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Learning1

# Learning1 - Learning With Bayesian Networks Learning Are...

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10/25/10 Learning With Bayesian Networks

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10/25/10 Learning Is Trivial If All Variables Are Observed X Y Z
10/25/10 Inference as Learning Suppose you have a coin with an unknown bias, θ ≡ P(head). You flip the coin multiple times and observe the outcome. From observations, you can infer the bias of the coin This is learning. This is inference.

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10/25/10 independent events Sufficient statistics
10/25/10 Simple Case: Coin Flipping With Discrete Hypotheses § Two hypotheses h 0 : θ=.5 h 1 : θ=.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 -> narrowing hypothesis space

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10/25/10
10/25/10 Infinite Hypothesis Spaces Consider all values of θ, 0 <= θ <= 1 Inferring θ is just like any other sort of Bayesian inference Likelihood is as before: With uniform priors on θ and we obtain This is a beta distribution: Beta(NH+1, NT+1)

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10/25/10 Beta Distribution Beta(α,β)
10/25/10 Incorporating Priors Suppose we have the prior distribution p(θ) = Beta(VH, VT) Then posterior distribution is also Beta p(θ|d) = Beta(VH+NH, VT+NT) Example of conjugate priors Beta distribution is the conjugate prior for a binomial or Bernoulli likelihood

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10/25/10
10/25/10

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10/25/10 Prior Distribution For θ Direct assessment Parametric distributions Conjugate distribution Mixtures of conjugate distributions
10/25/10 ALL BLACK SLIDES STOLEN FROM DAVID

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Learning1 - Learning With Bayesian Networks Learning Are...

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