# 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

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
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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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