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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

Note

  • 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

Idea

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

    Interpretation

  • 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

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
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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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