Coin Flipping Example

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Coin Flipping Example

• Two hypotheses
• h0 ?.5
• h1 ?.9
• Role of priors diminishes as number of flips
  increases
• Note weirdness that each hypothesis has an
  associated probability, and each hypothesis
  specifies a probability
• probabilities of probabilities!
• Setting prior to zero -gt narrowing hypothesis
  space

lt independent events
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Probability Densities

• For discrete events, probabilities sum to 1.
• For continuous events, probabilities integrate to
  1.
• e.g., uniform density
• p(x) 1/r if 0 lt x lt r 0 otherwise
• Probability densities should not be confused with
  probabilities.
• if p(Xx) is a density, p(Xx)dx is a probability
  the probability that x lt X lt xdx
• Notation
• P(...) for probability, p(...) for density

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Probability Densities

• Can treat densities as probabilities, because
  when you do the computation, the dx's cancel
• e.g., Bayes rule

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Using the Posterior Distribution

• What do we do once we have the posterior?
• maximum a posteriori (MAP)
• posterior mean, also known as model averaging
• MAP vs. Maximum likelihood
• p(? d) vs. p(d ?)

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Infinite Hypothesis Spaces

• Consider all values of ?, 0 lt ? lt 1
• Inferring ? is just like any other sort of
  Bayesian inference
• Likelihood is as before
• With uniform priors on ? andwe obtain
• This is a beta distribution Beta(NH1, NT1)

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Beta Distribution
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Incorporating Priors

• Suppose we have the prior distribution
• p(?) Beta(VH, VT)
• Then posterior distribution is also Beta
• p(?d) Beta(VHNH, VTNT)
• Example of conjugate priors
• Beta distribution is the conjugate prior for a
  binomial or Bernoulli likelihood

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Latent Variables

• In many probabilistic models, some variables are
  latent (a.k.a. hidden, nonobservable).
• e.g., Gaussian mixture model
• Given an observation x, we might want to infer
  which component produced it.
• call this latent variable z

z
?
x
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Learning Graphical Model Parameters

• Trivial if all variables are observed.
• If model contains latent variables, need to infer
  their values before you can learn.
• Two techniques described in article
• Expectation Maximization (EM)
• Markov Chain Monte Carlo (MCMC)
• Other technique that I don't understand
• Spectral methods

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Expectation Maximization

• First, compute the expectation of z given current
  model parameters ?
• P(Zz x, ?) p(x z, ?) P(z)
• Second, compute parameter values that maximize
  the joint likelihood
• argmax? p(x, z ?)
• Iterate
• Guaranteed to converge to a local optimum of p(x
  ?)

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MCMC Gibbs Sampling

• Resample each latent variable conditional on the
  current values of all other latent variables
• Start with random assignments for the Zi
• Loop through all latent variables in random order
• Draw a sample P(Ziz x, ?) p(x z, ?) P(z)
• Reestimate ? (either after each sample or after
  an entire pass through the variables?). Same as M
  step of EM
• Gibbs sampling is a special case of
  Metropolis-Hastings algorithm
• make it more likely to move uphill in the overall
  likelihood
• can be more accurate/efficient than Gibbs
  sampling if you know something about the
  distribution you're sampling from

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Important Ideas in Bayesian Models

• Generative models
• Consideration of multiple models in parallel
• Inference
• prediction via model averaging
• importance of priors
• explaining away
• Learning
• distinction from inference not always clear cut
• parameter vs. structure learning
• Bayesian Occam's razor trade off between model
  simplicity and fit to data (next class)

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Important Technical Issues

• Representing structured data
• grammars
• relational schemas (e.g., paper authors, topics)
• Hierarchical models
• different levels of abstraction
• Nonparametric models
• flexible models that grow in complexity as the
  data justifies
• Approximate inference
• Markov chain Monte Carlo
• particle filters
• variational approximations