<< Chapter < Page Chapter >> Page >

n n 1 N x n A W n W n 0 2 iid. A 1 N n 1 N x n MVUB and MLE estimator. Now suppose that we have prior knowledge that A 0 A A 0 . We might incorporate this by forming a new estimator

A A 0 A A 0 A A 0 A A 0 A 0 A A 0
This is called a truncated sample mean estimator of A . Is A a better estimator of A than the sample mean A ?

Let p a denote the density of A . Since A 1 N x n , p a A 2 N . The density of A is given by

p a A A 0 a A 0 p a I { - A 0 A 0 } A A 0 a A 0
Now consider the MSE of the sample mean A .
MSE A a a A 2 p a


  • A is biased ( ).
  • Although A is MVUB, A is better in the MSE sense.
  • Prior information is aptly described by regarding A as a random variable with a prior distribution U A 0 A 0 , which implies that we know A 0 A A 0 , but otherwise A is abitrary.
Mean of A A .
Mean of A A .

The bayesian approach to statistical modeling

Where w is the noise and x is the observation.

n n 1 N x n A W n

Prior distribution allows us to incorporate prior information regarding unknown paremter--probable values of parameter aresupported by prior. Basically, the prior reflects what we believe "Nature" will probably throw at us.

Elements of bayesian analysis

  • (a)

    joint distribution p x , p x p
  • (b)

    marginal distributions p x p x p p x p x p where p is a prior .
  • (c)

    posterior distribution p x p x , p x p x p x p x p

0 1 p x n x x 1 n x which is the Binomial likelihood. p 1 B 1 1 1 which is the Beta prior distriubtion and B

This reflects prior knowledge that most probable values of are close to .

Joint density

p x , n x B x 1 1 n x 1

Marginal density

p x n x x n x n

Posterior density

p x x 1 n x 1 B x n x where B x n x is the Beta density with parameters x and n x

Selecting an informative prior

Clearly, the most important objective is to choose the prior p that best reflects the prior knowledge available to us. In general, however, our prior knowledge is imprecise andany number of prior densities may aptly capture this information. Moreover, usually the optimal estimator can't beobtained in closed-form.

Therefore, sometimes it is desirable to choose a prior density that models prior knowledge and is nicely matched in functional form to p x so that the optimal esitmator (and posterior density) can be expressed in a simple fashion.

Choosing a prior

    1. informative priors

  • design/choose priors that are compatible with prior knowledge of unknown parameters

    2. non-informative priors

  • attempt to remove subjectiveness from Bayesian procedures
  • designs are often based on invariance arguments

Suppose we want to estimate the variance of a process, incorporating a prior that is amplitude-scaleinvariant (so that we are invariant to arbitrary amplitude rescaling of data). p s 1 s satisifies this condition. 2 p s A 2 p s where p s is non-informative since it is invariant to amplitude-scale.

Conjugate priors


Given p x , choose p so that p x p x p has a simple functional form.

Conjugate priors

Choose p , where is a family of densities ( e.g. , Gaussian family) so that the posterior density also belongsto that family.

conjugate prior
p is a conjugate prior for p x if p p x

n n 1 N x n A W n W n 0 2 iid. Rather than modeling A U A 0 A 0 (which did not yield a closed-form estimator) consider p A 1 2 A 2 -1 2 A 2 A 2

With 0 and A 1 3 A 0 this Gaussian prior also reflects prior knowledge that it is unlikely for A A 0 .

The Gaussian prior is also conjugate to the Gaussian likelihood p A x 1 2 2 N 2 -1 2 2 n 1 N x n A 2 so that the resulting posterior density is also a simple Gaussian, as shown next.

First note that p A x 1 2 2 N 2 -1 2 2 n 1 N x n -1 2 2 N A 2 2 N A x - where x - 1 N n 1 N x n .

p x A p A x p A A p A x p A -1 2 1 2 N A 2 2 N A x - 1 A 2 A 2 A -1 2 1 2 N A 2 2 N A x - 1 A 2 A 2 -1 2 Q A A -1 2 Q A
where Q A N 2 A 2 2 N A x - 2 A 2 A 2 2 A A 2 2 A 2 . Now let A | x 2 1 N 2 1 A 2 A | x 2 N 2 x - A 2 A | x 2 Then by "completing the square" we have
Q A 1 A | x 2 A 2 2 A | x A A | x 2 A | x 2 A | x 2 2 A 2 1 A | x 2 A A | x 2 A | x 2 A | x 2 2 A 2
Hence, p x A -1 2 A | x 2 A A | x 2 -1 2 2 A 2 A | x 2 A | x 2 A -1 2 A | x 2 A A | x 2 -1 2 2 A 2 A | x 2 A | x 2 where -1 2 A | x 2 A A | x 2 is the "unnormalized" Gaussian density and -1 2 2 A 2 A | x 2 A | x 2 is a constant, independent of A . This implies that p x A 1 2 A | x 2 -1 2 A | x 2 A A | x 2 where A | x A | x A | x 2 . Now
A x A A A p x A A | x N 2 x - A 2 N 2 1 A 2 A 2 A 2 2 N x - 2 N A 2 2 N x - 1
Where 0 A 2 A 2 2 N 1


  • When there is little data A 2 2 N is small and A .
  • When there is a lot of data A 2 2 N , 1 and A x - .

Interplay between data and prior knowledge

Small N A favors prior.

Large N A favors data.

The multivariate gaussian model

The multivariate Gaussian model is the most important Bayesian tool in signal processing. It leads directly tothe celebrated Wiener and Kalman filters.

Assume that we are dealing with random vectors x and y . We will regard y as a signal vector that is to be estimated from an observation vector x .

y plays the same role as did in earlier discussions. We will assume that y is p1 and x is N1. Furthermore, assume that x and y are jointly Gaussian distributed x y 0 0 R xx R xy R yx R yy x 0 , y 0 , x x R xx , x y R xy , y x R yx , y y R yy . R R xx R xy R yx R yy

x y W , W 0 2 I p y 0 R yy which is independent of W . x y W 0 , x x y y y W W y W W R yy 2 I , x y y y W y R yy y x . x y 0 0 R yy 2 I R yy R yy R yy From our Bayesian perpsective, we are interested in p x y .

p x y p x , y p x 2 N 2 2 p 2 R -1 2 -1 2 x y R x y 2 N 2 R xx -1 2 -1 2 x R xx x
In this formula we are faced with R R xx R xy R yx R yy The inverse of this covariance matrix can be written as R xx R xy R yx R yy R xx 0 0 0 R xx R xy I Q R yx R xx I where Q R yy R yx R xx R xy . (Verify this formula by applying the right hand side above to R to get I .)

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistical signal processing' conversation and receive update notifications?