Probability Theory (revisited)

Transcription

1 Probablty Theory (revsted)

2 Summary Probablty v.s. plausblty Random varables Smulaton of Random Experments

3 Challenge The alarm of a shop rang. Soon afterwards, a man was seen runnng n the street, persecuted by a polceman. Is the man a thef? Can a machne reason lke us?

4 Jacob Bernoull ( )

5 Bernoull defnton Defnton of probablty attrbuted to Jacob Bernoull (1689): P m n m number of favorable cases n total number of cases Ths defnton establshes a lnk between probabltes and the output of experments. Queston: how to manpulate probabltes n a consstent way?

6 Kolmogorov ( )

7 Kolmogorov Axoms Kolmogorov defned a set of axoms for Probablty Calculus based on set theory and measure theory: He defnes: sgma algebra of sets, closed wth respect to complement and unon of a countable number of sets. a probablty measure for the sets belongng to the sgma feld, denoted as events. E A B Strong pont: all the operatons wth probabltes can be defned from the axoms n a consstent way. Queston: what s the relatonshp between probabltes and expermental data?

8 Probablty Space A probablty space conssts of: a sample space E, an event space F and a probablty measure P. E s the set of results of the random experment, F s a famly of subsets of E such that P s a functon from F nto [0,1] such that: 0,P(A ) tends to... A If dsjonts ), ( ) ( ) ( 1 ) (, 0 ) ( 2 1 A F A,B B P A P B A P E P F A A P F A F A F A F A F E countable set,

9 E. Jaynes ( )

10 Thnkng Robot How to buld a thnkng robot? Based on the works of Polya e Cox, Jaynes assgns a plausblty to each proposton and demonstrates that a consstent nductve logc must obey the rules of Probablty Calculus. In ths context, probabltes are assgned not to sets of a sgma feld but to propostons.

11 Random Varables A random varable assgns a numerc value to each experment. dscrete random varables contnuous

12 Dscrete Random Varables How to characterze a dscrete random varable? Probablty functon P( k) Pr{ x k} Propertes P( k) 0, k N k1 P( k) 1 Ex: S { 1,2,3,4}, P(1).1, P(2).3, P(3).4, P(4).2 Realzatons: Note: the same symbol wll be used to denote the random varable a realzaton. Dfferent symbols (e.g., captal and lower letters) could be used to make ths dfference more clear.

13 Bnomal Dstrbuton It answers the followng problem: what s the probablty of an event A beng observed k tmes n n random experments? n k P( k) (1 ) k nk =P(A) n= n= k k

14 Contnuous Random Varables How to characterze a contnuous random varable x? Probablty densty functon, p P{ x x o } x 0 p( x) dx Propertes p( x) 0, p( x) dx x R 1 n Ex: S R, p( x) 1, x [0,1[, p( x) 0, otherwse

15 Normal Dstrbuton Probablty densty functon N( x;, R) (2 ) n / 1 2 R 1/ 2 e 1 2 ( x)' R 1 ( x) R mean vector covarance matrx The level surfaces n R n are ellpsods centered n m and wth axs v where, v are egen values and egen vectors of R ( v 1)

16 Jont Dstrbuton The jont dstrbuton of x 1,..., x N, s defned on the set of values of the sequence, beng characterzed by Probablty functon P(x 1,..., x N ) (dscrete varables) Probablty densty functon p(x 1,..., x N ) (contnuous varables) Margnalzaton p( x1) P( x1, x2) x2 p( x2) P( x1, x2) x1

17 Independence Def: the r.v. x 1,..., x N are ndependent f and only f p( x 1,..., xn ) p( x ) ndependent varables dependent varables Covarances: R R Note: ndependent r.v. are converted nto dependent ones by applyng a non dagonal lnear transformaton

18 Correlaton Def: x 1,..., x N are correlated r.v. f ther covarance matrx s non dagonal. Notes: ndependent r.v. are always uncorrelated. The converse s not true. gven n r.vs. t s possble to decorrelate them by applyng a sutable lnear transformaton (e.g. KLT or PCA)

19 Condtonal Probabltes Defnton (condtonal probablty): P( x y) P( x, y)/ P( y) f P( y) 0 P(x y) s nterpreted as the probablty of occurrng x knowng that y occurred. Note: f x,y are contnuous random varables the condtonal probablty densty functon p(x y) s defned n an analogous way.

20 Expectaton Defnton (Expectaton): Let f: SR n E{ f ( x)} f ( x) P( x) x E { f ( x)} f ( x) p( x) dx (x - dscrete r.v.) (x - contnuous r.v.) Relatonshp wth the artmetc mean: E{ f ( x)} lm N N 1 N 1 f ( x ) x 1, x 2,... are realzaton of x Defnton (mean and covarance matrx): Let x be a random varable mean =E{x} covarance matrx: R=E{(x-)(x-)'}

21 Propertes of the Covarance Matrx A s a covarance matrx f and only f t s a square matrx, symmetrc and sem defnte postve. Other propertes: the egen values of a covarance matrx are non negatve. R, v 1 m v v ( v ' 1) m R 1 egen values and egen vectors

22 Propertes of Normal Dstrbuton If x 1,..., x n, are r.v. wth normal dstrbuton, any subset of varables x p1,..., x pm are also r.v. wth normal dstrbuton. Gven a r.v. x ~ N( x, R) the dstrbuton of y=ax+b s N( y, Ryy ) y Ax b, R ARA' yy Gven 2 varables x N( x, R ), y ~ N( y, R ) ~ xx yy then x y ~ N( x y, P) P R xx R yy f x, y are ndependent

23 Generaton of Random Values Dscrete varables: Splt [0,1[ nterval nto subntervals of length P(). generate a random value wth unform dstrbuton n [0,1[. The value of x s the ndex of the subnterval whch was selected. P(1) P(2) P(N) 0 1 x=2 Contnuous varables: specfc algorthms for some dstrbutons Metropols algorthm mportance samplng Gbbs sampler

24 Metropols Algorthm How to generate random values wth a gven dstrbuton? Metropols Algorthm: - move x randomly - accept the new value x wth probablty P mn( 1, p( x') / p( x)) otherwse make x =x Example 3 2 p ( x) ( 2x -4x 6)/ Hstogram e p(x) x

25 Importance Samplng It s used to compute expected values when t s dffcult to generate random values wth the true dstrbuton p(x) but t s possble to generate samples wth an auxlary dstrbuton q(x). Algorthm: generate n ndependent realzatons x ~q(x) assgn a weght to each realzaton (mportance) w = p(x ) /q(x ). n 1 n 1 Expectaton: E{ f ( x)} w f ( x ) Note: q ( x) 0, x : p( x) 0 poor performance n hgh dmenson spaces

26 Example We wsh to estmate 2 moments of a dstrbuton N(0,1) usng mportance samplng. We consdered n=100; 100 estmaton experments were performed m n n 2 1 n 1 wx m2 n 1 wx 1 1 m 1 m m2 1 m 2 q=n(0,.25) q=n(0,1) q=n(0,4) tal msmatch

27 Gbbs Sampler Problem: generate random values wth a known dstrbuton P( x 1,..., x N ) Algorthm: begn generate x 1 wth dstrbuton generate x 2 wth dstrbuton P( x P( x 1 2 / x 2 / x 1, x, x 3 3,..., x,..., x N N ) ) repeat generate x N wth dstrbuton P( x N / x 1, x 2,..., x N1 ) Ths algorthm generates a Markov wth asymptotc dstrbuton P( x 1,..., x N )

28 Optmzaton wth the Gbbs Sampler p Generate realzatons of a r.v. x wth dstrbuton p(x) a p 10 Change a untl a domnant mode s observed p 20 In the lmt the algorthm wll only generate values whch maxmze p. p 30 Dffculty: there are no optmal crtera for the evoluton of a

29 Problems x y P 1. Gven a dstrbuton P(x,y) defned by: ) P(x) ) P(y) ) P(x/y) v) E{x} v) E{y} v) E{x+y} v) E{xy} 2. The meanng of varables x,y,z s the followng : x-the s gas n the tank; y battery s OK; z- motor starts at frst attempt. Defne a probablty dstrbuton for these varables A random varable x~n(0,r) has an uncertanty ellpsod wth sem axs [3 1], [-.2.6]. Compute the covarance matrx R knowng that E{x 12 }=1. 4. We known that a brdge falls wth probablty.8 f the man structure elements break and ths happens wth probablty.001. Whch s the break probablty knowng that the brdge has fallen? Dscuss f ths problem can be solved. 5. Three prsoners A, B, C are n separate cells. One s gong to be released and the other two wll be condemned to de. Prsoner A asks the jaler to delver a farewell letter to one of the other prsoners whch wll be condemned. The next day the jaler tells hm that he delvered the letter to prsoner B. What s the probablty of A beng set free before and after the jaler answer?

30 Work Let x be a random varable wth dstrbuton N(0,1). Determne n an exact or approxmate way: E{x 2 }, E{x 4 }, E{cos(x)}, E{tan(x)}, E{tan -1 (x)}

31 Bblography E. T. Jaynes, Probablty Theory: the Logc of Scence, J. Marques, Reconhecmento de Padrões. Métodos Estatístcos e Neuronas, IST Press, S. Geman and D. Geman. Stochastc relaxaton, Gbbs dstrbutons and the Bayesan restoraton of mages. IEEE Transactons on Pattern Analyss and Machne Intellgence, 6: , 1984.