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Probabilistic reasoning over time. This sentence is likely to be untrue in the future!. The basic problem. What do we know about the state of the world now given a history of the world before . The only evidence we have are probabilities.

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Probabilistic reasoning over time l.jpg

Probabilistic reasoning over time

This sentence is likely to be untrue in the future!

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The basic problem

  • What do we know about the state of the world now given a history of the world before.

  • The only evidence we have are probabilities.

  • “Past performance may not be a guide to future performance.”

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Simplifying assumptions and notations

  • States are our “events”.

  • (Partial) states can be measured at reasonable time intervals.

  • Xt unobservable state variables at t.

  • Et (“evidence”) observable state variables at t.

  • Vm:n : Variables Vm, Vm+1,…,Vn

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Stationary, Markovian (transition model)

  • Stationary: the laws of probability don’t change over time

  • Markovian: current unobservalbe state depends on a finite number of past states

    • First-order: current state depends only on the previous state, i.e.:

    • P(Xt|X0:t-1)=P(Xt|Xt-1)

    • Second-order: etc., etc.

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Observable variables (the sensor model)

  • Observable variables depend only on the current state (by definition, essentially), these are the “sensors”.

  • The current state causes the sensor values.

  • P(Et|X0:t,E0:t-1)=P(Et|Xt)

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Start it up (the prior probability model)

  • What is P(X0)?

  • At time t, the joint is completely determined:

  • P(X0,X1,…Xt,E1,…,Et) =P(X0) • ∏i  t P(Xi|Xi-1)P(Ei|Xi)

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Better predictions?

  • More state variables (temperature, humidity, pressure, season…)

  • Higher order Markov processes (take more of the past into account).

  • Tradeoffs?

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What’s it good for?

  • Belief/monitoring the current state

  • Prediction about the next state

  • Hindsight about previous states

  • Explanation of possible causes

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Hidden Markov Models (HMMs)

  • Further simplification:

  • Only one state variable.

  • We can use matrices, now.

  • Ti,j = P(Xt=j|Xt-1=i)

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Speech Recognition

  • P(words|signal) = P(signal|words)P(words)

  • P(words) “language model”

  • “Every time I fire a linguist, the recognition rate goes up.”

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Model 1: Speech

  • Sample the speech signal

  • Decide the most likely sequence of speech symbols

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Phonetic alphabet

  • Phonemes: minimal units of sound that make a meaning difference (beat vs. bit; fit vs. bit)

  • Phones: normalized articulation results paid vs. tap

  • English has about 40

  • Co-articulation effects modeled as new symbols. sweet = w(s,iy)

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Model 2,3: Words, Sentences

  • Given the phones, what is the most likely word/word in the sentence?

  • “Give me all your money. I have a gub.”

    • Gub is unlikely to be a word,

    • And if it were, it would be less likely than “gun.”