Introduction to Bayesian Inference

Transcription

1 University of Pennsylvania EABCN Training School May 10, 2016

2 Bayesian Inference Ingredients of Bayesian Analysis: Likelihood function p(y φ) Prior density p(φ) Marginal data density p(y ) = p(y φ)p(φ)dφ Bayes Theorem: p(φ Y ) = p(y φ)p(φ) p(y )

3 Linear Regression / AR Models Consider AR(1) model: y t = y t 1 φ + u t, u t iidn(0, 1). Let x t = y t 1. Write as or y t = x tφ + u t, u t iidn(0, 1), Y = X φ + U. We can easily allow for multiple regressors. Assume φ is k 1. Notice: we treat the variance of the errors as know. The generalization to unknown variance is straightforward but tedious. Likelihood function: p(y φ) = (2π) T /2 exp { 1 } 2 (Y X φ) (Y X φ).

4 A Convenient Prior Prior: φ N ) { (0 k 1, τ 2 I k k, p(φ) = (2πτ 2 ) k/2 exp 1 } 2τ 2 φ φ Large τ means diffuse prior. Small τ means tight prior.

5 Deriving the Posterior Bayes Theorem: p(φ Y ) p(y φ)p(φ) { exp 1 } 2 [(Y X φ) (Y X φ) + τ 2 φ φ]. Guess: what if φ Y N( φ T, V T ). Then { p(θ Y ) exp 1 } 2 (φ φ T ) 1 V T (φ φ T ). Rewrite exponential term Y Y φ X Y Y X φ + φ X X φ + τ 2 φ φ = Y Y φ X Y Y X φ + φ (X X + τ 2 I)φ ( ) ( ) = φ (X X + τ 2 I) 1 X Y X X + τ 2 I ( ) φ (X X + τ 2 I) 1 X Y +Y Y Y X (X X + τ 2 I) 1 X Y.

6 Deriving the Posterior Exponential term is a quadratic function of φ. Deduce: posterior distribution of φ must be a multivariate normal distribution φ Y N( φ T, V T ) with φ T = (X X + τ 2 I) 1 X Y τ : τ 0: V T = (X X + τ 2 I) 1. φ Y approx ( ) N ˆφ mle, (X X ) 1. φ Y approx Pointmass at 0

7 Marginal Data Density Plays an important role in Bayesian model selection and averaging. Write p(y θ)p(θ) p(y ) = p(θ Y ) { = exp 1 } 2 [Y Y Y X (X X + τ 2 I) 1 X Y ] (2π) T /2 I + τ 2 X X 1/2. The exponential term measures the goodness-of-fit. I + τ 2 X X is a penalty for model complexity.

8 Posterior We will often abbreviate posterior distributions p(φ Y ) by π(φ) and posterior expectations of h(φ) by E π [h] = E π [h(φ)] = h(φ)π(φ)dφ = h(φ)p(φ Y )dφ. We will focus on algorithms that generate draws {φ i } N i=1 from posterior distributions of parameters in time series models. These draws can then be transformed into objects of interest, h(φ i ), and under suitable conditions a Monte Carlo average of the form h N = 1 N N h(φ i ) E π [h]. i=1 Strong law of large numbers (SLLN), central limit theorem (CLT)...

9 Direct Sampling In the simple linear regression model with Gaussian posterior it is possible to sample directly. For i = 1 to N, draw φ i from N ( φ, Vφ ). Provided that V π [h(φ)] < we can deduce from Kolmogorov s SLLN and the Lindeberg-Levy CLT that a.s. h N E π [h] N ( h N E π [h] ) = N ( 0, V π [h(φ)] ).

10 Decision Making The posterior expected loss associated with a decision δ( ) is given by ρ ( δ( ) Y ) = L ( θ, δ(y ) ) p(θ Y )dθ. Θ A Bayes decision is a decision that minimizes the posterior expected loss: δ (Y ) = argmin d ρ ( δ( ) Y ). Since in most applications it is not feasible to derive the posterior expected risk analytically, we replace ρ ( δ( ) Y ) by a Monte Carlo approximation of the form ρ N ( δ( ) Y ) = 1 N N L ( θ i, δ( ) ). i=1 A numerical approximation to the Bayes decision δ ( ) is then given by δ N(Y ) = argmin d ρ N ( δ( ) Y ).

11 Inference Point estimation: Quadratic loss: posterior mean Absolute error loss: posterior median Interval/Set estimation P π {θ C(Y )} = 1 α: highest posterior density sets equal-tail-probability intervals

12 Forecasting Example: h 1 y T +h = θ h y T + θ s u T +h s s=0 h-step ahead conditional distribution: y T +h (Y 1:T, θ) N (θ h y T, 1 ) θh. 1 θ Posterior predictive distribution: p(y T +h Y 1:T ) = p(y T +h y T, θ)p(θ Y 1:T )dθ. For each draw θ i from the posterior distribution p(θ Y 1:T ) sample a sequence of innovations u i T +1,..., ui T +h and compute y i T +h as a function of θ i, u i T +1,..., ui T +h, and Y 1:T.

13 Model Uncertainty Assign prior probabilities γ j,0 to models M j, j = 1,..., J. Posterior model probabilities are given by γ j,t = γ j,0 p(y M j ) J j=1 γ j,0p(y M j ), where p(y M j ) = p(y θ (j), M j )p(θ (j) M j )dθ (j) Log marginal data densities are one-step-ahead predictive scores: ln p(y M j ) T = ln p(y t θ (j), Y 1:t 1, M j )p(θ (j) Y 1:t 1, M j )dθ (j). t=1 Model averaging: J p(h Y ) = γ j,t p(h j (θ (j) ) Y, M j ). j=1