Probabilistic reasoning over time

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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.
• Vmn Variables Vm, Vm1,,Vn

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Stationary, Markovian (transition model)

• Stationary the laws of probability dont 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(XtX0t-1)P(XtXt-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(EtX0t,E0t-1)P(EtXt)

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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(XiXi-1)P(EiXi)

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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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Whats 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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Example
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Hidden Markov Models (HMMs)

• Further simplification
• Only one state variable.
• We can use matrices, now.
• Ti,j P(XtjXt-1i)

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Speech Recognition

• P(wordssignal) P(signalwords)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.

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Lots of tricky bits