Machine Learning for Data Science (CS4786) Lecture 24
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1 Machine Learning for Data Science (CS4786) Lecture 24 HMM Particle Filter Course Webpage :
2 Rejection Sampling
3 Rejection Sampling Sample variables from joint distribution (both latent and observed variables)
4 Rejection Sampling Sample variables from joint distribution (both latent and observed variables) Throw away all ones that don t match observation
5 Rejection Sampling Sample variables from joint distribution (both latent and observed variables) Throw away all ones that don t match observation Now compute empirical frequencies
6 Rejection Sampling Sample variables from joint distribution (both latent and observed variables) Throw away all ones that don t match observation Now compute empirical frequencies Problem: too wasteful (too many Rejections)
7 REJECTION SAMPLING Algorithm: Topologically sort variables (parents first children later) For t = 1 to n (number of samples) For i = 1 to N (number of variables in model) End For End For Sample x t i P(X i Parents(X i ) already sampled) Discard samples that do not match observations Compute empirical frequencies
8 IMPORTANCE SAMPLING We really want to draw from distribution P. But we can only draw from distribution Q easily Trick: Draw x 1,...,x n Q Re-weight each sample x t by P(X = x t )Q(X = x t )
9 Importance Sampling
10 Importance Sampling For bayesian networks:
11 Importance Sampling For bayesian networks: What we want: P (X Latent1,...,X Latentm X Observed1,...X Observedn )
12 Importance Sampling For bayesian networks: What we want: P (X Latent1,...,X Latentm X Observed1,...X Observedn ) What we sample from: my P (X Parent(X )) Latentj Latentj j=1
13 Importance Sampling For bayesian networks: What we want: P (X Latent1,...,X Latentm X Observed1,...X Observedn ) What we sample from: my P (X Parent(X )) Latentj Latentj Weight: j=1 ny P (X Parent(X )) Observedi Observedi i=1
14 IMPORTANCE SAMPLING Likelihood weighting: Topologically sort variables (parents first children later) For t = 1 to n (number of samples) Set w t = 1 For i = 1 to N (number of variables) End For Output, End For If X i is observed, Set w t w t P(X i = x i Parents(X i ) = already sampled) Set x t i = x i (the observed value) Else, sample x t i P(X i Parents(X i ) = already sampled) P(Variable = valueobservation) = n t=1 w t1{variable = value} n t=1 w t
15 HIDDEN MARKOV MODEL (HMM)
16 HIDDEN MARKOV MODEL (HMM)
17 Example: HIDDEN MARKOV MODEL (HMM)
18 HIDDEN MARKOV MODEL (HMM) Example: But you don t observe location (dark room)
19 HIDDEN MARKOV MODEL (HMM) Example: But you don t observe location (dark room) You hear how close the bot is!
20 HIDDEN MARKOV MODEL (HMM) Example: But you don t observe location (dark room) You hear how close the bot is! What you hear: + noise
21 HIDDEN MARKOV MODEL (HMM) Example: But you don t observe location (dark room) You hear how close the bot is! What you hear: + noise Can you catch the Bot? In time?
22 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 m x m grid we have, K = m 2 states Number of alphabets = # colors you can observe
23 HIDDEN MARKOV MODEL (HMM) S 1 S 2 S 3 X 1 X 2 X 3 Eg: for m x m grid we have, K = m 2 states
24 HIDDEN MARKOV MODEL (HMM) S 1 S 2 S 3 X 1 X 2 X 3 Eg: for m x m grid we have, K = m 2 states Transition matrix is K x K (too large)
25 HIDDEN MARKOV MODEL (HMM) S 1 S 2 S 3 X 1 X 2 X 3 Eg: for m x m grid we have, K = m 2 states Transition matrix is K x K (too large) Use sampling to do approximate inference Number of samples n << m 4
26 Inference Question Can we compute (efficiently and approximately) P (S t x 1,...,x t 1 ) We cant afford too much time to compute since we need to move the bot in time
27 HIDDEN MARKOV MODEL (HMM)
28 HIDDEN MARKOV MODEL (HMM)
29 HIDDEN MARKOV MODEL (HMM)
30 HIDDEN MARKOV MODEL (HMM)
31 HIDDEN MARKOV MODEL (HMM)
32 HIDDEN MARKOV MODEL (HMM)
33 HIDDEN MARKOV MODEL (HMM)
34 HIDDEN MARKOV MODEL (HMM)
35 HIDDEN MARKOV MODEL (HMM)
36 HIDDEN MARKOV MODEL (HMM)
37 HIDDEN MARKOV MODEL (HMM)
38 HIDDEN MARKOV MODEL (HMM)
39 HIDDEN MARKOV MODEL (HMM)
40 HIDDEN MARKOV MODEL (HMM)
41 HIDDEN MARKOV MODEL (HMM)
42 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
43 HIDDEN MARKOV MODEL (HMM) Eg: say observations were Rejection sampling: Reject samples that don t match observations
44 HIDDEN MARKOV MODEL (HMM) Eg: say observations were Rejection sampling: Reject samples that don t match observations We can do this sequentially!
45 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
46 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
47 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
48 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
49 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
50 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
51 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
52 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
53 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
54 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
55 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
56 HIDDEN MARKOV MODEL (HMM) Eg: say observations were Problem: Most samples rejected
57 HIDDEN MARKOV MODEL (HMM) Eg: say observations were Multiple samples simultaneously. Problem: Most samples rejected
58 HIDDEN MARKOV MODEL (HMM) Eg: say observations were
59 HIDDEN MARKOV MODEL (HMM) Eg: say observations were Importance weighting: weight samples
60 HIDDEN MARKOV MODEL (HMM) Eg: say observations were Importance weighting: weight samples
61 HIDDEN MARKOV MODEL (HMM) Eg: say observations were P( X1=13) x P( X2=8) x P( X3=9) x P( X5=24) x P( X5=19) xp( X4=14) Importance weighting: weight samples
62 HMM PARTICLE FILTER Use multiple samples and track each ones weights.
63 HMM PARTICLE FILTER Use multiple samples and track each ones weights.
64 HMM PARTICLE FILTER Use multiple samples and track each ones weights.
65 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1)
66 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1)
67 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2)
68 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2)
69 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2) P( X3)
70 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2) P( X3) P( X5) P( X5) P( X4)
71 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2) P( X3) P( X5) P( X5) P( X4) This is same as 6 separate samples
72 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2) P( X3) P( X5) P( X5) P( X4) This is same as 6 separate samples Instead of tracking each sample s weight, resample according to weights
73 HMM PARTICLE FILTER Use multiple samples and track each ones weights. P( X1) P( X2) P( X3) P( X5) P( X5) P( X4) This is same as 6 separate samples Instead of tracking each sample s weight, resample according to weights Problem: Too many samples have negligible weight!
74 HMM PARTICLE FILTER Instead of tracking each one, resample!
75 HMM PARTICLE FILTER Instead of tracking each one, resample!
76 HMM PARTICLE FILTER Instead of tracking each one, resample!
77 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1)
78 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1)
79 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1)
80 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2)
81 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2)
82 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2)
83 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2) P( X3)
84 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2) P( X3)
85 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2) P( X3) P( X5) P( X5) P( X4)
86 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2) P( X3) P( X5) P( X5) P( X4) On every round, transfer particles from previous states according to transition probability
87 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2) P( X3) P( X5) P( X5) P( X4) On every round, transfer particles from previous states according to transition probability Resample particles according to P(observation state)
88 HMM PARTICLE FILTER Instead of tracking each one, resample! P( X1) P( X2) P( X3) P( X5) P( X5) P( X4) On every round, transfer particles from previous states according to transition probability Resample particles according to P(observation state) Use new particles to proceed
89 Particle Filtering
90 Particle Filtering Without resampling, we carry many particles with very small probabilities
91 Particle Filtering Without resampling, we carry many particles with very small probabilities too many samples needed for a good estimate
92 Particle Filtering Without resampling, we carry many particles with very small probabilities too many samples needed for a good estimate By resampling, we got rid of samples with very small probabilities
93 Particle Filtering Without resampling, we carry many particles with very small probabilities too many samples needed for a good estimate By resampling, we got rid of samples with very small probabilities Hence fewer samples suffice
94 HMM PARTICLE FILTER Inference time only depends on number of samples Of course more the samples the better accuracy Often we don t need too many samples. Why?
95 Gibbs Sampling Repeat n times for, n samples, Start with arbitrary value for variables Replace each variable by new sample from P(Variable all other variables) Go over all variables multiple times Return final sample of the N variables
96 VARIATIONAL INFERENCE Basic idea: we want to infer P(UnobservedObserved) We create a new parametric distribution Q (Unobserved) where is picked based on Obervations We pick such that, Q is close to P(UnobservedObserved) Closeness measured using KL divergence Mean-field approximation, Q (X 1,...,X m ) = m Q j (X j ) j=1
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