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Bayesian Estimation

Module by: Clayton Scott. E-mail the author

We are interested in estimating θθ given the observation xx. Naturally then, any estimation strategy will be based on the posterior distribution pθ| x p x θ . Furthermore, we need a criterion for assessing the quality of potential estimators.

Loss

The quality of an estimate θ ^ θ is measured by a real-valued loss function: Lθ θ ^ L θ θ . For example, squared error or quadratic loss is simply Lθ θ ^=(θ θ ^)T(θ θ ^) L θ θ θ θ θ θ

Expected Loss

Posterior Expected Loss: ELθ θ ^| x =Lθ θ ^pθ| x d θ x L θ θ θ L θ θ p x θ Bayes Risk:

ELθ θ ^=Lθ θ ^pθ| x pxd θ d x =Lθ θ ^px| θ pθd x d θ =EELθ θ ^| x L θ θ x θ L θ θ p x θ p x θ x L θ θ p θ x p θ x L θ θ
(1)
The "best" or optimal estimator given the data xx and under a specified loss is given by θ ^=argminθELθ θ ^| x θ θ x L θ θ

Example 1: Bayes MSE

BMSE θ ^θ θ ^2pθ| x d θ pxd x BMSE θ x θ θ θ 2 p x θ p x Since px0 p x 0 for every x x, minimizing the inner integral θEθ2pθ| x d θ =ELθ θ ^| x θ θ θ 2 p x θ x L θ θ (where ELθ θ ^| x x L θ θ is the posterior expected loss) for each x x, minimizes the overall BMSE.

θ θ ^2pθ| x d θ θ ^ =(θ θ ^2pθ| x ) θ ^ d θ =-2(θ θ ^)pθ| x d θ θ θ θ θ 2 p x θ θ θ θ θ 2 p x θ -2 θ θ θ p x θ
(2)
Equating this to zero produces θ ^=θpθ| x d θ Eθ| x θ θ θ p x θ x θ The conditional mean (also called posterior mean) of θθ given xx!

Example 2

x n =A+ W n   ,   n1N    n n 1 N x n A W n W n 𝒩0σ2 W n 0 σ 2 prior for unknown parameter AA: pa=U A 0 A 0 p a U A 0 A 0 px| A =12πσ2N2e-12σ2 n =1N x n A2 p A x 1 2 σ 2 N 2 -1 2 σ 2 n 1 N x n A 2 pA| x ={12 A 0 2πσ2N2e-12σ2 n =1N x n A2 A 0 A 0 12 A 0 2πσ2N2e-12σ2 n =1N x n A2d A   if  |A| A 0 0  if  |A|> A 0 p x A 1 2 A 0 2 σ 2 N 2 -1 2 σ 2 n 1 N x n A 2 A A 0 A 0 1 2 A 0 2 σ 2 N 2 -1 2 σ 2 n 1 N x n A 2 A A 0 0 A A 0 Minimum Bayes MSE Estimator:

A ^=EA| x =apA| x d A = A 0 A 0 A12 A 0 2πσ2N2e-12σ2 n =1N x n A2d A A 0 A 0 12 A 0 2πσ2N2e-12σ2 n =1N x n A2d A A x A A a p x A A A 0 A 0 A 1 2 A 0 2 σ 2 N 2 -1 2 σ 2 n 1 N x n A 2 A A 0 A 0 1 2 A 0 2 σ 2 N 2 -1 2 σ 2 n 1 N x n A 2
(3)

Notes

  1. No closed-form estimator
  2. As A 0 A 0 , A ^1N n =1N x n A 1 N n 1 N x n
  3. For smaller A 0 A 0 , truncated integral produces an A ^ A that is a function of xx, σ2 σ 2 , and A 0 A 0
  4. As NN increases, σ2N σ 2 N decreases and posterior pA| x p x A becomes tightly clustered about 1N x n 1 N x n . This implies A ^1N n x n A 1 N n x n as n n (the data "swamps out" the prior)

Other Common Loss Functions

Absolute Error Loss

(Laplace, 1773) Lθ θ ^=|θ θ ^| L θ θ θ θ

ELθ θ ^| x =|θ θ ^|pθ| x d θ = θ ^( θ ^θ)pθ| x d θ + θ ^(θ θ ^)pθ| x d θ x L θ θ θ θ θ p x θ θ θ θ θ p x θ θ θ θ θ p x θ
(4)
Using integration-by-parts it can be shown that θ ^( θ ^θ)pθ| x d θ = θ ^Pθ<y| x d y θ θ θ θ p x θ y θ P x θ y θ ^(θ θ ^)pθ| x d θ = θ ^Pθ>y| x d y θ θ θ θ p x θ y θ P x θ y where Pθ<y| x P x θ y and Pθ>y| x P x θ y are a cumulative distributions. So, ELθ θ ^| x = θ ^Pθ<y| x d y + θ ^Pθ>y| x d y x L θ θ y θ P x θ y y θ P x θ y Take the derivative with respect to θ ^ θ implies Pθ< θ ^| x =Pθ> θ ^| x P x θ θ P x θ θ which implies that the optimal θ ^ θ under absolute error loss is posterior median.

'0-1' Loss

Lθ θ ^={0  if   θ ^=θ1  if   θ ^θ= I { θ ̂ θ } L θ θ 0 θ θ 1 θ θ I { θ ̂ θ } ELθ θ ^| x =E I { θ ̂ θ } | x =Pr θ ^θ| x x L θ θ x I { θ ̂ θ } x θ θ which is the probability that θ ^θ θ θ given xx. To minimize '0-1' loss we must choose θ ^ θ to be the value of θθ with the highest posterior probability, which implies θ ^θ θ θ with the smallest probability.

Figure 1
Figure 1 (MAP.png)
The optimal estimator θ ^ θ under '0-1' loss is the maximum a posteriori (MAP) estimator--the value of θθ where pθ| x p x θ is maximized.

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