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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. • “Past performance may not be a guide to future performance.”
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
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.
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)
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)
Better predictions? • More state variables (temperature, humidity, pressure, season…) • Higher order Markov processes (take more of the past into account). • Tradeoffs?
What’s it good for? • Belief/monitoring the current state • Prediction about the next state • Hindsight about previous states • Explanation of possible causes
Hidden Markov Models (HMMs) • Further simplification: • Only one state variable. • We can use matrices, now. • Ti,j = P(Xt=j|Xt-1=i)
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.”
Model 1: Speech • Sample the speech signal • Decide the most likely sequence of speech symbols
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)
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.”