Machine Learning for Data Science (CS4786) Lecture 19
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1 Machine Learning for Data Science (CS4786) Lecture 19 Hidden Markov Models Course Webpage :
2 Quiz
3 Quiz Two variables can be marginally independent but not conditionally independent given a third variable? True False
4 Quiz Two variables can be marginally independent but not conditionally independent given a third variable? True False
5 Quiz Two variables can be marginally independent but not conditionally independent given a third variable? True False Two variables can be conditionally independent given a third variable but not marginally independent? True False
6 Quiz Two variables can be marginally independent but not conditionally independent given a third variable? True False Two variables can be conditionally independent given a third variable but not marginally independent? True False
7 EXAMPLE: CIAND MI
8 EXAMPLE: CIAND MI Marginally independent Genetic Info of Mother Genetic Info of Father
9 EXAMPLE: CIAND MI Marginally independent but Conditionally dependent given child Genetic Info of Mother Genetic Info of Father Genetic Info of Child
10 EXAMPLE: CIAND MI Marginally independent but Conditionally dependent given child Genetic Info of Mother Genetic Info of Father Genetic Info of Child Given genetic info of child, with genetic info about mother, we can infer something about father
11 EXAMPLE: CIAND MI Marginally independent but Conditionally dependent given child Marginally dependent Genetic Info of Mother Genetic Info of Father Genetic Info of Child Genetic Info of Sister Genetic Info of Brother Given genetic info of child, with genetic info about mother, we can infer something about father
12 EXAMPLE: CIAND MI Marginally independent but Conditionally dependent given child Marginally dependent but Conditionally independent given Parent Genetic Info of Mother Genetic Info of Father Genetic Info of Parents Genetic Info of Child Genetic Info of Sister Genetic Info of Brother Given genetic info of child, with genetic info about mother, we can infer something about father
13 EXAMPLE: CIAND MI Marginally independent but Conditionally dependent given child Marginally dependent but Conditionally independent given Parent Genetic Info of Mother Genetic Info of Father Genetic Info of Parents Genetic Info of Child Genetic Info of Sister Genetic Info of Brother Given genetic info of child, with genetic info about mother, we can infer something about father Given genetic info about parents, genetic info of sibling reveals nothing new about myself.
14 BAYESIAN NETWORKS
15 BAYESIAN NETWORKS Bayes net: directed acyclic graph + P(node parents)
16 BAYESIAN NETWORKS Bayes net: directed acyclic graph + P(node parents) Directed acyclic graph G = (V,E) Edges going from parent nodes to child nodes Direction indicates parent generates child
17 BAYESIAN NETWORKS Bayes net: directed acyclic graph + P(node parents) Directed acyclic graph G = (V,E) Edges going from parent nodes to child nodes Direction indicates parent generates child Provide conditional probability table/distribution P(node parents)
18 BAYESIAN NETWORKS Bayes net: directed acyclic graph + P(node parents) Directed acyclic graph G = (V,E) Edges going from parent nodes to child nodes Direction indicates parent generates child Provide conditional probability table/distribution P(node parents) P (X 1,...,X N )= NY P (X i Parents(X i )) i=1
19 GRAPHICAL MODELS Two main questions
20 GRAPHICAL MODELS Two main questions Learning/estimation: Given observations, can we learn the parameters for the graphical model?
21 GRAPHICAL MODELS Two main questions Learning/estimation: Given observations, can we learn the parameters for the graphical model? Inference: Given model parameters, can we answer queries about variables in the model
22 GRAPHICAL MODELS Two main questions Learning/estimation: Given observations, can we learn the parameters for the graphical model? Inference: Given model parameters, can we answer queries about variables in the model Eg. what is the most likely value of a latent variable given observations
23 GRAPHICAL MODELS Two main questions Learning/estimation: Given observations, can we learn the parameters for the graphical model? Inference: Given model parameters, can we answer queries about variables in the model Eg. what is the most likely value of a latent variable given observations Eg. What is the distribution of a particular variable conditioned on others
24 HIDDEN MARKOV MODEL (HMM) Speech recognition Natural language processing models Robot localization User attention modeling Medical monitoring Time! sequence of observations
25 MARKOV MODEL S 1 S 2 S 3 Each node is identically distributed given its predecessor (stationary) The values the nodes take are called states Parameters? P(S1) the initial probability table P(St St-1) the transition probabilities
26 MARKOV MODEL Bot tends to follow outlined path, but with some probability jumps to arbitrary neighbor Number of states: 25 (one for each location) For white boxes probability of jumping to any of the 4 neighbors is same 1/4 For Blue boxes, probability of following path is 0.9 and jumping to some other neighbor is
27 MARKOV MODEL Bot tends to follow outlined path, but with some probability jumps to arbitrary neighbor Number of states: 25 (one for each location) For white boxes probability of jumping to any of the 4 neighbors is same 1/4 For Blue boxes, probability of following path is 0.9 and jumping to some other neighbor is
28 MARKOV MODEL If we observe the bot long enough, we get an estimate of its behavior (the transition table of jumping from state to state) If we observe enough number of times, we can also estimate initial distribution over states
29 MARKOV MODEL Inference question: what is probability that we will be in state k at time t? P (S t = k)? Answer:
30 MARKOV MODEL Inference question: what is probability that we will be in state k at time t? P (S t = k)? Answer: P (S t = k) = KX... KX P (S 1 = s 1,...,S t 1 = s t 1,S t = k) s 1 =1 s t 1 =1 = KX... KX ty 1 (P (S i = s i S i 1 = s i 1 ) P (S t = k S t 1 = s t 1 )) s 1 =1 s t 1 =1 i=1
31 MARKOV MODEL Inference question: what is probability that we will be in state k at time t? P (S t = k)? Answer: P (S t = k) = KX... KX P (S 1 = s 1,...,S t 1 = s t 1,S t = k) s 1 =1 s t 1 =1 = KX... KX ty 1 (P (S i = s i S i 1 = s i 1 ) P (S t = k S t 1 = s t 1 )) s 1 =1 s t 1 =1 i=1 For every t we can repeat the above or
32 MARKOV MODEL Inference question: what is probability that we will be in state k at time t? P (S t = k)? Answer: P (S t = k) = KX... KX P (S 1 = s 1,...,S t 1 = s t 1,S t = k) s 1 =1 s t 1 =1 = KX... KX ty 1 (P (S i = s i S i 1 = s i 1 ) P (S t = k S t 1 = s t 1 )) s 1 =1 s t 1 =1 i=1 For every t we can repeat the above or P (S t = k) = KX s t 1 =1 P (S t = k S t 1 = s t 1 )P (S t 1 = s t 1 ) recursively compute probability of previous state
33 MARKOV MODEL As time goes by, P(St = k) approaches a fixed distribution called stationary distribution Without any further observations, you are unlikely to find the bot on a new run (only by luck)
34 HIDDEN MARKOV MODEL (HMM)
35 Same example: HIDDEN MARKOV MODEL (HMM)
36 Same example: HIDDEN MARKOV MODEL (HMM) But you don t observe location (dark room)
37 Same example: HIDDEN MARKOV MODEL (HMM) But you don t observe location (dark room) You hear how close the bot is!
38 Same example: HIDDEN MARKOV MODEL (HMM) But you don t observe location (dark room) You hear how close the bot is! S 1 S 2 S 3 X 1 X 2 X 3 X t s are loudness of what you hear
39 HIDDEN MARKOV MODEL (HMM)
40 HIDDEN MARKOV MODEL (HMM) Both during the initial training/estimation phase, you never see the bot you only hear it
41 HIDDEN MARKOV MODEL (HMM) Both during the initial training/estimation phase, you never see the bot you only hear it But you hear it at any point in time
42 HIDDEN MARKOV MODEL (HMM) Both during the initial training/estimation phase, you never see the bot you only hear it But you hear it at any point in time We will come back to learning next class.
43 HIDDEN MARKOV MODEL (HMM) Both during the initial training/estimation phase, you never see the bot you only hear it But you hear it at any point in time We will come back to learning next class. What is probability that bot will be in state k at time t given the entire sequence of observations?
44 HIDDEN MARKOV MODEL (HMM) Both during the initial training/estimation phase, you never see the bot you only hear it But you hear it at any point in time We will come back to learning next class. What is probability that bot will be in state k at time t given the entire sequence of observations? P (S t = k X 1,...,X N )?
45 Same example: HIDDEN MARKOV MODEL (HMM) But you don t observe location (dark room) You hear how close the bot is! What you hear:
46 Same example: HIDDEN MARKOV MODEL (HMM) But you don t observe location (dark room) You hear how close the bot is! What you hear: Can you catch the Bot?
47 HIDDEN MARKOV MODEL (HMM) Xt s are what you hear (observation) St s are the unseen locations (states) Eg: for n x n grid we have, K = n 2 states Number of alphabets = 5 (colors you can observe)
48 HIDDEN MARKOV MODEL (HMM) S 1 S 2 S 3 X 1 X 2 X 3 Xt s are what you hear (observation) St s are the unseen locations (states) Eg: for n x n grid we have, K = n 2 states Number of alphabets = 5 (colors you can observe)
49 HIDDEN MARKOV MODEL (HMM) What are the parameters?
50 HIDDEN MARKOV MODEL (HMM) S 1 S 2 S 3 X 1 X 2 X 3 What are the parameters?
51 HIDDEN MARKOV MODEL (HMM)
52 HIDDEN MARKOV MODEL (HMM) What is probability that bot will be in location k at time t given the entire sequence of observations?
53 HIDDEN MARKOV MODEL (HMM) What is probability that bot will be in location k at time t given the entire sequence of observations? P (S t = k X 1,...,X N )?
54 INFERENCE IN HMM P (S t = k X 1,...,X N ) / P (X t+1,...,x N S t = k, X 1,...,X t )P (S t = k X 1,...,X t ) / P (X t+1,...,x N S t = k, X 1,...,X t )P (S t = k, X 1,...,X t ) / P (X t+1,...,x N S t = k, X 1,...,X t )P (X t S t = k, X 1,...,X t 1 )P (S t = k, X 1,...,X t 1 ) = / P (X t+1,...,x N S t = k)p (X t S t = k)p (S t = k, X 1,...,X t 1 )
55 INFERENCE IN HMM P (S t = k X 1,...,X N ) / P (X t+1,...,x N S t = k, X 1,...,X t )P (S t = k X 1,...,X t ) / P (X t+1,...,x N S t = k, X 1,...,X t )P (S t = k, X 1,...,X t ) / P (X t+1,...,x N S t = k, X 1,...,X t )P (X t S t = k, X 1,...,X t 1 )P (S t = k, X 1,...,X t 1 ) = / P (X t+1,...,x N S t = k)p (X t S t = k)p (S t = k, X 1,...,X t 1 ) We know P (X t S t = k) s and P (S t S t 1 ) Compute P (X t+1,...,x N ) and P (S t = k, X 1,...,X t 1 )recursively.
56 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 message St 1 7!S t (k) =P (S t = k, X 1,...,X t 1 ) message St+1 7!S t (k) =P (X n,...,x t+1 S t = k) P (S t = k X 1,...,X n ) / message St 1 7!S t (k) message St+1 7!S t (k) P (X t S t = k)
57 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 Forward: message St 1 7!S t (k) =P (S t = k, X 1,...,X t 1 ) message St+1 7!S t (k) =P (X n,...,x t+1 S t = k) P (X 1,...,X t 1,S t = k) = KX j=1 P (S t = k S t 1 = j)p (X t 1 S t 1 = j)p (X 1,...,X t 2,S t 1 = j) message St 1 7!S t (k) = KX P (S t = k S t 1 = j)p (X t 1 S t 1 = j)message St 2 7!S t 1 (j) j=1
58 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 Forward: message St 1 7!S t (k) =P (S t = k, X 1,...,X t 1 ) message St+1 7!S t (k) =P (X n,...,x t+1 S t = k) P (X 1,...,X t 1,S t = k) = KX j=1 P (S t = k S t 1 = j)p (X t 1 S t 1 = j)p (X 1,...,X t 2,S t 1 = j) message St 1 7!S t (k) = KX P (S t = k S t 1 = j)p (X t 1 S t 1 = j)message St 2 7!S t 1 (j) j=1
59 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 Forward: message St 1 7!S t (k) =P (S t = k, X 1,...,X t 1 ) message St+1 7!S t (k) =P (X n,...,x t+1 S t = k) P (X 1,...,X t 1,S t = k) = KX j=1 P (S t = k S t 1 = j)p (X t 1 S t 1 = j)p (X 1,...,X t 2,S t 1 = j) message St 1 7!S t (k) = KX P (S t = k S t 1 = j)p (X t 1 S t 1 = j)message St 2 7!S t 1 (j) j=1
60 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 message St 1 7!S (k) =P (S t t = k, X 1,...,X t 1 ) message St+1 7!S (k) =P (X t n,...,x t+1 S t = k) Backward: KX P (X n,...,x t+1 S t = k) = P (X n,...,x t+2 S t+1 = j)p (X t+1 S t+1 = j)p (S t+1 = j S t = k) j=1 KX message St+1 7!S t (k) = message St+2 7!S t+1 (j)p (X t+1 S t+1 = j)p (S t+1 = j S t = k) j=1
61 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 message St 1 7!S (k) =P (S t t = k, X 1,...,X t 1 ) message St+1 7!S (k) =P (X t n,...,x t+1 S t = k) Backward: KX P (X n,...,x t+1 S t = k) = P (X n,...,x t+2 S t+1 = j)p (X t+1 S t+1 = j)p (S t+1 = j S t = k) j=1 KX message St+1 7!S t (k) = message St+2 7!S t+1 (j)p (X t+1 S t+1 = j)p (S t+1 = j S t = k) j=1
62 INFERENCE IN HMM S 1 S 2 S 3 X 1 X 2 X 3 message St 1 7!S (k) =P (S t t = k, X 1,...,X t 1 ) message St+1 7!S (k) =P (X t n,...,x t+1 S t = k) Backward: KX P (X n,...,x t+1 S t = k) = P (X n,...,x t+2 S t+1 = j)p (X t+1 S t+1 = j)p (S t+1 = j S t = k) j=1 KX message St+1 7!S t (k) = message St+2 7!S t+1 (j)p (X t+1 S t+1 = j)p (S t+1 = j S t = k) j=1
63 LEARNING PARAMETERS FOR HMM Now that we have algorithm for inference, what about learning Given observations, how do we estimate parameters for HMM? Three guesses...
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