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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 we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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