Bayesian Machine Learning
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1 Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 2: Bayesian Basics Cornell University August 25, / 17
2 Canonical Machine Learning Problems Linear Classification Polynomial Regression Clustering with Gaussian Mixtures 2 / 17
3 Linear Classification Data: D = {(x i, y i)} n i=1. x i R D y i {+1, 1}. Model: p(y i = +1 θ, x) = { 1 if θ T x i 0, 0 otherwise (1) Parameters: θ R D Goal: To infer θ from data and to predict future labels p(y D, x). 3 / 17
4 Polynomial Regression Data: D = {(x i, y i)} n i=1. x i R y i R. Model: y i = a 0 + a 1x i + a 2x 2 i + + a mx m i + ɛ i, where ɛ i N (0, σ 2 ). Parameters: θ = (a 0,..., a m, σ 2 ) T. Goal: To infer θ from the data and to predict future outputs p(y D, x, m). 4 / 17
5 Clustering with Gaussian Mixtures (Density Estimation) Data: D = {x i} n i=1 x i R D Model: x i m w jp j(x i) (2) j=1 Parameters: θ = ((µ 1, Σ 1),..., (µ m, Σ m), w) where p j(x i) = N (x i; µ i, Σ i). (3) Goal: To infer θ from the data, predict the density p(x D, m), and infer which points belong to which cluster. 5 / 17
6 Bayesian Modelling (Theory of Everything) Ghahramani (2015) 6 / 17
7 Statistics from Scratch Basic Regression Problem Training set of N targets (observations) y = (y(x 1),..., y(x N)) T. Observations evaluated at inputs X = (x 1,..., x N) T. Want to predict the value of y(x ) at a test input x. For example: Given CO 2 concentrations y measured at times X, what will the CO 2 concentration be for x = 2024, 10 years from now? Just knowing high school math, what might you try? 7 / 17
8 Statistics from Scratch 700 Airline Passengers (Thousands) Year 8 / 17
9 Statistics from Scratch Guess the parametric form of a function that could fit the data f (x, w) = w T x [Linear function of w and x] f (x, w) = w T φ(x) f (x, w) = g(w T φ(x)) [Linear function of w] (Linear Basis Function Model) [Non-linear in x and w] (E.g., Neural Network) φ(x) is a vector of basis functions. For example, if φ(x) = (1, x, x 2 ) and x R 1 then f (x, w) = w 0 + w 1x + w 2x 2 is a quadratic function. Choose an error measure E(w), minimize with respect to w E(w) = N i=1 [f (xi, w) y(xi)]2 9 / 17
10 Statistics from Scratch A probabilistic approach We could explicitly account for noise in our model. y(x) = f (x, w) + ɛ(x), where ɛ(x) is a noise function. One commonly takes ɛ(x) = N (0, σ 2 ) for i.i.d. additive Gaussian noise, in which case p(y(x) x, w, σ 2 ) = N (y(x); f (x, w), σ 2 ) Observation Model (4) N p(y x, w, σ 2 ) = N (y(x i); f (x i, w), σ 2 ) Likelihood (5) i=1 Maximize the likelihood of the data p(y x, w, σ 2 ) with respect to σ 2, w. For a Gaussian noise model, this approach will make the same predictions as using a squared loss error function: log p(y X, w, σ 2 ) 1 2σ 2 N [f (x i, w) y(x i)] 2 (6) i=1 10 / 17
11 Statistics from Scratch The probabilistic approach helps us interpret the error measure in a deterministic approach, and gives us a sense of the noise level σ 2. Probabilistic methods thus provide an intuitive framework for representing uncertainty, and model development. Both approaches are prone to over-fitting for flexible f (x, w): low error on the training data, high error on the test set. Regularization Use a penalized log likelihood (or error function), such as model fit {}}{ complexity penalty log p(y X, w) 1 n {}}{ (f (x i, w) y(x i) 2 ) λw T w. (7) 2σ 2 i=1 But how should we define complexity, and how much should we penalize complexity? Can set λ using cross-validation. 11 / 17
12 Bayesian Inference Bayes Rule p(a b) = p(b a)p(a)/p(b), p(a b) p(b a)p(a). (8) posterior = likelihood prior marginal likelihood, p(w y, X, σ2 ) = p(y X, w, σ2 )p(w) p(y X, σ 2 ). (9) Predictive Distribution p(y x, y, X) = p(y x, w)p(w y, X)dw. (10) Think of each setting of w as a different model. Eq. (15) is a Bayesian model average, an average of infinitely many models weighted by their posterior probabilities. No over-fitting, automatically calibrated complexity. Eq. (10) is intractable for many likelihoods p(y X, w, σ 2 ) and priors p(w). Typically more interested in the induced distribution over functions than in parameters w. Can be hard to have intuitions for priors on p(w). 12 / 17
13 Parametric Regression Review Deterministic E(w) = N (f (x i, w) y i) 2. (11) i=1 Maximum Likelihood Bayesian p(y(x) x, w) = N (y(x); f (x, w), σn) 2, (12) N p(y X, w) = N (y(x i); f (x i, w), σn) 2. (13) i=1 posterior = likelihood prior marginal likelihood, p(y X, w)p(w) p(w y, X) =. (14) p(y X) p(y x, y, X) = p(y x, w)p(w y, X)dw. (15) 13 / 17
14 Worked Example: Basis Regression (Chalkboard) We have data D = {(x i, y i)} N i=1 We use the model: y = w T φ(x) + ɛ (16) ɛ N (0, σ 2 ). (17) We want to make predictions of y for any x. We will now explore this question on the whiteboard using maximum likelihood and Bayesian approaches. We will consider topics such as regularization, cross-validation, Bayesian model averaging and conjugate priors. 14 / 17
15 Rant: Regularisation = MAP Bayesian Inference Example: Density Estimation Observations y 1,..., y N drawn from unknown density p(y). Model p(y θ) = w 1N (y µ 1, σ 2 1) + w 2N (y µ 2, σ 2 2), θ = {w 1, w 2, µ 1, µ 2, σ 1, σ 2}. Likelihood p(y θ) = N i=1 p(yi θ). Can learn all free parameters θ using maximum likelihood / 17
16 Regularisation = MAP Bayesian Inference Regularisation or MAP Find argmax θ log p(θ y) c = model fit {}}{ complexity penalty {}}{ log p(y θ) + log p(θ) Choose p(θ) such that p(θ) 0 faster than p(y θ) as σ 1 or σ 2 0. Bayesian Inference Predictive Distribution: p(y y) = p(y θ)p(θ y)dθ. Parameter Posterior: p(θ y) p(y θ)p(θ). p(θ) need not be zero anywhere in order to make reasonable inferences. Can use a sampling scheme, with conjugate posterior updates for each separate mixture component, using an inverse Gamma prior on the variances σ 2 1, σ / 17
17 Learning with Stochastic Gradient Descent Chalkboard. 17 / 17
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