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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

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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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