The design of a hypothesis test/detector often
involves constructing the solution to an optimizationproblem. The optimality criteria used fall into two classes:
Bayesian and frequent.
In the Bayesian setup, it is assumed that the
a priori probability of
each hypothesis occuring(
) is known. A cost
is assigned to each possible outcome:
The optimal test/detector is the one that minimizes the Bayes
risk, which is defined to be the expected cost of anexperiment:
In the event that we have a binary problem, and both
hypotheses are
simple , the decision rule that
minimizes the Bayes risk can be constructed explicitly. Let usassume that the data is continuous (
i.e. ,
it has a density) under each hypothesis:
Let
and
denote the
decision
regions corresponding to the optimal test. Clearly,
the optimal test is specified once we specify
and
.
The Bayes risk may be written
Recall that
and
partition the input space: they are
disjoint and their union is the full input space. Thus, everypossible input
belongs to precisely one of these regions. In order to minimize
the Bayes risk, a measurement
should belong to the decision
region
for which the corresponding integrand in the preceding equationis smaller. Therefore, the Bayes risk is minimized by assigning
to
whenever
and assigning
to
whenever this inequality is reversed. The resulting rule may beexpressed concisely as
Here,
is called the
likelihood ratio ,
is called the threshold, and
the overall decision rule is called the
Likelihood Ratio Test (LRT). The expressionon the right is called 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
. Failure may not indicate studiousness:
conscientious students may fail the test. Define the modelsas
: did not study
: did study
The conditional densities of the grade are shown in
.
Conditional densities for the grade distributions
assuming that a student did not study(
) or did
(
) 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
and
. The costs
are chosen to reflect the instructor's sensitivity
to student feelings:
(an erroneous decision either way is given the
same cost) and
. The likelihood ratio is plotted in
and the threshold value
, which is computed from the
a
priori probabilities and the costs to be
, is indicated. The calculations of this
comparison can be simplified in an obvious way.
or
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!
A special case of the minimum Bayes risk rule, the
minimum probability of error rule , is
used extensively in practice, and is discussed at length inanother module.
Problems
Denote
and
. Express the Bayes risk
in terms of
and
,
, and
. Argue that the optimal decision rule is not
altered by setting
.
Suppose we observe
such that
. Argue that it doesn't matter whether we assign
to
or
.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
