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From a grander viewpoint, these expressions represent an achievable lower bound on performance (as assessed by the probability oferror). Furthermore, this probability will be non-zero if the conditional densities overlap over some range of values of r , such as occurred in the previous example. Within regions of overlap, the observed values are ambiguous: eithermodel is consistent with the observations. Our "optimum" decision rule operates in such regions by selecting that modelwhich is most likely (has the highest probability) of generating the measured data.

Neyman-pearson criterion

Situations occur frequently where assigning or measuring the a priori probabilities π i is unreasonable. For example, just what is the a priori probability of a supernova occurring in any particular region of the sky? We clearlyneed a model evaluation procedure that can function without a priori probabilities. This kind of test results when the so-called Neyman-Pearson criterion is used to derive the decision rule.

Using nomenclature from radar, where model 1 represents the presence of a target and 0 its absence, the various types of correct and incorrect decisions have the following names.

In statistics, a false-alarm is known as a type I error and a miss a type II error .
  • Detection probability

    we say it's there when it is; P D 1 true say 1
  • False-alarm probability

    we say it's there when it's not; P F 0 true say 1
  • Miss probability

    we say it's not there when it is; P M 1 true say 0
The remaining probability 0 true say 0 has historically been left nameless and equals 1 P F . We should also note that the detection and miss probabilities are related by P M 1 P D . As these are conditional probabilities, they do not depend on the a priori probabilities. Furthermore, the two probabilities P F and P D characterize the errors when any decision rule is used.

These two probabilities are related to each other in an interesting way. Expressing these quantities in terms of thedecision regions and the likelihood functions, we have P F r Z 1 p R 0 r P D r Z 1 p R 1 r As the region Z 1 shrinks, both of these probabilities tend toward zero; as Z 1 expands to engulf the entire range of observation values, they both tend toward unity. This rather directrelationship between P D and P F does not mean that they equal each other; in most cases, as Z 1 expands, P D increases more rapidly than P F (we had better be right more often than we are wrong!). However, the "ultimate" situation where a rule isalways right and never wrong ( P D 1 , P F 0 ) cannot occur when the conditional distributions overlap. Thus, to increase the detection probability we mustalso allow the false-alarm probability to increase. This behavior represents the fundamental tradeoff in detection theory .

One can attempt to impose a performance criterion that depends only on these probabilities with the consequent decision rulenot depending on the a priori probabilities. The Neyman-Pearson criterion assumes that the false-alarm probability is constrained to be less than orequal to a specified value α while we maximize the detection probability P D . P F P F α Z 1 P D A subtlety of the solution we are about to obtain is that the underlying probability distribution functions may not becontinuous, with the consequence that P F can never equal the constraining value α . Furthermore, a (unlikely) possibility is that the optimum value for the false-alarm probability is somewhat lessthan the criterion value. Assume, therefore, that we rephrase the optimization problem by requiring that the false-alarmprobability equal a value α that is the largest possible value less than or equal to α .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
Difference between extinct and extici spicies
Amanpreet 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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