Machine Learning 4Hidden Markov Models

The Problem to Be Solved

More Specifically

Given a sequence of acoustic observations

Most probable sequence of wordsCorresponding to speaker’s intent

Two Items

SequenceThe signal is observable, the output is not.

Framed a Different Way: The Ice Cream Task

A climatologist in 2799 wants to reconstruct the weather in Baltimore during 2012

Baltimore is now under water Jacob Eisner, who lived in Baltimore in the early 21st century kept a diary.

His diary, through much historical drama, became the property of the Missouri Historical Society, a short walk from Washington University where the climatologist works.

This diary, besides containing lots of dreary stuff about emotional states, contains a record of how many ice cream cones Jason ate each day that summer.

What was the sequence of hot and cold days during the eventful summer of 2012?

Note two items:

Sequence: ice cream comesObservation: sequence of ice cream conesHidden: sequence of hot and cold days

We presume:There is a probabilistic relationship between the sequence of ice cones and the sequence of hot and cold days

Dr. Eisner 2012 (not eating ice cream)

Dr. Markov circa 1900 (not eating ice cream either)

Model of Newspaper Vending Machine as FSA

Markov Chains

• Each aij is an index into a table• Gives transition probabilities

Weather Model from Luger (p. 375)

S1 = sunny, s2 = cloudy, s3 = foggy, s4 = rainy

Invented Gender/Handedness data

Male (M) Female (F) TotalLeft (L) 5 8 13Right (R) 3 4 7Total 8 12 20

As a Hidden Markov Model

P(LLL)

+

)

(!) There must be a better way

P(LLL) = (.625 * 625 * .625 * .4 * .4 * .4) + (.625 * .625 * .667 * .4 * .4 * .6) + (.625 * .667 * .625 * .4 * .6 * .4)

(.625 * .667 * .667 * .4 * .6 * .6) + (.667 * .625 * .625 * .6 * .4 * .4) + (.667 * .625 * .625 * .6 * .4 * .6) + (.667 * .667 * .625 * .6 * .6 * .4) + (.667 * .667 * .667 * .6 * .6 * .6) = .015625 + .0250125 + .0250125 + .02669334 + .0250125 + .03751875 + .04004001 + .064096048

= .259010648

The Ice Cream HMM

A: priorsmatrix

Start Hot Cold

Start 0.0 .8 .2

Hot 0.0 .7 .3

Cold 0.0 .4 .6

B: likelihoods matrix

1 Cone 2 Cones 3 Cones

Hot .2 .4 .4Cold .5 .4 .1

Ice Cream Task

Rows labeled by prior state/conditioning event

A: priors matrix

start female male

start 0.0 .6 .4

female 0.0 .6 .4

male 0.0 .6 .4

B: likelihoods matrix

left right

Female .67 .33

Male .625 .375

Gender Task

Rows labeled by prior state/conditioning event

Forward Algorithm

Forward Trellis

bj(ot)

.0464