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The criterion used in the previous section---minimize the average cost of an incorrect decision---may seem to be acontrived way of quantifying decisions. Well, often it is. For example, the Bayesian decision rule depends explicitly on the a priori probabilities. A rational method of assigning values to these---either by experiment or through trueknowledge of the relative likelihood of each model---may be unreasonable. In this section, we develop alternative decisionrules that try to respond to such objections. One essential point will emerge from these considerations: the likelihood ratio persists as the core of optimal detectors asoptimization criteria and problem complexity change . Even criteria remote fromperformance error measures can result in the likelihood ratio test. Such an invariance does not occur often in signal processing andunderlines the likelihood ratio test's importance.

Maximizing the probability of a correct decision

As only one model can describe any given set of data (the models are mutually exclusive), the probability of beingcorrect P c for distinguishing two models is given by P c say 0 when 0 true say 1 when 1 true We wish to determine the optimum decision region placement.Expressing the probability of being correct in terms of the likelihood functions p R i r , the a priori probabilities and the decision regions, we have P c r Z 0 π 0 p R 0 r r Z 1 π 1 p R 1 r We want to maximize P c by selecting the decision regions Z 0 and Z 1 . Mimicking the ideas of the previous section, we associate each value of r with the largest integral in the expression for P c . Decision region Z 0 , for example, is defined by the collection of values of r for which the first term is largest. As all of the quantities involved are non-negative, the decision rulemaximizing the probability of a correct decision is

Given r , choose i for which the product π i p R i r is largest.
When we must select among more than two models, this result still applies (prove this for yourself). Simple manipulations lead to the likelihood ratio test when we must decide between two models. p R 1 r p R 0 r 0 1 π 0 π 1 Note that if the Bayes' costs were chosen so that C i i 0 and C i j C , ( i j ), the Bayes' cost and the maximum-probability-correct thresholds would be the same.

To evaluate the quality of the decision rule, we usually compute the probability of error P e rather than the probability of being correct. This quantity can be expressed in terms of the observations, thelikelihood ratio, and the sufficient statistic.

P e π 0 r Z 1 p R 0 r π 1 r Z 0 p R 1 r π 0 Λ Λ η p Λ 0 Λ π 1 Λ Λ η p Λ 1 Λ π 0 ϒ ϒ γ p ϒ 0 ϒ π 1 ϒ ϒ γ p ϒ 1 ϒ
These expressions point out that the likelihood ratio and the sufficient statistic can each be considered afunction of the observations r ; hence, they are random variables and have probability densities for each model.When the likelihood ratio is non-monotonic, the first expression is most difficult to evaluate. Whenmonotonic, the middle expression often proves to be the most difficult.No matter how it is calculated, no other decision rule can yield a smaller probability oferror . This statement is obvious as we minimized the probability of error implicitly by maximizing the probability of being correct because P e 1 P c .

Questions & Answers

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
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
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
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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, Elements of detection theory. OpenStax CNX. Jun 22, 2008 Download for free at http://cnx.org/content/col10531/1.5
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