Learning Bit by Bit

Hidden Markov Models

Weighted FSA

weatherweatherTheThe isis

outsideoutside

1.0

.7

.3

Markov Chain

• Computing probability of an observed sequence of events

Markov Chain

weatherweather

TheThe

isis

outsideoutside

.7

.3

Observation = “The weather outside”

windwind

.5

.5

.1

.9

Parts of Speech

• Grammatical constructs like noun, verb

POS examples• N noun chair, bandwidth, pacing• V verb study, debate, munch• ADJ adjective purple, tall, ridiculous• ADV adverb unfortunately, slowly• P preposition of, by, to• PRO pronoun I, me, mine• DET determiner the, a, that, those

Parts of Speech-uses

• Speech recognition• Speech synthesis• Data mining• Translation

POS Tagging

• Words often have more than one POS: back– The back door = JJ– On my back = NN– Win the voters back = RB– Promised to back the bill = VB

• The POS tagging problem is to determine the POS tag for a particular instance of a word.

POS Tagging

• Sentence = sequence of observations• Ie. “Secretariat is expected to race tomorrow”

Disambiguating “race”

Hidden Markov Model

• Observed• Hidden

Hidden Markov Model

• 2 kinds of probabilities:– Tag transitions – Word likelihoods

Hidden Markov Model

• Tag transition prob = P( tag | previous tag)– ie. P(VB | TO)

Hidden Markov Model

• Word likelihood probability = P(word | tag)– ie. P(“race” | VB)

• Actual probabilities:– P (NN | TO) = .00047– P (VB | TO) = .83

• Actual probabilities:– P (NR| VB) = .0027– P (NR| NN) = .0012

• Actual probabilities:– P (race | NN) = .00057– P (race | VB) = .00012

Hidden Markov Model

• Probability “to race tomorrow” =“TO VB NR”• P(VB|TO) * P(NR|VB) * P(race|VB)• .83 * .0027 * .00012 = 0.00000026892

Hidden Markov Model

• Probability “to race tomorrow” =“TO NN NR”• P(NN|TO) * P(NR|NN) * P(race|NN)• .00047* .0012* .00057 = 0.00000000032148

Hidden Markov Model

• Probability “to race tomorrow” =“TO NN NR” = 0.00000000032148• Probability “to race tomorrow” =“TO VB NR”

= 0.00000026892

Bayesian Inference

• Correct answer = max (P (hypothesis | observed))

Bayesian Inference

• Prior probability = likelihood of the hypothesis

Bayesian Inference

• Likelihood = probability that the evidence matches the hypothesis

Bayesian Inference

• Bayesian vs. Frequentists• Subjectivity

Examples