Probably Bracke Noao, Probably Vecors, Markov Chas ad Sochasc Processes Xg M. Wag Sherma Vsual Lab, Suyvale, CA, USA Table of Coes Absrac page1 1. Iroduco page. PBN ad Tme-depede Dscree Radom Varable.1. Eve-Evdece Bracke, Base ad Idey Operaor page 3.. Observable ad Expecao Value page 10.3. Idepede dscree radom varables page 1 3. PBN ad Tme-depede Couous Radom Varable 3.1. Base ad Idey Couous Probably page 15 3.. Expecao Value for Couous Radom varable page 18 3.3. Phase Space ad Paro Fuco Sasc Physcs page 0 4. PBN ad Tme-depede Probably Vecors ad Markov Chas 4.1. Probably Vecors Sample Space page 0 4.. Lef-acg Traso Operaor ad Markov Chas page 5. Sochasc Processes ad Tme Evoluo Dffereal Equaos 5.1. Sochasc Processes PBN page 5 5.. Tme Evoluo: he Schrodger ad Heseberg Pcures page 9 Summary page 3 Appedx A page 33 Appedx B page 34 Refereces page 36 Absrac Drac oao has bee wdely used for vecors Hlber spaces of Quaum Theores. I hs paper, we propose o use he Probably Bracke Noao (PBN), a ew se of symbols defed smlarly (bu o decally) as Drac oao. By applyg PBN o fudameal defos ad heorems for dscree ad couous radom varables, we show ha PBN could play a smlar role probably sample space as Drac oao Hlber vecor space. Applyg PBN o homogeeous Markov chas (MC) wh dscree me, we show ha our sysem sae P-kes are defed wh he probably vecors Markov chas (MC). The we apply PBN o geeral sochasc processes (SP). The maser equao of me-couous homogeeous MC he Schrodger pcures s dscussed. Our sysem sae P-bra s defed wh Do s sae fuco ad Pel s sadard bra. I he ed, we vesgaed he raso of probably desy from he Schrodger pcure o he Heseberg pcure for me-couous homogeeous MC. We Dr. Xg M Wag Probably Bracke Noao Page 1 of 36
summarze he smlares ad dffereces bewee PBN ad Drac oao he wo ables of Appedx A. 1. Iroduco Drac oao ([1], 7.; [], Appedx I) s a very powerful ool o mapulae vecors Hlber space. I has bee wdely used Quaum Mechacs (QM) ad Quaum Feld Theores (QFT). Now has also bee roduced o Iformao Rereval (IR) [-3]. We call hs se of oao he Vecor Bracke Noao (VBN). The ma beauy of VBN s ha may formulas ca be preseed a symbolc absrac way, depede of sae expaso or base seleco, whch, whe eeded, s easly doe by serg a u operaor. Ispred by he grea success of VBN for vecors Hlber space, we ow propose Probably Bracke Noao (PBN), a ew se of symbols for probably modelg probably sample space. Frs, me-depede probably sample space, we defe symbols lke probably bra (P-bra), P-ke, P-bracke, P-bass, sysem sae P-ke/bra, u operaor ad more PBN as her couerpars of VBN. We show ha PBN has he smlar power as VBN: may probably formulas ow ca also be preseed absrac way, depede P-bass. I erms of PBN, we ca defy me-depede sysem sae P-kes wh so-called probably vecors ([4], 11.1), whch play very mpora roles MC (Markov chas, [4], 11 ad [5]). We show how o cosruc lef-acg or rgh-acg raso of dscree-me Markov chas (MC). Ths wll help us o udersad relaed opcs lke dffuso maps [8-9] daa cluserg [15, 19]. I he las seco, we apply PBN o some mpora sochasc processes, ad prese he me evoluo equao (or maser equao) of me-couous homogeeous MC (TCH-MC) Schrodger. We fd our sysem sae P-bra s decal o he sae fuco or sadard bra roduced Do- Pel Techques [16-18]. Fally, we roduce he Heseberg pcure of sochasc processes ad expla he mplcao of he shf from Schrodger o Heseberg pcure. I Appedx A, we gve a em-o-em comparso of PBN ad VBN wo shor ables, summarzg he smlares ad dffereces of he wo oaos. I Appedx B, we show how o derve maser equao for TCH-MC wh couous-sae.. PBN ad Tme-depede Dscree Radom Varable I hs seco, we roduce he basc symbols of PBN for me-depede dscree sample space. We defe he probably eve-bra (P-bra), probably evdece-ke (Pke) ad her bracke (P-bracke) o represe he codoal probably. I he process, we ofe use he defos, heorems ad samples he book by Grsead ad Sell [4], deoed as Ref [4]- or Based o Ref. [4]. Our defos, suggesos ad Dr. Xg M Wag Probably Bracke Noao Page of 36
heorems wll be saed as proposos. I hs arcle, he bra, ke ad bracke defed VBN are referred as v-bra, v-ke ad v-bracke..1. The Basc Symbols of Probably Bracke Noao Defo.1.1 (Dsrbuo Fuco, Based o Ref [4]-Defo.): Le X be a radom varable whch deoes he value of he oucome of a cera experme, ad assume ha hs experme has oly fely may possble oucomes. Le Ω be he sample space of he experme (.e., he se of all possble values of X, or equvalely, he se of all possble oucomes of he experme.) A dsrbuo fuco for X s a realvalued fuco m whose doma s ad whch sasfes: 1. m ( ) 0,. m( ) 1. for all, ad (.1.1) For ay subse E of Ω, he probably of E s gve by: E P( E) m( ) (.1.a) E E Defo.1. (Codoal Probably, [4], page 134): For ay subse A ad B of Ω, he codoal probably of eve A uder evdece B s defed by: P( A B) A B P( A B) (.1.b) P( B) B Proposo.1.1 (Probably eve-bra ad evdece-ke): Suppose ha A ad B are subses of a sample space Ω, we deoe: 1. The symbol P( A ( A represes a probably eve bra, or P-bra;. The symbol B) represes a probably evdece ke, or P-ke. Proposo.1. (Probably Eve-Evdece Bracke): The codoal probably of eve A gve evdece B sample space Ω s deoed by he probably eve-evdece bracke or P-bracke: P( A B) A B B P( A B) ( A B), f 0 1 P( B) B (.1.3a) By defo, he P-bracke has he followg properes (see [4], 1.): Dr. Xg M Wag Probably Bracke Noao Page 3 of 36
Dscree RV: P( A B) 1 f A B (.1.4a) Couous RV: P( A B) 1 f A a B & dx 0 (.1.4b) a P( A B) 0 f A B (.1.4c) P( A B) P( A ), f A ad B are muually depede, see Eq. (..1) (.1.4d) These are he mos mpora properes of P-bracke. We ca see ha P-bracke s o he er produc of wo vecors, whle VBN he v-bracke s. Proposo.1.3: For ay eve E sample space Ω, he probably P(E) ca be wre as: P( E) P( E ) (.1.5a) Proof: By defo ad wh Eq. (.1.-.1.3), P(E Ω) ca be valdaed as follows: E E P( E ) P( E) m( ) ( ) (.1.5b) (.1. a) E E The P-bracke defed (.1.3a) ow ca also be wre as: P( A B) P( A B) P( A B ) P( B) P( B ) (.1.3b) Le us use our PBN o rewre he proof of some heorems Ref. [4]. Theorem 1.1.1 (Ref. [4]-Theorem 1.1.4): If A ad B are wo dso subses of Ω, he: P( A B) P A P B or P( A B ) P( A ) P B (.1.6) Proof PBN: P( A B ) P( ) P( ) P( ) ( AB) A B P( A ) P( B ) (.1.8) Theorem.1. (Ref. [4]-Theorem 1.1.): If A 1,..., A are parwse dso subses of Ω (.e., o wo of he A s have a eleme commo), he P( A 1 A A ) P( A ) 1 Proof PBN: By defo (.1.5b): P( A1 A A ) P( ) P( A ) (.1.9a) 1A 1 Dr. Xg M Wag Probably Bracke Noao Page 4 of 36
Theorem.1.3 (Ref. [4]-Theorem 1.1.5): P( A ~ ) 1 P( A) Proof PBN: Usg defo (.1.5b), we have: P ( A ) P ( ) P ( ) P ( ) (.1.9b) A A P( ) P( A ) 1 P( A ) (.1.9b) Theorem.1.4: Bayes formula (see [4],.1) Proof PBN: A B A B A P( A B) (.1.3 a) B A B 1 P( B A) P( A) P( B A) P( A ) P( B ) P( B) (.1.10) By defo.1.1, he se of all elemeary eves ω, assocaed wh he radom varable X, s he sample space Ω, ad hey are muually dso: (.1.11), Usg Eq. (.1.4), we have followg P-brackes (orhoormaly) for base elemes: P( ) ( ) (.1.1) Proposo.1.4 (Probably Sample Base ad U Operaor): Lke a Hlber space, he complee muually-dso (CMD) eves (.1.1) form a probably sample base (or P-bass) of he sample space, assocaed wh he radom varable X. We ca use hem o defe he followg operaor, whch ca be sered o ad P-bracke as a u operaor: ) P( ) ( ˆ P I. 1 N (.1.13) Proof: Suppose ha A ad B are subses of a sample space Ω, we have: P( A Iˆ B) ( A ) P( B) P( B) (.1.4) A B B AB A B P( A B) (.1.14) (.1.5 b) A B B B Now we ca expad he sysem sae P-ke. Is rgh expaso s: Dr. Xg M Wag Probably Bracke Noao Page 5 of 36
(.1.15a) ) Iˆ ) ) P( ) m( ) ) Whle for he sysem P-bra, s lef expaso s: N N P( P( Iˆ P( )( P( (.1.15b) (.1.4) The wo expasos are que dffere, alhough her P-bracke s cosse wh he requreme of ormalzao: N N N (.1.15c) 1 P( ) P( ) P( m( ) ) m( ) m( ), 1, 1 1 Here we see he esseal dfferece bewee P-bra ad P-ke. The sysem P-ke, whe expeded, has he formao of he probably for each of s members accordg o he dsrbuo fuco. O he oher had, he expaso of sysem P-bra does o have ay formao of he probably dsrbuo. Ths s que dffere from he behavor of he v-bra ad v-ke Hlber space, where oe s he Herma cougao of he oher. Ths asymmery s o oly rue for sample space Ω, bu also rue for ay subse E Ω. As P-ke, s rgh expaso s: ) P( ) ˆ P( E ) P( ) E E) I E) ) P( E) ) (.1.16a) (.1.10) P( E ) P( E ) We see P-ke E) gves wha base eves coas, ad he codoal probably of each eve uder evdece E. Noe ha P( E) 1, as Eq. (.1.4a) predced: P( ) P( ) P( ) E E P( E ) P( E) 1 P( E ) P( E ) P( E ) As P-bra, s lef expaso s: (.1.16b) P( E P( E Iˆ P( E ) P( P( E We see P-bra P( E oly gves wha base eves coas. Aga, he bracke P( E ) gves us he rgh probably value by usg Eq. (.1.5b). If he dsrbuo fuco s me-depede, as Markov chas (see 4), we wll use ( ) he sysem P-ke ) ) ( )) as probably vecor, represeg he oucomes ad her probables a me, whle he P-bra P(Ω represe he se of all possble Dr. Xg M Wag Probably Bracke Noao Page 6 of 36
oucomes a all me. Ther expaso ad ormalzao are smlar o me-depede sample space: (.1.17a) ) ) ) )( ) (, ) ) ( ) ˆ I m (.1.17b) P( P( I P( ) P( P( (.1.17c) P( ) P( ) m(, ) m(, ) 1 Ths leads us o followg proposo: Proposo.1.5 (Sysem sae P-ke ad P-bra): A dsrbuo fuco s a sae of he sysem of he sample space. Because he P-ke Ω) represes such a sysem sae, we call Ω) he sysem sae P-ke (or sysem P- ke) of he sample space; we call (Ω he sysem sae P-bra (or sysem P-bra). If a operao volves he kowledge of dsrbuo fuco, we should always sar wh P-sae ke ad s rgh expaso. Usg he dey operaor of Eq. (.1.14), he Bayes formula (.1.10) ow ca be wre: P( B A) P( A ) P( B A) P( A ) P( A B) ( A B) P( B ) ( B ) P( ) (.1.18) Now le us cosder some examples (see [4], 1.). Example.1.1 (Rollg a De, Ref. [4]-Example.6-.8): A de s rolled oce. We le X deoe he oucome of hs experme. The he sample space for hs experme s he 6-eleme se Ω = {1,, 3, 4, 5, 6}. We assumed ha he de was far, ad we chose he dsrbuo fuco defed by m() =1/6, for = 1,..., 6. Usg PBN, we have he dey operaor for hs sample space: 6 ) P( 1 (.1.19a) 1 Ad, because he 6 oucomes have he same probably p, we ca calculae he probably for each oucome: 6 6 (.1.19b) 1 P( ) P( ) P( ) P( ) 6 p 1 1 Dr. Xg M Wag Probably Bracke Noao Page 7 of 36
Hece he probably for each oucome has he same value: 1 P( ) P( ) p (.1.19c) 6 Example.1.: (Rollg a De, Ref. [4]-Example.8, Example.1 coued): If E s he eve ha he resul of he roll s a eve umber, he E = {, 4, 6} ad P(E) = m() + m(4) + m(6) =1/6 + 1/6 + 1/6 = ½ Usg PBN, he probably of eve E ca be easly calculaed usg Eqs. (.1.4-9) as: 6 P( E) P( E ) P( E Iˆ ) P( E ) P( ) 1 1 P( ) p 3 6 E,4,6 1 (.1.19d) Applyg he Bayes formula (.1.10a) o he eve ad E our de sample space ad usg Eqs. (.1.3a), (.1.7) ad (.1.19), we ca easly calculae he codoal probably P( E) as follows: P( ) 1/ 6 1 P ( E ) P ( ), ( eve ) P( E) 1/ 1/ 3 (.1.0) P( E ) 0, ( odd) The expasos of he sysem P-ke ad sysem P-bra of hs sample space are que dffere: 6 6 ˆ 1 ) I ) ) P( ) ) 6 (.1.1a) 1 1 6 6 (.1.1b) P( P( Iˆ P( )( P( 1 1 Bu her P-bracke s cosse wh he fudameal propery: 6 6 6 1 1 P( ) P( ) 1 6 6 (.1.1c) 1 1, 1 Nex le us have a bref dscusso o depede eves. Defo.1. (Idepede eves, Ref. [4]-Defo 4.1): Le E ad F be wo eves. We say ha hey are depede f eher 1) Boh eves have posve probably ad P(E F) = P(E) ad P(F E) = P(F), or ) A leas oe of he eves has probably 0. Dr. Xg M Wag Probably Bracke Noao Page 8 of 36
Theorem.1.5 (Ref. [4]-Theorem 4.1): Two eves E ad F are depede f ad oly f P( E F) P( E) P( F). Proof PBN: From defo (.1.3), ad Defo.1., he proof s que smple: P( E F) P( E F) P( F) P( E ) P( F ) (.1.a) (.1.3 a) P( E F ) P( E ) P( F ) P( E F) P( E ) P( F ) P( F ) (.1.b) (.1.3 b) (.1. a) Theorem.1.6 (Ref. [4], Theorem 1.3): For ay complee muually dso (CMD) se {H } Ω ad ay eve E Ω, we have: P( E ) P( E H ) P( H ) (.1.3) Proof PBN: A CMD se has followg properes: P( H H ), H (.1.4) Usg above properes we have: E E H ) A, P( E ) P( E H ) P( E H ) P( H ) (.1.9 a) (.1.3 b) ( A A A, hece: (.1.5) Ths mples ha he CMD ses {H } defed Eq. (.1.4) cosruc aoher base of he sample space, ad we ca buld a u operaor from hese ses o ser o a P-bracke: Iˆ H ) P( H (.1.6) Proposo.1.6 (U operaor from ay CMD ses): If ses {H } are CMD, as Eq. 9.14), he we ca buld a u operaor as Eq. (.1.6). Proof: P( A H ) P( H B) Bayes formula P( H A) P( A ) P( B H ) P( H ) P( H ) P( B ) P( B H ) P( H A) P( A ) P( B A) P( A ) P( A B) P( B ) P( B ) (.1.7) Bayes (.1.5) formula Usg Eq. (.1.6), he Bayes formula Eq. (.1.10,.1.18) ow ca be wre as: Dr. Xg M Wag Probably Bracke Noao Page 9 of 36
P( B A) P( A ) P( B A) P( A ) P( A B) ( A B) ( B ) ( B H ) P( H ) (.1.8) Ths s decal o he verso gve by Ref. [4],.1... Observable ad Expecao Value Defo..1 (Expecao Value, Ref. [4]-Defo 6.1): Le X be a umercallyvalued dscree radom varable wh sample space Ω ad dsrbuo fuco m(x). The expeced value E(X) s defed as E( X ) x m( x) (..1) x provded hs sum coverges absoluely. If he above sum does o coverge absoluely, he we say ha X does o have a expeced value. Proposo..1 (Observable ad s ege-ke ad ege-bra): We call he radom varable X of sample space Ω a observable. The fac ha X akes value x a P-ke x) or P-bra (x ca be deoed PBN as a operaor acg o hem: X x) x x), ( x X ( x x (..a) Accordg o Proposo.1.4, hey form a base of Ω assocaed wh X: P( x x ') xx', x) P ( x 1 (..b) x Now we have he followg proposo for compac expresso of expecao value. Proposo.. (Expecao Value): The expeced value of he observable X sample space Ω ca be expressed as: X X E( X ) P( X ) (..3) Proof: P( X ) P( X x) P( x ) P( x x) P( x ) x x (..4) P( x) x P( x ) x P( x ) x m( x) E( X ) x x x If F(X) s a couous fuco of observable X, he s easy o show ha: Dr. Xg M Wag Probably Bracke Noao Page 10 of 36
F( X ) E( F( X )) P( F( X ) ) F( x) m( x) (..5) Defo.. (Varace, Based o Ref. [4]-Defo 6.5): Le X be a real-valued radom varable wh desy fuco f(x). The varace σ = V (X) s defed by x V ( X ) P( ( X X ) ) (..6) I ca be easly see ha: P( ( X XX X ) ) X XX X X X X X X X (..7) Example..1 (Rollg a De, Example.1.1 coued): We have he followg observable he de sample space, as based o Eqs. (.1.19), X ) ), {1,,...6} (..8a) Is expecao value ca be readly calculaed: 6 6 P( X ) P( X ) P( ) P( ) P( ) 1 1 6 6 1 7 ( ) 1 6 1 6 6 (..8b) Ad he varace ca be calculaed as: 6 1 49 91 49 45 X X (..8c) 6 4 6 4 4 1 Defo..3 (Codoal Expecao Value, Based o Ref. [4]-Defo 6.): If F s ay eve ad X s a radom varable wh sample space = {x 1, x,...}, he he codoal expecao gve F s defed by E ( X F) x P( X x F) (..9) Proposo..: I PBN, we ca express codoal varace (..6) as: E( X F) P( X F) (..10) Proof: Dr. Xg M Wag Probably Bracke Noao Page 11 of 36
x x P( x) x P( x F) x P( x F) x P( x F) (..11) P ( X F ) P ( X x ) P ( x F ) P ( x x ) P ( x F ) x x Codoal expecao s used mos ofe he form provded by he followg heorem. Theorem..3 (Ref. [4]-Theorem 6.5): Le X be a radom varable wh sample space. If F 1, F,..., F r are eves such ha F F = ø; for ad Ω = U F, he E( X ) E( X F ) P( F ) (..1) Proof PBN: Usg Eq. (..11), (.1.6) ad (..4), we have: E( X F ) P( F ) P( X F ) P( F ) (..10) x P( x F ) P( F ) x P( x ) X k k k k (..11) k (.1.6) k (..1) Example..4 (Rollg Two De): We have he followg observable he -de sample space, as he exeso of Eqs. (.1.19), X, ), ), Y, ), ),, {1,,...6} (..8a) Is expecao value of X Y ca be readly calculaed: 6 6 P( X Y ) P( X Y, ) P(, ) P(, ) P(, ), 1, 1 6 6 7 P(, ) 49, 1 36 1 6 4 (..8b).3. Idepede Dscree Radom Varables Defo.3.1 (Jo radom varable, based o Ref. [4]-Defo 4.3): Le X 1, X,..., X be radom varables assocaed wh a experme. Suppose ha he sample space (.e., he se of possble oucomes) of X s he se Ω 1. The he o radom varable (or radom vecor) X = (X 1, X,..., X ) s defed o be he radom varable whose oucomes coss of ordered -uples of oucomes, wh he h coordae lyg he se Ω. The sample space of X s he Caresa produc of he Ω s: Ω = Ω 1 Ω Ω. (.3.1) Dr. Xg M Wag Probably Bracke Noao Page 1 of 36
The o dsrbuo fuco of X s he fuco whch gves he probably of each of he oucomes of X. Proposo.3.1: I PBN, he sample space of o varable X ca be wre as: ) ) (.3.a) 1 The facor sample space Ω ) have he followg properes: P( ) 1, ) ) ) ), P( P( P( P( (.3.b) The base eve ke of o radom varables ca be wre as: 1, r1,, r ) r ), r ) r ) 1 r r ) r ) (.3.c) Ths expresso mgh be cosdered as he couerpar of Fock space [3]. Defo.3. (Idepede Radom Varables, based o Ref. [4]-Defo 4.4): The radom varables X 1, X,..., X are muually depede f for ay choce of r 1, r,..., r : 1 r1, X r,, X r ) P( X 1 r1 ) 1 P( X (.3.4) Thus, f X 1, X,..., X are muually depede, he he o dsrbuo fuco of he radom vecor X = (X 1, X,..., X ) s us he produc of he dvdual dsrbuo fucos. Whe wo radom varables are muually depede, we shall say more brefly ha hey are depede. Proposo.3.3: I PBN, usg proposo.3.1, Eq. (.3.4a) ca be wre as: P( r, r,, r ) P( r ) m ( r ) (.3.5) 1 1 1 1 From Eq. (.3.5), we ca derve followg properes for depede radom varables: P( r ) P( r, r,, r ) P ( r ) P( r ) P ( r ) (.3.6) 1 1 1 1 1 1 1 1 r r r P( r, r ) P( r r ) P( r ) P( r ) P( r ) P ( r ) (.3.7a) P( r r ) P( r ) P( r ) P( r r ) P( r ) P ( r ) P( r ) P( r ) Eq. (.3.7) s equvale o Eq. (.1.11). (.3.7b) Dr. Xg M Wag Probably Bracke Noao Page 13 of 36
As observables, we have he followg ege-bras ad ege-kes: X, r r (.3.8) r1,, r ) r r1,, r ), ( r1,, r X ( r1, Hece, for aalycal fucos F (x) ad k(x), we have (.3.8a) c F ( X ) P ( c F ( X ) ) c F ( X ) (.3.8b) X P( X ) X k ( ) k ( ) k ( ) Here, we have used he followg expecao value: X P( X ) P( X ) (.3.9) Example.. (rollg wo de) acually s a example of wo depede radom varables X ad Y. Usg Eq. (.3.8b), we ca easly recalculae (..8b) as follows: 7 49 P( X Y ) X Y 4 (.3.10) 3. PBN ad Tme-depede Couous Radom Varable I hs seco, we defe PBN for couous sample space. Because may oaos ad proofs are smlar o dscree sample space, we oly dscuss a few seleced coceps. 3.1. The Sample-base ad he Idey Operaor Defo 3.1.1 (Couous Dsrbuo or Desy Fuco): Le X be a radom varable whch deoes he value of he oucome of a cera experme, ad assume ha hs experme has oly fely may possble oucomes. Le Ω be he sample space of he experme (.e., he se of all possble values of X, or equvalely, he se of all possble oucomes of he experme.) A dsrbuo fuco for X s a real-valued fuco whose doma s Ω ad whch sasfes: 1. f ( x) 0, for all. dx f ( x) 1. x x, ad (3.1.1) For ay subse E of Ω, we defe he probably of E o be he umber P(E) gve by: Dr. Xg M Wag Probably Bracke Noao Page 14 of 36
( E P E) ( ) dx f x (3.1.) x E Suppose ha A ad B are subses of a sample space Ω. The codoal probably of eve A uder evdece B sample space Ω, ad he properes of eve-evdece are he same as dscree sample space. The heorems.1.1-.1.4 ca be proved smlar way for couous sample space. Noe he codo Eq. (.1.4b): Couous RC: P( A B) 1 f A a B & dx 0 (.1.4b) Ths s because oherwse we may have Drac dela fuco, as show Eq. (3.1.3a) below. For orhoormaly, we exed Eq. (.1.1) o couous case: P( x x') ( x x') ( x x' ) (3.1.3a) The, lke a dscree sample space, hese eves form a bass of he sample space. They ca be used o defe a dey operaor he sample space: dx x ) P ( x Iˆ (3.1.3b) x P( A B) The defo Eq. (.1.3a), ( A B) P( A B), requres P(B) > 0. Bu from P( B) Eq. (3.1.), we have P(B) = 0 f B x. Ths does o cause problem ow, because: a P( A x ) P( x ) P( A x) 1, f x A P( x ) P( x ) (3.1.3c) Proposo 3.1.1: I PBN, he dsrbuo fuco (or probably desy) s deoed by: f ( x) P( x ) (3.1.4a) The proof s smlar o ha he dscree sample case. We check here f s cosse wh he ormalzao requreme: (3.1.4b) P( ) P( x) dx P( x ) dx P( x ) dx f ( x) 1 x x Example 3.1.1 (Dars, based o Ref. [4], Example 3.8-.9): A game of dars volves hrowg a dar a a crcular arge of u radus. Suppose we hrow a dar oce so ha hs he arge, ad we observe where lads. To descrbe he possble oucomes of hs experme, s aural o ake as our sample space he se of all he pos he arge. I s covee o descrbe hese pos by her recagular coordaes, relave o a coordae sysem wh org a he ceer of he arge, so ha each par (x, y) of coordaes wh x + y 1 descrbes a possble oucome of he experme. The Ω = {(x, Dr. Xg M Wag Probably Bracke Noao Page 15 of 36
y): x + y 1} s a subse of he Eucldea plae, ad he eve E = {(x, y): y > 0}, for example, correspods o he saeme ha he dar lads he upper half of he arge, ad so forh. Assumg uform dsrbuo, he probably of he eve ha he dar lads ay subse E of he arge should be deermed by wha fraco of he arge area les E. Thus, we ca calculae P(E Ω): Area of E Area of E P( E ) dx dy f ( x, y) Area of x, ye dxdy f ( x, y) f (0,0) dxdy f (0,0) Area of E E E (3.1.5a) Hece, we ge he desy fuco: 1 P( x, y ) f ( x, y) f (0,0) (3.1.5b) I couous probably, he Bayes formula (.1.10) ow ca be wre: P( A B) P( B A) dx P( x ) P( B A) P( A ) xa P( B ) dx P( x ) xb Noe s vald eve for sgular case lke: (3.1.6) P( x ' x) P( x ) ( x x ') P( x ) P( x x ') ( x x ') P( x ' ) P( x ' ) Defo 3.1. (Codoal Couous Desy Fuco, Ref. [4], 4.) f ( x) f x E f ( x E) P( E) (3.1.7) 0 f x E Proposo 3.1.: Usg PBN, we ca deoe he codoal desy fuco as: f ( x E) P( x E) (3.1.8) Proof: Usg Bayes formula (.1.10), we have: Dr. Xg M Wag Probably Bracke Noao Page 16 of 36
P( x E) P( x ) f ( x) P( E x) P( x ), f x E P( E ) P( E) P( E ) 0, f x E Proposo 3.1.3 (Codoal Probably of Eve E gve F, see Ref. [4], 4.): (3.1.9) P( F E) P( F x) dx P( x E) dx P( x E) xf Proof: We ca check ha hs s cossece wh our defo of codoal probably, Eq. (.1.3a): P( x ) P( E F ) P( F E) dx P( x E) dx P ( E ) P ( E ) xf xe F (3.1.10) Example 3.1. (Dars, Based o Ref. [4], Example 5.19): I he dar game (cf. Example 3.8, our Example 3.1.1), suppose we kow ha he dar lads he upper half of he arge. Wha s he probably ha s dsace from he ceer s less ha ½? Here E = {(x, y): y 0}, ad F = {(x, y): x + y < (1/) }. Hece, E / 1 P( E ) P( E) (3.1.11a) P( E F ) E F (1/ )( / 4) 1 P( F E) P( E ) E / 4 (3.1.11b) Here aga, he sze of F E s 1/4 he sze of E. The codoal desy fuco s: P( x, y ) 1 /, f ( x, y ) E f ( x, y E) P( x, y E) P( E ) 1 / 0, f ( x, y) E (3.1.11c) Example 3.1.3 (Expoeal Desy, Ref. [4]-Example 3.17): There are may occasos where we observe a sequece of occurreces, whch occur a radom mes. For example, we mgh be observg emssos of a radoacve soope, or cars passg a mlepos o a hghway, or lgh bulbs burg ou. I such cases, we mgh defe a radom varable X o deoe he me bewee successve occurreces. Clearly, X s a couous radom varable whose rage cosss of he o-egave real umbers. I s ofe he case ha we ca model X by usg he expoeal desy. Ths desy s gve by he formula: Dr. Xg M Wag Probably Bracke Noao Page 17 of 36
e f ( ) 0,, f f 0 0 (3.1.1a) Usg PBN, we have he followg base: P( ') ( '), d ) P( 1, X ) ), P( ) f ( ) (3.1.1b) 0 We ca see ha sample space s ormalzed: P( ) P( ) d P( ) d P( ) 1 (3.1.13) 0 0 Example 3.1.4 (Expoeal Desy, Ref. [4]-Example 5.0): We reur o he expoeal desy (cf. Example 3.17). Suppose we are observg a lump of pluoum- 39. Our experme cosss of wag for a emsso, he sarg a clock, ad recordg he legh of me X ha passes ul he ex emsso. Experece has show ha X has a expoeal desy wh some parameer λ, whch depeds upo he sze of he lump. Suppose ha whe we perform hs experme, we oce ha he clock reads r secods, ad s sll rug. Wha s he probably ha here s o emsso a furher s secods? Le G() be he probably ha he ex parcle s emed afer me. The (3.1.14) G( ) P( G ) P( G ') d ' P( ' ) d ' f ( ') 0 ' G ' ' ' d e e e Le E be he eve he ex parcle s emed afer me r ad F he eve he ex parcle s emed afer me r + s. The P( E F ) G( r s) P( F E) e P( E ) G( r) s (3.1.15) 3.. Expecao Value for Couous Radom varable Defo 3..1 (Expecao Value, Ref. [4]-Defo 6.4): Le X be a umercallyvalued couous radom varable wh sample space ad dsrbuo fuco f(x). The expeced value E(X) s defed as E ( X ) dx x f ( x) (3..1) x Dr. Xg M Wag Probably Bracke Noao Page 18 of 36
provded hs egral coverges absoluely. If he above egral does o coverge absoluely, he we say ha X does o have a expeced value. As he dscree case, he propery ha radom varable X akes value x evdece x) ca be wre PBN as follows: X x) x x) (3..) As defed., X s a observable of he sample space, ad x) are s ege-kes. The expresso of expeced value of couous X s decal o he dscree case: X X E( X ) P( X ) (3..3) We ca see s cossece wh defo (3..1): P( X ) P( X x) dx P( x ) x P( x) x dx P( x ) dx x f ( x) E( X ) x x (3..4) If F(X) s a couous fuco of observable X, he we also have Eq. (..5). The equaos (..6-1..7) relaed o Varace are also he vald. Example 3..1 (Dars, see Example 3.1.1): We have wo observables: X x, y) x x, y), Y x, y) y x, y) (3..5a) Ther expecao values ca be easly calculaed: P( X ) P( X x, y) dydy P( x, y ) ( x, y) 1 P( X x, y) dydy P( x, y ) xdxdy 0 ( x, y) ( x, y) (3..5b) I he las sep we used he propery of he egral of a odd fuco symmerc boudary. Example 3.. (Expeced Lfe-Tme): Le us cosder Example 3.1.1 as he lgh bulbs burg ou problem. The he expeced value of he me wll be he average lfe me of a bulb: P( X ) P( X ) d P( ) 1 d f ( ) d e (3..6) Dr. Xg M Wag Probably Bracke Noao Page 19 of 36
3.3. Phase Space ad Paro Fuco of Ideal Gas The formulas relaed o couous radom varables are smlar o he dscree radom varables dscussed.3. I hs seco, we use PBN o dscuss a sysem of N oeracg dsgushable molecules (see [6], 10.1). The dsrbuo desy of sgle parcle depeds o s eergy (as [6], Eq. (10.1b) ad (10.9)): kt p e e f ( ) P( x, y, z, px, py, pz 1 ) P( x, p 1 ) f m z z (3.3.1) z 1 3 h d 3 x d 3 p e V h 3 d 3 p e p m m V h 3 / (3.3.) The expecao value of he eergy of a sgle parcle s gve by: P( ) P( x, p) d x d p P ( x, p ) d x d p f ( ) 3 3 3 3 1 1 1 1 3 3 1 3 3kT d x d p ( e ) z l z z z (3.3.3) Because he N parcles are dsgushable, usg Eq. (.3.8a), we have he expecao value of oal eergy as: 3NkT N N P( 1 1 1) (3.3.4) 1 4. Probably Vecors ad Markov Chas 4.1. Tme-Depede Probably Vecors Sample Space I Ref. [3], we have dscussed he ormalzed Weghed Term-space, whch s a bouded N-dmesoal couous space over he feld of [0, 1], resrced he u cube: N w w k, w [0,1] 1 These vecors are ormalzed as: (4.1.1) N w 1 1 w w (4.1.) Dr. Xg M Wag Probably Bracke Noao Page 0 of 36
I probably heory, we wll face aoher kd of vecors Probably Vecors (see [4], 11.1). There are probably row vecors (PRV) ad probably colum vecors (PCV), defed sample space, resrced he same u cube as Eq. (4.1.1), bu s ormalzed as follows: N w P( ), P( ) 1, P( ) 0 (4.1.3) 1 Because of hs ormalzao requreme, probably vecors do o form a closed vecor space, or do hey form vecors Hlber space. Bu me-depede probably vecors have very mpora applcaos probably heores lke Markov cha ([4], chaper 11; [5]) ad IR models lke dffuso maps [8-9], we wa o show how o buld me-depede probably vecors from a sample space wh a me-depede dsrbuo fuco, as descrbed by Eq. (.1.17). From he sysem P-ke of Eq. (.1.17a), we ca easly form a PCV by mappg ke-o-ke ad bra-o-bra as follows: m( 1, ) N N m(, ) I ) m(, ) (4.1.4a) m( N, ) Is couer-par, a row vecor, ca also be mapped from he sysem P-bra (Ω Eq. (.1.17b) as: N (4.1.4b) 1, 1,, 1 I s me-depede ad s o a PRV, because does o sasfy Eq. (4.1.3). Bu, from Eq. (4.1.4a-b) ad (.1.17c), we see he ormalzao s correc: N m(, ) m(, ) 1, N (4.1.5) As we wll see ex seco, he raso marx of Markov cha s defed o acg o a row vecor from rgh. We eed a PRV wh me-depede dsrbuo fuco, whch ca be obaed as he raspose of PCV (4.1.4a): N m(, ) [ m(, ), m(, ),, m(, )] (4.1.7) 1 N I s correcly ormalzed because of Eq. (.1.17). We would lke o po ou, ha he er produc of (Eq. 4.1.4a) ad (4.1.7) usually s o equal o 1, ad does o have ay meag erms of probably, sce: N ( m(, )) 1 (4.1.8) Dr. Xg M Wag Probably Bracke Noao Page 1 of 36
I summary, o buld VCR ad PRV a sample space, we sar from he expaso of sysem P-ke Ω). Afer we have bul he PCV, we do a raspose o ge he PRV. We wll see more dealed applcao of probably vecors ex seco. 4.. The Lef-acg Traso Operaor ad Markov Cha I hs seco, we gve a bref dscuss of me-dscree ad sae-dscree Markov cha of dscree sae spaces (see [4] Chap.11, or [5]). Our goal s o demosrae how o use PBN o descrbe he sample space of a Markov cha. I our dscusso, we assume our Markov cha s me-homogeeous. We also assume our sample space has he followg dscree P-bass: P( ), ) P( I 1 r (4..1) The raso marx eleme P s defed as he raso probably from sae o sae a me (a eger, measurg seps). Is elemes are o-egave real umbers ad are me-depede for a me-homogeeous Markov cha [4, 8]: P P( X X ) P( X X ) P(, 1, ) (4..a) 1 1 r 1 P 1 (4..b) I marx form, f we defe a probably row vecor (PRV) a = 0 as u (0), he P acg o he PRV from rgh mes gves he PRV a me = ([4], heorem 11.): u u P, or : u u P (4..c) ( ) (0) ( ) ( ) Example 4..1 (The Lad of Oz, Based o Ref. [4], example 11.1-11.3): Accordg o Kemey, Sell, ad Thompso [7], he Lad of Oz s blessed by may hgs, bu o by good weaher. They ever have wo ce days a row. If hey have a ce day, hey are us as lkely o have sow as ra he ex day. If hey have sow or ra, hey have a eve chace of havg he same he ex day. If here s chage from sow or ra, oly half of he me s hs a chage o a ce day. Wh hs formao we form a Markov cha as follows. We ake as saes he kds of weaher R, N, ad S. From he above formao we deerme he raso probables. These are mos coveely represeed a square array as Dr. Xg M Wag Probably Bracke Noao Page of 36
Le he al probably vecor u equal (1/3, 1/3, 1/3). The we ca calculae he dsrbuo of he saes afer hree days usg Theorem 11. ad our prevous calculao of P3. We oba From (4..c) ad he above example, we see ha he raso marx s acg o he row vecor o s lef sde. Ths mples ha we eed o deal wh a lef-acg operaor ad a PRV. To use our P-bass, we defe followg raso operaor sample space based o he elemes {p }: r Pˆ ') P P( ' (4..d) ', ' 1 ' ' The marx eleme of he operaor he P-bass s he eleme of raso marx: r P( Pˆ ) P( ') p P( ' ) p p ' ' ' ' ' ' ' ' ', ' 1 ', ' 1 r (4..e) As dscussed 4.3, o buld a PRV, we eed o sar wh he PCV. Because we are alkg abou he me-evoluo of saes, he dsrbuo fuco ow s medepede, so we use he followg sysem P-ke as Eq. (.1.17): r r r ( ) ( ) ( ) ( ) ) ) ) P( ) ) 1 (4..3a) Now we buld he v-base from he P-bass by oe-o-oe-map: r, I 1 (4..3b) From Eq. (4..3) ad Eq. (4.1.7), we oba he PRV as: Dr. Xg M Wag Probably Bracke Noao Page 3 of 36
( ) r r ( ) ( ) ( ) ( ) ( ) [ 1,,, r ], 1 (4..4) The lef-acg operaor, o ac o a PRV from rgh, s defed as r P ' p' ' ' ', ' 1 Proposo 4..1 (Tme evoluo lef acg case): P ( ) ( 1) (4..5a) (4..5b) Proof: r ( ) P ( ) ' p ' r ', ', 1 ', ', 1 ' ' r ( ) ( ) ( 1) ' ' ' ' ' ' (4..6) p p The las sep comes from Eq. (4..c). I geeral, we have: P ( ) (0) Usg PBN, we ca express Example 4.1.1 our row vecor ad operaor: (4..7) 1 1 1 P { 1 3 } P 3 3 3 (3) (0) 3 3.4011.198.4013 [.401,.198,.401] (4..8) We ca also check ha he lef-acg operaor P acg o a PCV o s rgh does o ras he sae as expeced, ad o always produce a vald PCV (o ormalzed o 1). For example, whe he lef-acg gves desred probably vecor, he rgh-acg may make o sese: 1/ [ 1,0,0] 1/ 1/ 4 1/ 4 0 1/ 4 1/ 4 1/ 1/ [1/,1/ 4,1/ 4], 1/ 1/ 1/ 4 1/ 4 0 1/ 4 1/ 4 1/ 1/ 1 1/ 0 1/ 0 1/ 4 The fac ha he raso marx of Markov cha s lef-acg o a PRV s us he resul of he coveoal defo (4..a). If we defe a raso marx as he raspose of (4..a): Dr. Xg M Wag Probably Bracke Noao Page 4 of 36
T p p (4..9) The we ca buld a rgh-acg raso operaor acg o a PCV, already gve by Eq. (4..3a). Example 4.. (The Lad of Oz, rgh-acg case): We rewre he marx Example 4.1.1 ad Eq. (4..8), usg her raspose. The marx ad he VCR become: 1/ 1/ 1/ 4 T P 1/ 4 0 1/ 4 (4..10) 1/ 4 1/ 1/ ( 3) ˆ T 3 (0) ˆ T 3 1 1 ( P ) ( P ) { 1 3 3.401.401 1.198.401 3.198.401 1 3 3} (4..11) If he raso marx s symmerc, he he correspodg operaor becomes bdrecoal. Such a marx may have mpora applcao dffuso maps proposed by Lafo for daa cluserg [8-9]. Ther sarg po s a raso marx of Markov cha wh some specal codos. We fd ha symmerc raso marces may provde good examples for dffuso maps. Bu hs s beyod he scope of hs paper. We leave o our fuure work. 5. Probably Bracke Noao ad Sochasc Processes 5.1. Sochasc Processes PBN The Markov cha we have dscussed s a specal case (dscree-me, dscree-oucome ad homogeous) of Markov Process. May sochasc processes are also Markov chas. I hs seco, we wa o apply PBN o some basc formulas of some mpora sochasc processes [10-13]. We wll see ha PBN o oly does smplfy he formulao, bu also make possble o represe he evoluo equao boh he Heseberg pcure ad he Schrodger Pcure, as used QM [14]. Basc Noaos for Sochasc Process (SP): The base P-ke of SP X ( ) ( T ) s a me depede observable ad we ca geeralze Proposo..1 (Observable ad s egeke) for dscree or couous radom varables (R.V): Dr. Xg M Wag Probably Bracke Noao Page 5 of 36
X ( ) X ( ) x ) X ( ) x, ) x x, ) P( x, x, ) P( x x ) (Dscree R.V.) (5.1.1a) X ( ) X ( ) x) X ( ) x, ) x x, ) P( x, x ', ) P( x x ') ( x x ') (Couous R.V.) (5.1.1b) The me-depede sample space Ω() s a sapsho of whole sample space, coas all possble oucomes observed a me =. The oal sample space, Ω coas all possble ou comes a all me. By defo, we have: ( ), P( ( )) 1, P( x, ) 1; P( x, ) P( x ( )) (5.1.a) The me-depede probably dsrbuo ow ca be wre as: P( x, ) P( x ( )) m( x, ) ( Dscree RV. ) P( x, ) P( x ( )) f ( x, ) ( Couous RV. ) (5.1.b) The me depede expecao value of observable X() ow s: P( X ( ) ) P( X ( ) x, ) P( x, ) m( x, ) x ( Dscree RV. ) P( X ( ) ) dxp( X ( ) x, ) P( x, ) dx f ( x, ) x ( Couous RV. ) Here we have used he me-depede dey operaor: Iˆ( ) x, ) P ( x, ( Dsacree RV. ) Iˆ( ) dx x, ) P ( x, ( Couous RV. ) (5.1.3) (5.1.4a) We have a me parameer here, because he raso probably or me-creme s always defed he me-cremeal dreco. Whe we ser he dey operaor, we eed o choose approprae me (see Eq. (5.1.6)). For example, a me, he measureme pcks up he value from ) : X X Iˆ ( ) Iˆ ( ) X, X ( ) ) X Iˆ ( ) ) X ) (5.1.4b) P( X ) P( X I( ) ) P( X ) (5.1.4c) The shf of me depedece from he observable o he sae P-ke ca be hough as a shf from he Heseberg pcure o Schrodger pcure (see Eq. (5..5) or.5, [19]). A S.P. X() has depede cremes, f for 1 < < < m < m+1 he {1, m 1} : Dr. Xg M Wag Probably Bracke Noao Page 6 of 36
P( X X x X X x ) P( X X x ), (5.1.5a) m1 m m 1 1 m1 m m If S.P has depede-me creme, we ca always se X 0 0 ad have: P( X x c X c) P( X X x X X c) P( X X x ) (5.1.5b) s s s 0 s A SP X() may have Markov propery, whch assumes ha he fuure probably dsrbuo ca be predced from he curre sysem sae, bu o he pas sysem sae. Ths meas, for 1 < < < m < m+1, P( X ( ) x X ( ) x, X ( ) x,, X ( ) x ) m1 m1 m m m1 m1 1 1 P( x, x, ; x, ; ; x, ) P( x, x, ) m1 m1 m m m1 m1 1 1 m1 m1 m m (5.1.5c) A SP s homogeeous f has he followg propery for s 0 : P([ X ( ) X ( s) x] ) P([ X ( ) X ( s ) x] ) (5.1.5d) If he SP s homogeeous ad X(0) = 0, he we have he followg propery: P([ X ( s) X ( s) x] ) P([ X ( ) X (0) x] ) P([ X ( ) x] ) P( x ( )) (5.1.5e) The Chapma-Kolmogorov Theorem ([10], p174, p13; [11]-[13]): Ths equao ca be derved by usg Codoal Toal Probably Law (TPL) ad Markov propery. Bu we ca derve hem smply usg our dey operaor ad Eq. (5.1.4): m p P(, m,0) P(, m I ( m),0) P(, m k, m) P( k, m,0) p m p k ( Dsacreeme, dscree R. V ) (5.1.6a) k k p ( s) P(, s,0) P(, s I( s),0) P(, s k, s) P( k, s 0, ) p ( s) p ( ) ( Couous me, dscree R. V ) (5.1.6b) k k k P( x, y, s) P( x, Iˆ ( ) y, s) P( x, z, ) dz P( z, y, s) where s k k ( Couous me, couous R. V ) (5.1.6c) I geeral, f a S.P. has Markov propery, he we ca ser a Idey operaor (5.1.4a) sde he raso marx (a P-bracke), wh a me less ha he me o he lef ad greaer ha he me o he rgh. Le us ls some mpora examples of sochasc processes. Dr. Xg M Wag Probably Bracke Noao Page 7 of 36
Posso Process ([10], p.161; [11-13]): I s a coug process, N(), havg followg properes: (1). { N( ), 0} s o-egave process wh depede cremes ad N(0) 0 ; (). I s homogeeous ad s probably dsrbuo s gve by: m( k, ) P([ N( s) N( s) k] ) P([ N( ) N(0) k] ) k ( ) P([ N( ) k] ) P( k ( )) e Posso k! N (0) 0 depede creame Dsrbuo (5.1.7a) Usg dey operaor, oe ca easly fd ha: ( ) N( ) P( N( ) ) P( N( ) k, ) P( k, ) k m( k, ) ; ( ) P( [ N( ) N( )] ) k (5.1.7b) I ca be show (see [10], p.15; [11], 3, p.1) ha Posso Process has Markov propery, ad s raso probably s: ( ) p ( ) P([ N( s) ] N( ) ) P([ N( s) N( ) ] ) e, f ( )! ( ) 0, f (5.1.8) p Weer Process (see [10], p.159; [11], 8, p.1; [1] 1.1): I s also a homogeeous process { W ( ), 0} wh depede cremes ad W (0) 0. Is probably desy s a ormal dsrbuo N (0, ) : f ( x, ) P([ W ( s) W ( s) x] ) P([ W ( ) W (0) x] ) X (0) 0 homogeeous 1 x P([ W ( ) x] ) P( x, ) P( x ( )) exp[ ] Normal Dsrbuo (5.1.9) Usg dey operaor, oe ca easly fd ha: ( ) P( W ( ) ) 0, ( ) P( W ( ) ) (5.1.10) Browa Moo ([1] 1.3): I s assocaed wh a sadard Weer process W s () (wh σ =1) as follows: X ( ) X (0) W ( ) (5.1.11) s Dr. Xg M Wag Probably Bracke Noao Page 8 of 36
Usg Eq. (5.10) (wh σ = 1), oe ca easly fd ha: Drf : ( ) P( [ X ( ) X (0)] ), Varace : ( ) P( [ X ( ) ( )] ) (5.1.1) If we defe: Y ( ) X ( ) X (0) W ( ) (5.1.13) s The he probably desy f(y, ) of Y() s gve by: f ( y, ) dy P( y ( )) dy P( y, ) dy P([ Y ( ) W ( )] ) dy P([ W ( ) ( y ) / ] ) dx P([ x ( y ) / ], ) dx 1 ( y ) P([( y ) / ], ) dx dx exp[ ] (5.1.9) dx 1 1 ( y ), f ( y, ) P( y ( )) exp[ ] N(, ) (5.1.14) dy Therefore, Browa moo s us a Weer process correspodg o a ormal dsrbuo N (, ). Browa moos also have Markov propery ([13], 1.6). 5.. Tme Evoluo: The Schrodger ad Heseberg Pcures Tme evoluo of sochasc processes s a very mpora subec mahemacs, physcs ad also IR ([8-10]). I hs seco, we apply our PBN o prese Tme evoluo or maser equao for TCH-MC wh dscree sae space. We wll see ha PBN ca make maser lke he Schrodger Equao QM. Kolmogorov Forward ad Backward Equaos: We assume ha he Markov chas are sochascally couous: for fesmal h, he raso probably has he Talor expasos ([10], p. 17; [11], 5, p.5; ad [14], 6.8): p ( h) p (0) p '(0) h o( h ) q h o( h ) (5..1) The, usg Eq. (4.1.6b), we have: p p ( h) ( ) k p k k p k ( ) ( q ( ) p k k ( h) h o( h )) k p k ( ) ( k q k h o( h )) (5..) Dr. Xg M Wag Probably Bracke Noao Page 9 of 36
Therefore, we ge followg Forward equaos: p ' ( ) lm[( p ( h) p ( )) / h] p ( ) q k h0 Smlarly, we ca derve he Backward equaos: p ( h ) p ( h) p ( ) p ' ( ) q p ( ) k k k k k k Ther marx forms are: k k (5..3) (5..4) Forward: P '( ) P( ) Q; Backward: P '( ) QP( ); (5..5) They boh have he followg formal soluo wh al codo P(0) I : P( ) P(0) exp[ Q] exp[ Q] ( Q ) k 0 k! As dscussed 4., we roduce raso operaor: p ( ) P( Pˆ ( ) ), q P( Q ) k (5..6) The we have he followg dffereal equao ([10], p 0): d P ˆ( ) P ˆ( ) Q ˆ QP ˆ ˆ( ); (5..7) d The Schrodger Pcure: We ca fd he absolue probably ([10], p1) as follows: p ( ) P ( X ( ) ) P(, ) P( ) p (0) p ( ) (5..8) k k k I sasfes followg dffereal equaos: T p ( ) P( ) pk ( ) qk ( Q ) k P( k ) (5..9) k k From he above equaos, we oba he TEDE (or maser equao) PBN: T ˆ T ˆ ˆ Q dl 0 ) Q ) L ), ) U ( ) 0) e 0) e 0) (5..10) We kow ha he sapsho of sample space a me ca be mapped as o a Probably Colum Vecor (PCV), ad ca be expaded wh a v-base, as expressed Eq. (4.1.4): ˆ I p U 0 (5..11) ) ( ), ( ) Dr. Xg M Wag Probably Bracke Noao Page 30 of 36
Ths s o ew. I s decal o he maser equao used Do s formalsm [16-18] for dscree-sae homogeeous Markov chas (lke brh-deah process): ˆ ( ) ˆ ( ), ( ) ˆ L L U ( ) (0) e (0) (5..1) Do s defo of a «sae fuco» (see [16] ad [18]) ca be readly defed as our sysem sae P-bra: P ( P ( s, F ˆ ( ) s F ˆ ( ) ( ) P ( F ˆ ( ) ) (5..16) Here, he bass s formed by he egevecors of occupao operaors a Fock space: ˆ, 1, ', ', ' 1 (5..17) I Pel s formalsm [17], he base (from populao ) s ormalzed a specal way: 1 ˆ! I m,! m, (5..18) Therefore, he sysem sae P-bra s ow expaded as: ˆ 1 1 P( P( I P( ) ( P(! (5..19)! (.1.4) Mappg o vecor space, s ohg else, bu he «sadard bra» roduced [17]: 1 1 P( P(,!! E[ Fˆ ] Fˆ Fˆ ( ) P( Fˆ ) (5..0) We call Eq. (5..11-1) he maser equao Schrodger pcure, because hey are smlar o he Schrodger equao of QM Drac (VBN) oao. ˆ ˆ / ( ) ( ), H ( ) ( ) (0) (0) H U e (5..1) Noe Eq. (5..1) s depede of represeaos. I s also rue for homogeeous MC of couous saes (see App. B). Now we roduce he Heseberg pcure as used QM ([15], 11.1): ˆ ˆ ˆ 1 ˆ ˆ 0 ) U ( ) ), X ( ) U ( ) X U ( ) (5..) Dr. Xg M Wag Probably Bracke Noao Page 31 of 36
P( X ˆ ) P( U ˆ ( ) X ˆ U ˆ ( ) ) P( X ˆ ( ) ) P( Xˆ ( ) ) 1 0 0 0 0 (5..3) I he las sep, we have used he fac ha 0 he Heseberg pcure. Based o X ˆ ( ), we ca roduce followg relaos: x Uˆ x P x Uˆ x P x x x x 1, ) ( ) ), ( ( ) (,, (, ', ) ( ') ˆ ˆ ˆ 1 ˆ ˆ ˆ 1 ˆ P( x ', X ( ) x, ) P( x ' U ( ) U ( ) X U ( ) U ( ) x) P( x ' X x) x P( x ' x) (5..4) Usg he fac ha 0 he Heseberg pcure, we ge he shf Eq. (5.1.): P( x ) P( x Uˆ ( ) ) P ( x, ) P( x, ) (5..5) 0 0 Summary I hs paper, we proposed he ew se of symbols of PBN (Probably Bracke Noao) probably sample space. We showed ha, by usg PBN, mos defos ad formulas of probably heory ow could be represeed ad mapulaed us lke her couerpars QM usg Drac oao or VBN (Vecor Bracke Noao). We also defed me-depede sysem sae P-ke wh probably vecors used raso marx of Markov chas (MC). Nex, we appled PBN o geeral sochasc processes (S.P.), especally, dscussed he me evoluo equao or maser equao of mecouous MC wh dscree saes, represeed hem Schrodger pcures. We defed our sysem sae P-bra wh he sae fuco Do s formalsm or he sadard bra Pel s Techques. I he ed, we vesgaed he raso from Schrodger pcure o Heseberg pcure of me-couous MC as sochasc process. To show he smlares ad dffereces bewee PBN ad VBN, a dealed comparso s gve he ables of Appedx A. The dervao of maser equao for homogeeous MC wh Couous-saus usg PBN s gve Appedx B. Of course, more vesgaos eed o be doe o verfy he cossece (or correcess), usefuless ad lmaos of our proposos. We have doe some of hem. I Ref. [0], we have suded he basc coceps of probably space, mpora properes of codoal expecao ad roducory margales by usg PBN; Ref. [1], we have demosraed ha, uder Wck roao, he Schrodger equao (5.5.1) Drac oao s aurally shfed o he maser equao (5.5.10) PBN; Ref. [], we have derved he ufed expressos of codoal probably ad codoal expecao, defed boh Hlber space (usg Drac oao) ad probably space (usg PBN) for varous quaum sysems. Dr. Xg M Wag Probably Bracke Noao Page 3 of 36
Appedx A Table 1: Probably Bracke vs. Vecor Bracke (dscree case) Space Bra Ke Bracke PBN Sample space, assocaed wh a radom varable X P(A : a eve se P( : sae P-bra B): a evdece se ): sae P-ke P(A B) (A B): P-bracke (Codoal probably) VBN Hlber space H, assocaed wh a Herma operaor H A : a (row) vecor H () : v-sae bra B : a (colum) vecor H () = () v-sae ke A B A, B : v-bracke (Ier produc) Bracke raspose Usg Bayes formula B A = A B * Specal relaos (PBN oly) Base org P(A B) = 1 f A B P(A B) = 0 f A B P( A B) P( A ), f A ad B are muually depede Complee muual-dso ses assocaed wh varable X: = δ, Σ = Egevecors of a Herma Operaor H: Ĥ = E Orhoormaly P( )= δ = δ U operaor Î = Σ )P( Î = Σ Rgh expaso Lef expaso ) = Σ )P( ) = Σ m () ) P( = Σ ( )P( = Σ P( () = Σ () = Σ c () ) () = () Sae ormalzao P( ) = Σ m () = 1 () () = Σ c () Observable X )= x ) Ĥ = E Expecao value X = P( X ) = Σ m () x H = () Ĥ () = Σ c () E Dr. Xg M Wag Probably Bracke Noao Page 33 of 36
Table : Probably Bracke vs. Vecor Bracke (couous case) PBN VBN Specal relaos (PBN oly) Observable desy/ dsrbuo Fuco P( A B) 1 f A a B & dx 0 P(A B) = 0 f A B P ( A B) ( x x ') ( x x ') f A x, B x ', as base eves P( A B) P( A ), f A ad B are muually depede X x )= x x) f(x, ) = P(x ) a ˆp p = p p c(p, ) = p () Orhoormaly P(x x ) = δ(x -x ) p p = δ(p -p ) U operaor Î = x) dx P(x Î = p dp p Rgh expaso Lef expaso ) = x) dx P(x ) = x) f(x, ) dx P( = P( x) dx (x = dx P(x () = p dp p () = p c(p, ) dp () = () Sae ormalzao P( ) = f(x, ) dx = 1 () () = c(p, ) dp = 1 Expecao value X = P( X ) = f(x, ) x dx ˆp = () ˆp () = c(p, ) p dp Appedx B Maser Equao of Homogeeous Markov Cha wh Couous Saes Eq. (5..9) ad (5..10) are represeao-depede. They ca be easly exeded o M.C of couous-saes. Le us assume ha he sysem s -h sae f s locaed he rage of ( x, x x), herefore, P( ( )) P( x ( )) x ad: ( ) ( ) ( ˆ T P x P x xp x Q x ) x P( x ) ˆ T ( ) ' ( ') ( ' ) ( ˆ T P x dx P x Q x P x P x Q ) x0 (, ) ' ( ˆ T P x dx P x Q x ') P( x ', ) dx ' L( x, x ') P( x ', ) (B.1) Dr. Xg M Wag Probably Bracke Noao Page 34 of 36
Ths s a specal case o he maser equao (Eq. (4.6) of [1]): P( x, ) dx '[ W ( x x ') P( x ', ) W ( x ' x) P( x, )] (B.) For M.C of dscree saes, he maser equao reduces o (see Eq. (4.7) of [1]): P ( ) [ ( ) ( ) ( ) ( )] W P ' ' ' W P ' (B.3) If W s me-depede, we ca rewre o he form of Eq. (5..3) by defg: 1 P ( ) 1 ( ) h P h v lm w, w lm h0 h h0 h ( ) (B.4) w ( ) w 0, L w v v ( ) (B.5) The Eq. (B.3) ca be rewre as he akes form P ( ) [ ( ) ( )] ( ) W P ' ' ' v P ' L P ' ' ' (B.6) To exed o couous saes MC, we roducg: v( x) dx ' w( x ' x) (B.7) x' The he maser equao for homogeeous MC becomes: P( x, ) dx '[ w( x x ') ( x x ') v( x)] P( x ', ) dx ' L( x, x ') P( x ', ) (B.8) Combg Eq. (5..10) ad Eq. (B.8) ad usg PBN, we have maer equao for homogeeous couous-me M.C of boh dscree ad couous saes: ) ˆ L ) (B.9) ˆ ˆ ) U (,0) 0) exp[ L] 0) (B.10) Dr. Xg M Wag Probably Bracke Noao Page 35 of 36
Refereces [1]. Herber Kroemer, Quaum Mechacs, for Egeerg, Maeral Scece ad Appled Physcs. Prece Hall, 1994. []. Keh Va Rsberge. The Geomery of Iformao Rereval. Cambrdge, 004. [3]. X. Wag. Drac Noao, Fock Space, Rema Merc ad IR Models, arxv.org/abs/cs.ir/0701143 [4]. Charles M. Grsead ad J. Laure Sell. Iroduco o probably. d revsed edo, Amerca Mahemacal Socey, 1997. Also see: hp://www.mah.darmouh.edu/~prob/prob/prob.pdf [5]. Thomas Espola. Iroduco o Thermophyscs. Wm. C. Brow, 1994. [6]. Markov Chas: hp://e.wkpeda.org/wk/markov_cha [7]. J. G. Kemey, J. L. Sell, G. L. Thompso. Iroduco o Fe Mahemacs, 3rd ed. Eglewood Clffs, NJ: Prece-Hall, 1974 [8]. S. Lafo e al., Dffuso Maps ad Coarse-Grag: A Ufed Framework for dmesoaly Reduco, Graph Parog ad Daa Se Parameerzao. Paer Aalyss ad Mache Iellgece, IEEE Trasacos o Volume 8, Issue 9, Sep. 006 Page(s): 1393-1403 [9]. S. Lafo. Dffuso Maps ad Geomerc Harmocs. hp://www.mah.yale.edu/~sl349/publcaos/dsserao.pdf [10]. E. Ye ad D. Zhag. Probably Theores ad Sochasc Processes (I Chese), Beg, Cha, Scece Publcaos, 005 [11]. Oleg Selezev, Sochasc Processes, Lecure Noes, 006-007, see: hp://www.mah.umu.se/persoal/selezev_oleg/ [1]. J. L. Garca-Palacos, Iroduco o he Theory of Sochasc Processes ad Browa Moo Problem, Lecure Noes, 004, see hp://babbage.sssa./abs/cod-ma/07014 [13]. N. L. Sokey., Browa Models Ecoomcs, Lecure Noes, 006, see hp://home.uchcago.edu/~sokey/course.hm [14]. A. M. Ross. Iroduco o Probably Modelg, 9 h edo, Elsever Ic., 007 [15]. R. Lboff, Iroducory Quaum Mechacs. d edo, Addso-Wesley, 199. [16]. M. Do, J.Phys.A: Mah. Ge. 9 1465, 1479 (1976). [17]. L. Pel, Pah Iegral Approach o Brh-deah Processes o a Lace, J. Physque 46 (1985) 1469-1483, or hp://people.a.f./~pel/pah.pdf [18]. S. Trmper, Maser Equao ad Two Hea Reservors, Phys. Rev., E 74, 05111(006), or arxv.org/abs/cod-ma/0608354; [19]. X. Wag, Iduced Hlber Space, Markov Cha, Dffuso Map ad Fock Space Thermophyscs, arxv.org/abs/cs/07011 [0]. X. Wag, Probably Bracke Noao: Probably Space, Codoal Expecao ad Iroducory Margales, arxv.org/abs/0707.36 [1]. X. Wag, From Drac Noao o Probably Bracke Noao: Tme Evoluo ad Pah Iegral uder Wck Roaos, arxv.org/abs/0901.4816 []. X. Wag, Probably Bracke Noao: he Ufed Expressos of Codoal Expecao ad Codoal Probably Quaum Modelg, arxv.org/abs/0911.146 Dr. Xg M Wag Probably Bracke Noao Page 36 of 36