Bayesian Estimation: Chapters 10-12 by Natasha Devroye, Study notes of Electrical and Electronics Engineering

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Estimation: parts of Chapters 10-

Bayesian Estimation

Natasha Devroye

[email protected]

http://www.ece.uic.edu/~devroye

Spring 2010

Summary

Bayesian philosophy

Bayesian risk

Bayesian risk

Hit-or-miss cost (MAP)

MMSE estimators

MMSE estimators

Example of MMSE

Bayesian linear model

Bayesian linear model

Examples: MMSE in Gaussian noise

Examples: MMSE in Gaussian noise

Maximum a Posteriori (MAP) estimation

Example

MAP properties

Linear MMSE (LMMSE)

Geometric LMMSE

LMMSE example

LMMSE properties

LMMSE properties

Sequential LMMSE: geometric approach

3

“Hit-or-Miss”

Cost Function

1! 1 !

ˆ |

Err.Cov.:

Estimate:

“Squared” Cost Function

(Nonlinear Estimate)

7,+-$ Linear Estimate

:',;' : E{$},E{ 1 }, C

Jointly Gaussian 1 and $

(Yields Linear Estimate)

!!! 1 11 1! !

! 1 11

1

ˆ

1

Err.Cov. :

Estimate:

'

'

!!! 1 11 1! !

! 1 11

1

ˆ

1

Err.Cov. :

'

'

Estimate #! ( '!

Bayesian Linear Model

(Yields Linear Estimate)

!!!! !

!!!

1

ˆ

1

Err.Cov. :

Estimate:

'

'

"

# #

"

# #

!

See http://www.ws.binghamton.edu/fowler/fowler%20personal%20page/EE522_files/EECE%20522%20Notes_28%20Ch_12B.pdf