<< Chapter < Page Chapter >> Page >

In statistics, hypothesis testing is some times known as decision theory or simply testing. The key result around whichall decision theory revolves is the likelihood ratio test.

The likelihood ratio test

In a binary hypothesis testing problem, four possible outcomes can result. Model 0 did in fact represent the best model for the data and the decision rule said it was (a correct decision) or saidit wasn't (an erroneous decision). The other two outcomes arise when model 1 was in fact true with either a correct or incorrect decision made. The decision process operates by segmentingthe range of observation values into two disjoint decision regions 0 and 1 . All values of r fall into either 0 or 1 . If a given r lies in 0 , for example, we will announce our decision

"model 0 was true"
; if in 1 , model 1 would be proclaimed. To derive a rational method of deciding which model best describes the observations, we needa criterion to assess the quality of the decision process. Optimizing this criterion will specify the decision regions.

The Bayes' decision criterion seeks to minimize a cost function associated with making a decision. Let C i j be the cost of mistaking model j for model i ( i j ) and C i i the presumably smaller cost of correctly choosing model i : C i j C i i , i j . Let i be the a priori probability of model i . The so-called Bayes' cost C is the average cost of making a decision.

C i j i j C i j j say i when H j true i j i j C i j j H j true say i
The Bayes' cost can be expressed as
C i j i j C i j j 0 true r i i j i j C i j j r i p r H j r r 0 C 0 0 0 p r 0 r C 0 1 1 p r 1 r r 1 C 1 0 0 p r 0 r C 1 1 1 p r 1 r
p r i r is the conditional probability density function of the observed data r given that model i was true. To minimize this expression with respect to the decision regions 0 and 1 , ponder which integral would yield the smallest value if its integration domain included a specificobservation vector. This selection process defines the decision regions; for example, we choose 0 for those values of r which yield a smaller value for the first integral. 0 C 0 0 p r 0 r 1 C 0 1 p r 1 r 0 C 1 0 p r 0 r 1 C 1 1 p r 1 r We choose 1 when the inequality is reversed. This expression is easily manipulated to obtain the decision rule known as the likelihood ratio test .
p r 1 r p r 0 r 0 1 0 C 1 0 C 0 0 1 C 0 1 C 1 1
The comparison relation means selecting model 1 if the left-hand ratio exceeds the value on the right; otherwise, 0 is selected. Thus, the likelihood ratio p r 1 r p r 0 r symbolically represented by r , is computed from the observed value of r and then compared with a threshold equaling 0 C 1 0 C 0 0 1 C 0 1 C 1 1 . Thus, when two models are hypothesized, the likelihood ratio test can be succinctly expressed as thecomparison of the likelihood ratio with a threshold.
r 0 1

The data processing operations are captured entirely by the likelihood ratio p r 1 r p r 0 r . Furthermore, note that only the value of the likelihood ratio relative to the threshold matters; to simplify the computation of thelikelihood ratio, we can perform any positively monotonic operations simultaneously on the likelihood ratio and the threshold without affecting thecomparison. We can multiply the ratio by a positive constant, add any constant, or apply a monotonically increasing functionwhich simplifies the expressions. We single one such function, the logarithm, because it simplifies likelihoodratios that commonly occur in signal processing applications. Known as the log-likelihood, we explicitlyexpress the likelihood ratio test with it as

r 0 1
Useful simplifying transformations are problem-dependent; by laying bare that aspect of the observations essential to themodel testing problem, we reveal the sufficient statistic r : the scalar quantity which best summarizes the data ( Lehmann, pp. 18-22 ). The likelihood ratio test is best expressed in terms of thesufficient statistic.
r 0 1
We will denote the threshold value by when the sufficient statistic is used or by when the likelihood ratio appears prior to its reduction to a sufficient statistic.

As we shall see, if we use a different criterion other than the Bayes' criterion, the decision rule often involves thelikelihood ratio. The likelihood ratio is comprised of the quantities p r i r , termed the likelihood function , which is also important in estimation theory. It is thisconditional density that portrays the probabilistic model describing data generation. The likelihood functioncompletely characterizes the kind of "world" assumed by each model; for each model, we must specify the likelihood functionso that we can solve the hypothesis testing problem.

A complication, which arises in some cases, is that the sufficient statistic may not be monotonic. If monotonic, thedecision regions 0 and 1 are simply connected (all portions of a region can be reached without crossing into the other region). If not,the regions are not simply connected and decision region islands are created (see this problem ). Such regions usually complicate calculations of decision performance. Monotonic ornot, the decision rule proceeds as described: the sufficient statistic is computed for each observation vector and comparedto a threshold.

An instructor in a course in detection theory wants to determine if a particular student studied for his last test.The observed quantity is the student's grade, which we denote by r . Failure may not indicate studiousness: conscientious students may fail the test. Define the modelsas

  • 0 :did not study
  • 1 :did study
The conditional densities of the grade are shown in .
Conditional densities for the grade distributions assuming that a student did not study( 0 ) or did ( 1 ) are shown in the top row. The lower portion depicts the likelihood ratio formed from these densities.
Based on knowledge of student behavior, the instructor assigns a priori probabilities of 0 1 4 and 1 3 4 . The costs C i j are chosen to reflect the instructor's sensitivity to student feelings: C 0 1 1 C 1 0 (an erroneous decision either way is given the same cost) and C 0 0 0 C 1 1 . The likelihood ratio is plotted in and the threshold value , which is computed from the a priori probabilities and the costs to be 1 3 , is indicated. The calculations of this comparison can be simplified in an obvious way. r 50 0 1 1 3 or r 0 1 50 3 16.7 The multiplication by the factor of 50 is a simple illustration of the reduction of the likelihood ratio to asufficient statistic. Based on the assigned costs and a priori probabilities, the optimum decision rule says the instructor must assume that thestudent did not study if the student's grade is less than 16.7; if greater, the student is assumed to have studieddespite receiving an abysmally low grade such as 20. Note that as the densities given by each model overlap entirely:the possibility of making the wrong interpretation always haunts the instructor. However, no other procedure will be better!

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signal and information processing for sonar' conversation and receive update notifications?

Ask