + B b B + C b C 1/ 3 1/ 2 1/ 3 1/ 2 + 1/ / Component falure example alure statstcs of a partcular component are collected for every 100 days of operaton, and the followng data were obtaned. valuate the falure rate. Interval Tmes of falures Average /4/ day -1 STD 0.02 day -1 12

13 robablty densty functon Cumulatve dstrbuton functon x robablty that a random varable X takes a value not greater than x x < X x robablty densty functon fx d x x f x x f x dx dx f x dx x < X x + dx Bayesan nference on dstrbuton f θ f θ A : ror densty functon of θ : osteror densty functon of θ Bayesan update of the densty functon after event A has been observed f θ A A θ f θ A θ f θ dθ 13

14 Applcaton of Bayesan approach to component falure example osson dstrbuton k λt k λ e k! λt λ : alure rate T : Interval of observaton 100 days k : Tmes of falures observed Unform dstrbuton n [0, 0.1] s assumed for f λ at the begnnng. Bayesan update of falure rate λa k 6 2 k 3 3 k 5 4 k λ day -1 14

15 Concepts of probablty 1 Normatve defnton ascal racton of number of cases among the sample space p A / U Statstcal defnton d Alembert Asymptotc value of event frequency p lm f n n / n Concepts of probablty 2 Subectve defnton Bayes Degree of ndvdual confdence on occurrence of the event Informatcs defnton Shannon Amount of nformaton that mples occurrence of the event S log 2 p 15