# 0.1 Detection performance criteria

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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 ${P}_{c}=(\text{say}{ℳ}_{0}\text{when}{ℳ}_{0}\text{true})+(\text{say}{ℳ}_{1}\text{when}{ℳ}_{1}\text{true})$ We wish to determine the optimum decision region placement.Expressing the probability of being correct in terms of the likelihood functions $p(R, {ℳ}_{i}, r)$ , the a priori probabilities and the decision regions, we have ${P}_{c}=\int {\pi }_{0}p(R, {ℳ}_{0}, r)\,d r+\int {\pi }_{1}p(R, {ℳ}_{1}, r)\,d r$ 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 ${ℳ}_{i}$ for which the product ${\pi }_{i}p(R, {ℳ}_{i}, r)$ 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. $\frac{p(R, {ℳ}_{1}, r)}{p(R, {ℳ}_{0}, r)}\underset{{ℳ}_{0}}{\overset{{ℳ}_{1}}{\gtrless }}\frac{{\pi }_{0}}{{\pi }_{1}}$ Note that if the Bayes' costs were chosen so that ${C}_{ii}=0$ and ${C}_{ij}=C$ , ( $i\neq j$ ), 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.

${P}_{e}={\pi }_{0}\int p(R, {ℳ}_{0}, r)\,d r+{\pi }_{1}\int p(R, {ℳ}_{1}, r)\,d r={\pi }_{0}\int p(\Lambda , {ℳ}_{0}, \Lambda )\,d \Lambda +{\pi }_{1}\int p(\Lambda , {ℳ}_{1}, \Lambda )\,d \Lambda ={\pi }_{0}\int p(\Upsilon , {ℳ}_{0}, \Upsilon )\,d \Upsilon +{\pi }_{1}\int p(\Upsilon , {ℳ}_{1}, \Upsilon )\,d \Upsilon$
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 ${P}_{e}=1-{P}_{c}$ .

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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