application：
語音識別
分詞
1、隐藏状态 (天气)：Sunny，Cloudy，Rainy；
2、观察状态（海藻湿度）：Dry，Dryish，Damp，Soggy；
3、初始状态概率： Sunny（0.63）， Cloudy（0.17）， Rainy（0.20）；
4、状态转移矩阵：
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today weather
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sunny
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cloudy
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rainy
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yesterday weather
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sunny
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0.5
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0.375
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0.125
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cloudy
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0.25
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0.125
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0.625
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rainy
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0.25
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0.375
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0.375
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5、混淆矩阵：
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observed states
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dry
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dryish
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damp
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soggy
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hidden states
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sunny
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0.6
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0.2
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0.15
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0.05
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cloudy
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0.25
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0.25
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0.25
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0.25
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rainy
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0.05
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0.1
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0.35
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0.5
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Finding most probable sequence of hidden states
brute force
p(hidden state|(dry,damp,soggy)) = max(p(dry|sunny,damp|sunny,soggy|sunny),p(dry|sunny,damp|sunny,soggy|cloudy),p(dry|sunny,damp|sunny,soggy|rainy)…… )
viterbi (dynamic programming)
calculate the partial best path based on each phase, and record the partial best path we walk through
phase 1(dry):
p(sunny) = initial hidden state * p(dry|sunny)=0.63 *0.6 = 0.378
p(rainy) = initial hidden state * p(dry|rainy) = 0.17 * 0.05 = 0.0425
p(cloudy) = initial hidden state * p(dry|cloudy) = 0.2 * 0.25 =0.01
phase 2(damp):
p(sunny) = max(p(previous phase sunny)*p(sunny|sunny),p(previous phase rainy)*p(sunny|rainy),p(previous phase cloudy)*p(sunny|cloudy)) *p(damp|sunny)=max(0.378*0.5,0.0425*0.25,0.01*0.25)*0.15=0.378*0.5*0.15 = 0.02835 prev-state : sunny
…
phase 3(soggy):
same as phase 2