CPS 270: Artificial Intelligence http://www.cs.duke.edu/courses/fall08/cps270/ Markov processes and Hidden Markov Models (HMMs) Instructor: Vincent Conitzer Motivation The Bayes nets we considered so far were static: they referred to a single point in time E.g., medical diagnosis Agent needs to model how the world evolves Speech recognition software needs to process speech over time Artificially intelligent software assistant needs to keep track of users intentions over time Markov processes We have time periods t = 0, 1, 2, In each period t, the world is in a certain state St The Markov assumption: given St, St+1 is independent of all Si with i < t P(St+1 | S1, S2, , St) = P(St+1 | St)

Given the current state, history tells us nothing more about the future S1 S2 S3 St Typically, all the CPTs are the same: For all t, P(St+1 = j | St = i) = aij (stationarity assumption) Weather example St is one of {s, c, r} (sun, cloudy, rain) Transition probabilities: .6 s .3 .3

c .1 .4 .2 .3 not a Bayes net! r .5 Also need to specify an initial distribution P(S 0) Throughout, assume P(S0 = s) = 1 .3 Weather example .6 s .3 .3

c .1 .4 .2 .3 r .3 .5 What is the probability that it rains two days from now? P(S2 = r) P(S2 = r) = P(S2 = r, S1 = r) + P(S2 = r, S1 = s) + P(S2 = r, S1 = c) = .1*.3 + .6*.1 + .3*.3 = .18 Weather example .6 s .3 .3

c .1 .4 .2 .3 .5 r .3 What is the probability that it rains three days from now? Computationally inefficient way: P(S3 = r) = P(S3 = r, S2 = r, S1 = r) + P(S3 = r, S2 = r, S1 = s) + For n periods into the future, need to sum over 3n-1 paths Weather example .6 s .3

.3 c .1 .4 .2 .3 r .3 .5 More efficient: P(S3 = r) = P(S3 = r, S2 = r) + P(S3 = r, S2 = s) + P(S3 = r, S2 = c) = P(S3 = r | S2 = r)P(S2 = r) + P(S3 = r | S2 = s)P(S2 = s) + P(S3 = r | S2 = c)P(S2 = c) Only hard part: figure out P(S2) Main idea: compute distribution P(S1), then P(S2), then P(S3) Linear in number of periods! example on board Stationary distributions

As t goes to infinity, generally, the distribution P(St) will converge to a stationary distribution A distribution given by probabilitiesi (where i is a state) is stationary if: P(St = i) =i means that P(St+1 = i) =i Of course, P(St+1 = i) = j P(St+1 = i, St = j) = j P(St = j) aji So, stationary distribution is defined by i = j j aji Computing the stationary distribution .6 s .3 .3 c .1 .4 .2 .3 .5

s = .6s + .4c + .2r c = .3s + .3c + .5r r = .1s + .3c + .3r r .3 Restrictiveness of Markov models Are past and future really independent given current state? E.g., suppose that when it rains, it rains for at most 2 days S1 S2 S3 S4 S3, S4 S4 , S 5

Second-order Markov process Workaround: change meaning of state to events of last 2 days S1 , S 2 S2 , S 3 Another approach: add more information to the state E.g., the full state of the world would include whether the sky is full of water Additional information may not be observable Blowup of number of states Hidden Markov models (HMMs) Same as Markov model, except we cannot see the state Instead, we only see an observation each period, which depends on the current state S1 S2 S3

St O1 O2 O3 Ot Still need a transition model: P(St+1 = j | St = i) = aij Also need an observation model: P(Ot = k | St = i) = bik Weather example extended to HMM Transition probabilities: .6

s .3 .3 c .1 .4 .2 .3 .5 Observation: labmate wet or dry bsw = .1, bcw = .3, brw = .8 r .3 HMM weather example: a question .6 bsw = .1 s

.3 .3 c bcw = .3 .1 .4 .2 .3 r brw = .8 .3 .5 You have been stuck in the lab for three days (!) On those days, your labmate was dry, wet, wet, respectively What is the probability that it is now raining outside? P(S2 = r | O0 = d, O1 = w, O2 = w) By Bayes rule, really want to know P(S2, O0 = d, O1 = w, O2 = w)

Solving the question .6 bsw = .1 s .3 .3 c bcw = .3 .1 .4 .2 .3 r .5 brw = .8 .3

Computationally efficient approach: first compute P(S1 = i, O0 = d, O1 = w) for all states i General case: solve for P(St, O0 = o0, O1 = o1, , Ot = ot) for t=1, then t=2, This is called monitoring P(St, O0 = o0, O1 = o1, , Ot = ot) = s P(St-1 = st-1, O0 = o0, O1 = o1, , Ot-1 = ot-1) P(St | St-1 = st-1) P(Ot = ot | St) t-1 Predicting further out .6 bsw = .1 s .3 .3 c bcw = .3 .1 .4 .2

.3 .5 r brw = .8 .3 You have been stuck in the lab for three days On those days, your labmate was dry, wet, wet, respectively What is the probability that two days from now it will be raining outside? P(S4 = r | O0 = d, O1 = w, O2 = w) Predicting further out, continued .6 bsw = .1 s .3 .3 c

bcw = .3 .1 .4 .2 .3 .5 r brw = .8 .3 Want to know: P(S4 = r | O0 = d, O1 = w, O2 = w) Already know how to get: P(S2 | O0 = d, O1 = w, O2 = w) P(S3 = r | O0 = d, O1 = w, O2 = w) = s2 P(S3 = r, S2 = s2 | O0 = d, O1 = w, O2 = w) s2 P(S3 = r | S2 = s2)P(S2 = s2 | O0 = d, O1 = w, O2 = w) Etc. for S4 So: monitoring first, then straightforward Markov process updates Integrating newer information .6 bsw = .1

s .3 .3 c bcw = .3 .1 .4 .2 .3 .5 r brw = .8 .3 You have been stuck in the lab for four days (!) On those days, your labmate was dry, wet, wet, dry respectively What is the probability that two days ago it was raining outside? P(S1 = r | O0 = d, O1 = w, O2 = w, O3 = d)

Smoothing or hindsight problem Hindsight problem continued .6 bsw = .1 s .3 .3 c bcw = .3 .1 .4 .2 .3 r brw = .8 .3

.5 Want: P(S1 = r | O0 = d, O1 = w, O2 = w, O3 = d) Partial application of Bayes rule: P(S1 = r | O0 = d, O1 = w, O2 = w, O3 = d) = P(S1 = r, O2 = w, O3 = d | O0 = d, O1 = w) / P(O2 = w, O3 = d | O0 = d, O1 = w) So really want to know P(S1, O2 = w, O3 = d | O0 = d, O1 = w) Hindsight problem continued .6 bsw = .1 s .3 .3 c bcw = .3 .1 .4 .2

.3 r brw = .8 .3 .5 Want to know P(S1 = r, O2 = w, O3 = d | O0 = d, O1 = w) P(S1 = r, O2 = w, O3 = d | O0 = d, O1 = w) = P(S1 = r | O0 = d, O1 = w) P(O2 = w, O3 = d | S1 = r) Already know how to compute P(S1 = r | O0 = d, O1 = w) Just need to compute P(O2 = w, O3 = d | S1 = r) Hindsight problem continued .6 bsw = .1 s .3 .3 c

bcw = .3 .1 .4 .2 .3 .5 r brw = .8 .3 Just need to compute P(O2 = w, O3 = d | S1 = r) P(O2 = w, O3 = d | S1 = r) = s2 P(S2 = s2, O2 = w, O3 = d | S1 = r) = s2 P(S2 = s2 | S1 = r) P(O2 = w | S2 = s2) P(O3 = d | S2 = s2) First two factors directly in the model; last factor is a smaller problem of the same kind Use dynamic programming, backwards from the future Similar to forwards approach from the past Backwards reasoning in general Want to know P(Ok+1 = ok+1, , Ot = ot | Sk) First compute

P(Ot = ot | St-1) =st P(St = st | St-1)P(Ot = ot | St = st) Then compute P(Ot = ot, Ot-1 = ot-1 | St-2) =st-1 P(St-1 = st-1 | St-2)P(Ot-1 = ot-1 | St-1 = st-1) P(Ot = ot | St-1 = st-1) Etc. Dynamic Bayes Nets Everything so far is just doing inference on a special kind of Bayes net! So far assumed that each period has one variable for state, one variable for observation Often better to divide state and observation up into multiple variables weather in Durham, 1 NC wind, 1 weather in Beaufort, 1 weather in Durham, 2 NC wind, 2 weather in Beaufort, 2

edges both within a period, and from one period to the next Some interesting things we skipped Finding the most likely sequence of states, given observations Not necessary equal to the sequence of most likely states! (example?) Viterbi algorithm Key idea: for each period t, for every state, keep track of most likely sequence to that state at that period, given evidence up to that period Continuous variables Approximate inference methods Particle filtering