Viterbi Algorithm: Background

Transcription

1 Vierbi Algorihm: Background Jean Mark Gawron March 24, The Key propery of an HMM Wha is an HMM. Formally, i has he following ingrediens: 1. a se of saes: S 2. a se of final saes: F 3. an iniial sae: s 0 4. an inpu alphabe: Σ 5. a ransiion funcion A which akes any pair of saes and reurns a probabiliy. The ransiion probabiliies LEAVING any sae add up o An observaion funcion B. B(s i,o i ) is he probabiliy of observing emission o i (his is he way hese guys alk) a sae s i. s i is a sae and o i is a member of he inpu (observaion) alphabe. The key propery: The Markov Assumpion: he probabliy of a ransiion o s j from s i depends only on s i (Markov: Oreder 1). 1.1 Charniak s Vierbi HMM example Figure1showsasimpleHMMdueoCharniakwihanalphabe ab. Todiscuss his example, all we need is he concep of a ransiion probabiliy. There is a probabiliy assigned o he ransiion from one sae o anoher on a paricular observaion (or someimes we say emission). 1

2 b:.2 a:.4 q a:.3 b:.1 a:.2 b:.1 b:.5 a:.2 r Figure 1: Charniak s simple example HMM (1.0) q(1.0) r(0.0) qq(.2) qr(.1) rq(0.0) rr(0.0) b qqq(.04) qqr(.02) qrq(.01) qrr(.05) b qqqq(.008) qqqr(.004) qrrq(.005) qrrr(.025) b qqqqq(.0032) qqqqr(.0024) qrrrq(.005) qrrrr(.005) a Figure 2: Compuaional Tree of Vierbi on Charniak s HMM 2

3 The key propery is ha he sum of all he ransiions probabiliies from a given sae is 1. The ask of he Vierbi algorihm is o find he opimal pah hrough he HMM for a given piece of inpu. In his case opimal pah means highes probabiliy. Figure 2 shows he applicaion of he Vierbi algorihm o he HMM. The view of he compuaion is a compuaional ree. Time is verical. Tha is, each sep down hrough he diagram reveals one more inpu symbol. The HMM has wo saes ha boh accep a s and b s. From each sae, for any inpu, here are wo possible saes one could go o nex. Thus, if we were doing a naive search, compuing he oal cos of all pahs covering he inpu, he number of acive pahs a each momen in ime would double. Insead, he number of acive pahs rees remains consan as we descend hrough he compuaion. Why? Tha s he rick of he algorihm. 2 Rob s coin-flipping HMM Figure 3 shows he simple HMM used in he Vierbi implemenaion in he code model vierbi (see code model lis in he assignmen). This is a simple coin flipping HMM. The sae H does no expec a ail. The sae T does no expec a head. However, when in H, you can accep an h eiher by remaining in H or, wih equal probabiliy, by moving o T. When in T you can accep a eiher by moving o H or by remaining in T. Figure 4 shows he same HMM wih zero-probabiliy ransiions removed. This makes he overall picure clearer. Only he sar sae 0 is flexible abou wha i acceps, Bu when i acceps an h i can do so eiher by moving ino an h-acceping or -acceping sae. Thus, each ime inpu is acceped, a predicion is made abou wha will come nex. The ransiion probabiliies are compued using he following probabiliy model: n Pr(o 1,...,o n,s 1,...,s n ) = Pr(o i s i 1,s i ) Pr(s i s i 1 ) (1) We call: i=1 Pr(s i s i 1 ) (2) he sae ransiion probabiliy. These probabiliies are kep in he array of arrays $A in he implemenaion. We call Pr(o i s i ) (3) he observaion probabiliy These probabiliies are kep in he hash of arrays $B in he implemenaion. 3

4 h:.25 h:.5 h:.5 :.25 H :.5 :.5 0 h:.25 :.25 T Figure 3: Rob Malouf Vierbi Implemenaion Example 4

5 h:.5 0 h:.25 :.25 H h:.25 :.25 h:.5 :.5 :.5 T Figure 4: Rob Malouf Vierbi Implemenaion Example We call he produc of hese wo probabiliies he ransiion probabiliy: Pr(o i s i ) Pr(s i s i 1 ) (4) A each sage in he Vierbi algorihm his is he cos of moving from a previous sae o he curren sae given he oupu a he curren momen in ime. The ransiion probabiliies of Figure 3 are compued as follows. (For each ransiion, he observaion probabiliy is given firs and he ransiion probabiliy second). 0 : h 0=>0: 0 * 0 = 0 0=>H: 0.5 * 0.5 = =>T: 0.5 * 0.5 = =>0: 0 * 0 = 0 0=>H: 0.5 * 0.5 = =>T: 0.5 * 0.5 = (Toal Probs) = 1 H : h 5

6 H=>0: 0 * 0 = 0 H=>H: 1 * 0.5 = 0.5 H=>T: 1 * 0.5 = 0.5 H=>0: 0 * 0 = 0 H=>H: 0 * 0.5 = 0 H=>T: 0 * 0.5 = H (Toal Probs) = 1 T : h T=>0: 0 * 0 = 0 T=>H: 0 * 0.5 = 0 T=>T: 0 * 0.5 = 0 T=>0: 0 * 0 = 0 T=>H: 1 * 0.5 = 0.5 T=>T: 1 * 0.5 = T (Toal Probs) = 1 Here is he resul of running he Vierbi implemenaion on his for he oupu h : Oupu: h Time Sae Pah Prob 0: 0 1 1: T : T : T : T : H Saes: 0TTTTH The HMM acceps an h by going o he T sae, hen remains here and acceps he final by ransiioning o he h-predicing sae H. (The final sae is always an arbirary choice since we have no seen he prediced oupu ye.). The pah probabiliy afer he firs ransiion is.25, as required by Figure 4, and i is halved by each succeeding ransiion. 6

7 (1.0) 0(1.0) H(0.0) T(0.0) 0H(.25) 0T(.25) HH(0.0) HT(0.0) h 0HH(0.0) 0HT(0.0) 0TH(.125) 0TT(.125) 0THH(0.0) 0THT(0.0) 0TTH(.0625) 0TTT(.0625) 0TTHH(0.0) 0TTHT(0.0) 0TTTH(.03125) 0TTTT(.03125) Figure 5: Rob Malouf Vierbi Implemenaion Example The applicaion of Vierbi, using a compuaional ree diagram, is shown in Figure 5. 3 The Markov Assumpion and he Vierbi Assumpion Markov Assumpion:nh order MM Theprobabiliyhaq (saeaime)willbeq i (saenumber i)depends only on saes q n hrough q 1. Le s wrie he firs r seps of a pah p as p[: r], and he rh sep of p as p[r], and remember ha vierbi(q i, ) is a pah (he bes pah ending a q i a ime ), and since i s a pah of lengh o q i, vierbi(q i, )[] = q i. Vierbi Assumpion:nh order MM vierbi(q i, )[: n] is a bes pah o q n a ime n. Tha is subpahs of bes pahs are bes pahs: vierbi(q i,)[: n] = vierbi(q n, n) 7

8 This follows from he Markov assumpion. Le s look a he specific case of 1 and a firs order model. According o he Vierbi assumpion, if vierbi(q i, )[ 1] = q j, hen he firs 1 seps of vierbi(q i,) are he bes pah o q j a ime 1: vierbi(q j, 1) = vierbi(q i,)[: 1] This is a he hear of wha we do in he algorihm. We ry exending bes pahs o all he saes a 1 o find bes pah ime, knowing ha bes pah a ime mus exend one of he bes pahs a 1. 8