Machine Learning 4771

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1 Machine Learning 4771 Instructor: Tony Jebara

2 Topic 11 Maximum Likelihood as Bayesian Inference Maximum A Posteriori Bayesian Gaussian Estimation

3 Why Maximum Likelihood? So far, assumed max (log) likelihood (IID or otherwise) Philosophical: Why? Also, why ignore p()? Hint: Recall Bayes rule: posterior max L = max p( x 1,,x ) = max p( x i ) likelihood p x Everyone agrees on probability theory: inference and use of probability models when we have computed p(x) But how get to p(x) from data? Debate Two schools of thought: Bayesians and Frequentists p = p x p x prior evidence

4 Bayesians & Frequentists Frequentists (eymann/pearson/wald). An orthodox view that sampling is infinite and decision rules can be sharp. Bayesians (Bayes/Laplace/de Finetti). Unknown quantities are treated probabilistically and the state of the world can always be updated. de Finetti: p( event ) = price I would pay for a contract that pays 1$ when event happens Likelihoodists (Fisher). Single sample inference based on maximizing the likelihood function and relying on the Birnbaum s Theorem. Bayesians But they don t know it.

5 Bayesians & Frequentists Frequentists: Data are a repeatable random sample- there is a frequency Underlying parameters remain constant during this repeatable process Parameters are fixed Bayesians: Data are observed from the realized sample. Parameters are unknown and described probabilistically Data are fixed

6 Bayesians & Frequentists Frequentists: classical / objective view / no priors every statistician should compute same p(x) so no priors can t have a p(event) if it never happened avoid p(), there is 1 true model, not distribution of them permitted: p (x,y) forbidden: p(x,y ) Frequentist inference: estimate one best model use the ML estimator (unbiased & minimum variance) do not depend on Bayes rule for learning Bayesians: subjective view / priors are ok put a distribution or pdf on all variables in the problem even models & deterministic quantities (i.e. speed of light) use a prior p(), on the model before seeing any data Bayesian inference: use Bayes rule for learning, integrate over all model () unknown variables

7 Bayesian Inference Bayes rule gives rise to maximum likelihood Assume we have a prior over models p() posterior likelihood p x p = p x p x prior evidence How to pick p()? Pick simpler is better p Pick form for mathematical convenience We have data (can assume IID): Want to get a model to compute: X = x 1,x,,x p( x) Want p(x) given our data How to proceed? { }

8 Bayesian Inference Want p(x) given our data p x X d = p( x, X ) = p( x,x ) p( X )d p p X = p( x,x ) p( X ) p( x X ) = p( x x 1,x,,x ) n p( x = p( x ) i ) p( ) d p( X ) p ( X ) p x d Prior = 1 = = 3 Many models Weight on each model

9 Bayesian Inference to MAP & ML The full Bayesian Inference integral can be mathematically tricky. Maximum likelihood is an approximation of it p( x i ) p( ) = p( x ) p x X where * = p X p( x ) δ( * )d arg max arg max p( X ) p( X ) p x i p Maximum A Posteriori (MAP) is like Maximum Likelihood (ML) with a prior p() which lets us prefer some models over others l MAP ( ) = l ML ( ) + log p( ) = log p( x i ) + log p( ) p x i uniform d MAP ML p( X ) *

10 Bayesian Inference Example For Gaussians, we CA compute the integral (but hard!) p( x p( x X ) = p( x ) i ) p( ) d p X p x p x i p( )d Example: assume 1d Gaussian & Gaussian prior on mean p x = Gaussian p( ) = Gaussian 1 p x X µ π e 1 x µ 1 ( x π e 1 i µ ) 1 ( µ µ π e 1 0 ) dµ

11 Bayesian Inference Example Solve integral over all Gaussian means with variance=1 µ= 1 p x X ( x µ ) π e 1 1 ( x π e 1 i µ ) 1 ( µ π e 1 0 µ ) dµ µ= µ= exp 1 x µ 1 x µ ( i ) 1 µ µ i ( 0 ) µ= dµ µ= exp 1 + µ µ ( x + µ 0 + i x i ) + x ( ) dµ µ= µ= exp 1 + µ µ ( x + µ 0 + i x i ) + x + µ= dµ exp 1 ( x +µ 0 + x i i ) + x + µ = µ 0 + x i i +1 = ( x µ, σ ) σ = + +1 Can integrate over µ and Σ for multivariate Gaussian (Jordan ch. 4 and Minka Tutorial) = Γ +1 p x X (/) Γ ( +1 d )/ 1 ( +1)π Σ 1 1/ T Σ 1 ( x µ ) x µ +1 ( +1)/

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