# Basic elements of statistical decision theory and statistical  (Page 3/5)

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The CRLB states that under some mild regularity assumptions about the conditional density function ${p}_{X|Y}\left(x|y\right)$ , the variance of any unbiased estimator is bounded from below by the inverse of the $I\left({y}^{*}\right)$ [link] , [link] , [link] . Recall that an unbiased estimator is any estimator $\stackrel{^}{Y}$ that satisfies $E\left[\stackrel{^}{Y}\right]={y}^{*}$ . The CRLB tells us is that

$\text{var}\left(\stackrel{^}{Y}\right)\phantom{\rule{4pt}{0ex}}\ge \phantom{\rule{4pt}{0ex}}\frac{1}{I\left({y}^{*}\right)}.$

If $Y$ is a vector-valued quantity, then the expected negative Hessian matrix(matrix of partial second derivatives) of the log-likelihood function is called the Fisher Information Matrix (FIM), and a similar inequality tells us that the varianceof each component of any unbiased estimator of ${y}^{*}$ is bounded below by the corresponding diagonal element of the inverse of the FIM.Since the MSE of an unbiased estimator is equal to its variance, we see that the CRLB provides a very useful lower bound on the best MSEperformance that we can hope to achieve. Thus, the CRLB is often used as a comparison point for evaluating estimators. It may or may not bepossible to achieve the CRLB, but if we find a decision rule that does, we know that it also minimizes the MSE risk among all possibleunbiased estimators. In general, it may be difficult to compute the CRLB, but in certain important cases it is possible to findclosed-form or computational solutions.

## Bayesian decision theory

Bayesian Decision Theory provides a formal system for integrating prior knowledge and observed observations. Forthe purposes of illustration we will focus on problems involving continuous variables and observations, but extensions to discretecases are straightforward (simple replace probability densities with probability mass functions, and integrals with summations). The keyelements of Bayesian methods are:

1. a prior probability density function ${p}_{Y}\left(y\right)$ describing a priori knowledge of probable states for the quantity $Y$ ;
2. the likelihood function ${p}_{X|Y}\left(x|y\right)$ , as described above;
3. the posterior density function ${p}_{Y|X}\left(y|x\right)$ .

The posterior density is a function of the prior and likelihood, obtained according to Bayes rule:

${p}_{Y|X}\left(y|x\right)=\frac{{p}_{X|Y}\left(x|y\right){p}_{Y}\left(y\right)}{\int {p}_{X|Y}\left(x|y\right){p}_{Y}\left(y\right)dy}.$

The posterior is an indicator of probable values for $Y$ , based on the prior knowledge and the observation. Several options exist for deriving a specific estimateof $Y$ using the posterior. The mean value of the posterior density is one common choice (commonly called the posterior mean ). The posterior mean is the decision rule that minimizes the expectedsquared error loss (MSE risk) function. The value $y$ where the posterior density is maximized is another popular estimator (commonlycalled the Maximum A Posteriori (MAP) estimator). Note that the denominator of the posterior is independent of $y$ , so the MAP estimator is simply the maximizer of the product of the likelihood andthe prior. Therefore, if the prior is a constant function, the MAP estimator and MLE coincide.

## Statistical learning

In all of the methods described above, we assumed some amount of knowledge about the distributions of the observation $X$ and quantity of interest $Y$ . Such knowledge can come from a careful analysis of the physical characteristics of the problem at hand, or it can begleaned from previous experience. However, there are situations where it is difficult to model the physics of the problem and we may nothave enough experience to develop complete and accurate probability models. In such cases, it is natural to adopt a statistical learning approach [link] , [link] .

#### Questions & Answers

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
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
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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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 learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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