Outline. simplest HMM (1) simple HMMs? simplest HMM (2) Parameter estimation for discrete hidden Markov models

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1 Oulie Parameer esimaio for discree idde Markov models Juko Murakami () ad Tomas Taylor (2). Vicoria Uiversiy of Welligo 2. Arizoa Sae Uiversiy Descripio of simple idde Markov models Maximum likeliood esimae (usig Baum- Welc algorim) mode Bayes (or Leas square error) esimae mea Compariso of e mode ad mea simples HMM () simple HMMs? 2 Y Y2 Y3 Sae sequece (Markov cai) Observaio sequece Y,, 3 ( K, ), 2, ( Y Y K, Y ), 2,,, {,} ad Y {,} simples HMM (2) Codiioal Probabiliies a a a a b b b b Baum-Welc Algorim o Maximize e Log Likeliood Also, le r P Noe : a i a + ( ) ad r P( ). i, b i b i, ad r r Y for i,.

2 Baum-Welc Algorim () Baum-Welc Algorim (2) Cosider e likeliood fucio L( θ ), e parameer se θ, were L,, ( θ ) P( Y, θ ), e probabiliy of avig sequeces θ Y { r a, a, b, }., b,, ad, give ( k) ( k+) Usig e k esimae, θ, we wa θ o be e θ - value a maximizes Q θ, θ defied below. ( k), Q( θ, θ ) E logl( θ) Y, θ So, e goal is o maximize e expeced value of e log likeliood fucio, give e observaio sequece ad e curre esimae. Baum-Welc Algorim (3) Te algorim fids wo ypes of probabiliies. Le i {,}. forward procedure, Recursively fidα () i P( Y, i θ ), sarig from ime up o. backward procedure +, ( i) P Y i, Recursively fid β θ sarig from ime dow o., Baum-Welc Algorim (4) Caracerisics A implemeaio of E-M algorim. VERY widely used i various field. Te, uses ( k+ ) { α ( i) }, β { β ( i) }, adθ o compue. α θ Baum-Welc Algorim (5) Baum-Welc Algorim (6) Advaages Maximizes e likeliood e majoriy of imes. Te covergece is quick eoug e majoriy of imes. Sill feasible we e sae ad observaio space size is large. Implemeaio is easy. Disadvaages Srog depedecy o e iiial esimae. Guaraeed oly o fid a local maximum. Overfiig problem: o close o e rue parameer se we e daa size is small. Covergece is someime very slow. Olie compuaio is o possible.

3 LSE (Bayes) Esimae () Leas Square Error (LSE) Esimae (or Bayes Esimae) Fids e expeced value of e parameer se give a observaio sequece; i.e., ad usig Bayes' eorem, we ave were E [ θ ] P Y,,, P( Y, θ ) dθ. P Y, [ θ ] θ P( θ Y ) d. ˆ, θ E θ, Ω, Ω θ P Y, (, θ ) dθ, ( θ) Assumig e uiform disribuio of θ (i.e., leig P ), LSE (Bayes) Esimae (2) LSE (Bayes) Esimae (3) Firs, we le wic as e size 2. NOTE Te summaio is over Ω, { all e possible values of }, {,}, were # ( eve) for i, j, u eves over {, 2, K, }. k l iu ( i ad j) ( i ad Y u) ij # + # meas ad e oal umber of { k } ad L { }. Le K ij l iu If we fix r a 2 LSE (Bayes) Esimae (4) as 2 for simpliciy, P Y,, (, θ ) is i e form k k k k l l l l ( a ) a ( a ) b ( b ) b ( b ),,,,, (, θ ) dθ ad P( Y, θ ) ad so bo θ P Y fucios of K ad L, θ { r, a, a, b, b }. NOTE: Because of e symmery i e probabiliy disribuio, e iegraio sould be uder some resricio; e.g., a a. dθ are Fac. 2. LSE (Bayes) Esimae (5) To evaluae e iegrals, all we eed o kow are e values of { K, L}. { K, L} ca be expressed as a fucio of { k, k, l,, }, isead, were k All we eed is ω { k k, l,, }. is e umber, of ' s i,.

4 LSE (Bayes) Esimae (6) LSE (Bayes) Esimae (7) Fac, Give a paricular Y, differe sae sequeces ca produce e same value of ω { k k, l,, }.,, ( ω ) If we fid e values of for all ω, e e summaios ca be doe over ω suc a ( ω ) >, isead of over all, possible values of Ω. Le ω o e ω be e umber of give., values a correspods Te algorim sows a e umber of ( ω ) > are polyomial of. ω values suc a LSE (Bayes) Esimae (7) ( ω ) If we fid e values of for all ω, e e summaios ca be doe over ω suc a ( ω ) >, isead of over all, possible values of Ω. Te algorim sows a e umber of ( ω ) > are polyomial of. NOTE : Te observaio sae space size ca be exeded from m 2 example) o ay ieger m i geeral. ω values suc a (is Le LSE (Bayes) Esimae (8) (,,,, ) for from o ed for wi all + ω ( k, k, l,, ) suc a ( ω ) + ( k, k, l,, ) ad ( k +, k +, l + +,,) ( ω ) icreme by e value (Because of e symmery, we ca fid oce e oes for is obaied.) ω > for LSE (Bayes) Esimae (9) LSE (Bayes) Esimae () Advaages Closer a B-W esimaes o e rue parameers we e daa size is small. Olie compuaio is possible. Fids e exac expeced values (ubiased). Oe-ime compuaio. Disadvaage Compuaioal complexiy grows sill expoeially i e sae space size.

5 Example Baum-Welc esimae LSE (Bayes) esimae Example 2: B-W ad LSE esimaes wi a small daa se () Oulie: Geerae 2 of A { a } ij For eac θ, esimaes. θ - values, radomly wi respec o e deermia ad o e differece b geerae a se of b.,, {, Y },, ad obai e a a As for B-W esimaes: Fid esimaes usig radomly picked iiial esimaes. Pick e oe wi e larges basi. Example 2: B-W ad LSE esimaes wi a small daa se (2) Referees J. A. Bilms, A gele uorial of e EM algorim ad is applicaio o parameer esimaio for Gaussia mixure ad idde Markov models, Ieraioal Compuer Sciece Isiue, Tec. Rep. ICSI-TR- 97-2, April 998. L. E. Baum, A iequaliy ad associaed maximizaio ecique i saisical esimaio for probabilisic fucios of Markov processes, Iequaliies, vol. 3, pp. -8, 972. J. Murakami, Parameer esimae of a idde Markov cai, Upublised P.D. Disseraio, Arizoa Sae Uiversiy, Tempe, AZ, USA, May 25. Te firs are ploed. O e average, e B-W esimaes (orage dos) were farer away from e rue parameers a LSE oes (gree dos) by.73 ad less sable. Ackowledgeme Te preseer (J. Murakami) would like o ak Keio Uiversiy ad NZIMA for eir suppor.