Lirong Xia

Probabilistic reasoning over time

Tue, March 25, 2014

• Average 17.6

• Submission

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and upload files you are asked to

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Project 2

Recap: Q-Learning with state abstraction

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• Using a feature representation, we can write a Q function (or value function) for any state using a few weights:

• Advantage: our experience is summed up in a few powerful numbers

• Disadvantage: states may share features but actually be very different in value!

Function Approximation

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• Q-learning with linear Q-functions:transition = (s,a,r,s’)

• Intuitive interpretation:– Adjust weights of active features– E.g. if something unexpectedly bad happens, disprefer all

states with that state’s features• Formal justification: online least squares

'

difference max ', ' ( , )a

r Q s a Q s a Exact Q’s

Approximate Q’s

Example: Q-Pacman

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Q(s,a)= 4.0fDOT(s,a)-1.0fGST(s,a)

fDOT(s,NORTH)=0.5

fGST(s,NORTH)=1.0

Q(s,a)=+1

R(s,a,s’)=-500

difference=-501

wDOT←4.0+α[-501]0.5

wGST ←-1.0+α[-501]1.0

Q(s,a)= 3.0fDOT(s,a)-3.0fGST(s,a)

α= Northr = -500

s

s’

Today: Reasoning over Time

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• Often, we want to reason about a sequence of observations– Speech recognition– Robot localization– User attention– Medical monitoring

• Need to introduce time into our models• Basic approach: hidden Markov models (HMMs)• More general: dynamic Bayes’ nets

Markov Models

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• A Markov model is a chain-structured BN– Conditional probabilities are the same (stationarity)– Value of X at a given time is called the state– As a BN:

– Parameters: called transition probabilities or dynamics, specify how the state evolves over time (also, initial probs)

p(X1) p(X|X-1)

Example: Markov Chain

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• Weather:– States: X = {rain, sun}– Transitions:

– Initial distribution: 1.0 sun– What’s the probability distribution after one step?

p(X2=sun)=p(X2=sun|X1=sun)p(X1=sun)+

p(X2=sun|X1=rain)p(X1=rain)

=0.9*1.0+1.0*0.0 =0.9

Forward Algorithm

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• Question: What’s p(X) on some day t?

1

1

1

1|

knownt

t tx

t tp x p xp x

p x

x

Forward simulation

Example

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• From initial observation of sun

• From initial observation of rain

p(X1) p(X2) p(X3) p(X∞)

p(X1) p(X2) p(X3) p(X∞)

Stationary Distributions

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• If we simulate the chain long enough:– What happens?– Uncertainty accumulates– Eventually, we have no idea what the state is!

• Stationary distributions:– For most chains, the distribution we end up in is

independent of the initial distribution– Called the stationary distribution of the chain– Usually, can only predict a short time out

• p(X=sun)=p(X=sun|X-1=sun)p(X=sun)+

p(X=sun|X-1=rain)p(X=rain)

• p(X=rain)=p(X=rain|X-1=sun)p(X=sun)+

p(X=rain|X-1=rain)p(X=rain)12

Computing the stationary distribution

Web Link Analysis

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• PageRank over a web graph– Each web page is a state– Initial distribution: uniform over pages– Transitions:

• With prob. c, uniform jump to a random page (dotted lines, not all shown)

• With prob. 1-c, follow a random outlink (solid lines)

• Stationary distribution– Will spend more time on highly reachable pages– E.g. many ways to get to the Acrobat Reader download

page – Somewhat robust to link spam

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

X1 X2 X3 X4 …• Second-order Markov process

• Workaround: change meaning of “state” to events of last 2 days

X1, X2 …X2, X3 X3, X4 X4, X5

• 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

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• Markov chains not so useful for most agents– Eventually you don’t know anything anymore– Need observations to update your beliefs

• Hidden Markov models (HMMs)– Underlying Markov chain over state X– You observe outputs (effects) at each time step– As a Bayes’ net:

Example

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• An HMM is defined by:– Initial distribution: p(X1)

– Transitions: p(X|X-1)

– Emissions: p(E|X)

Rt-1 p(Rt)

t 0.7f 0.3

Rt p(Ut)

t 0.9f 0.2

Conditional Independence

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• HMMs have two important independence properties:– Markov hidden process, future depends on past via the

present– Current observation independent of all else given current

state

• Quiz: does this mean that observations are independent given no evidence?– [No, correlated by the hidden state]

Real HMM Examples

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• Speech recognition HMMs:– Observations are acoustic signals (continuous values)– States are specific positions in specific words (so, tens of

thousands)

• Robot tracking:– Observations are range readings (continuous)– States are positions on a map (continuous)

Filtering / Monitoring

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• Filtering, or monitoring, is the task of tracking the distribution B(X) (the belief state) over time

• We start with B(X) in an initial setting, usually uniform

• As time passes, or we get observations, we update B(X)

• The Kalman filter was invented in the 60’s and first implemented as a method of trajectory estimation for the Apollo program

Example: Robot Localization

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Sensor model: never more than 1 mistakeMotion model: may not execute action with small prob.

Example: Robot Localization

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Example: Robot Localization

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Example: Robot Localization

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Example: Robot Localization

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Example: Robot Localization

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Another weather example• Xt is one of {s, c, r} (sun, cloudy, rain)

• Transition probabilities:

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.3

not a Bayes net!

• Throughout, assume uniform distribution over X1

Weather example extended to HMM

• Transition probabilities:

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.3

• Observation: labmate wet or dry• p(w|s) = .1, p(w|c) = .3, p(w|r) = .8

HMM weather example: a question

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.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 it is now raining outside?

• p(X3 = r | E1 = d, E2 = w, E3 = w)• By Bayes’ rule, really want to know p(X3, E1 = d, E2 = w, E3 = w)

p(w|s) = .1

p(w|c)

= .3 p(w|

r) = .8

Solving the question

• Computationally efficient approach: first compute

p(X1 = i, E1 = d) for all states i

• General case: solve for p(Xt, E1 = e1, …, Et = et) for t=1, then t=2, … This is called monitoring

• p(Xt, E1 = e1, …, Et = et) = ΣXt-1 p(Xt-1 = xt-1, E1 = e1, …,

Et-1 = et-1) P(Xt | Xt-1 = xt-1) P(Et = et | Xt)

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.3

p(w|s) = .1

p(w|c)

= .3 p(w|

r) = .8

Predicting further out

• 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(X5 = r | E1 = d, E2 = w, E3 = w)

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.3

p(w|s) = .1

p(w|c)

= .3 p(w|

r) = .8

Predicting further out, continued…

• Want to know: p(X5 = r | E1 = d, E2 = w, E3 = w)

• Already know how to get: p(X3 | E1 = d, E2 = w, E3 = w)

• p(X4 = r | E1 = d, E2 = w, E3 = w) =

ΣX3 P(X4 = r, X3 = x3 | E1 = d, E2 = w, E3 = w)

=ΣX3 P(X4 = r | X3 = x3)P(X3 = x3 | E1 = d, E2 = w, E3 = w)

• Etc. for X5

• So: monitoring first, then straightforward Markov process updates

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.3

p(w|s) = .1

p(w|c)

= .3 p(w|

r) = .8

Integrating newer information

s

c r

.1

.2

.6

.3.4

.3

.3

.5

.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(X2 = r | E1 = d, E2 = w, E3 = w, E4 = d)– Smoothing or hindsight problem

p(w|s) = .1

p(w|c)

= .3 p(w|

r) = .8