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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
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ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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Abhi
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Abhi
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salma
Commplementary angles
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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I'm interested in nanotube
Uday
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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