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In decision making problems, we know the value of the observation, but do not know the value y . Therefore, it is appealing to consider the conditional density or pmf as a function of the unknown values y , with X fixed at its observed value. The resulting function is called the likelihood function. As the name suggests, values of y where the likelihood function is largest are intuitively reasonable indicators of the true value of the unknown quantity, which we will denoteby y * . The rationale for this is that these values would produce conditional densities or pmfs that place high probability on theobservation X = x .

The Maximum Likelihood Estimator (MLE) is defined to be the value of y that maximizes the likelihood function; i.e., in the continuous case

y ^ ( X ) = arg max y p X | Y ( X | y )

with an analogous definition for the discrete case by replacing the conditional densitywith the conditional pmf. The decision rule y ^ ( X ) is called an “estimator,” which is common in decision problemsinvolving a continuous parameter. Note that maximizing the likelihood function is equivalent to minimizing the negative log-likelihoodfunction (since the logarithm is a monotonic transformation). Now let y * denote the true value of Y . Then we can view the negative log-likelihood as a loss function

L ( y , y * ) = - log p X | Y ( X | y )

where the dependence on y * on the right hand side is embodied in the observation X on the left. An interesting special case of the MLE results when the conditional density P X | Y ( X | y ) is a Gaussian, in which case the negative log-likelihood corresponds to a squared errorloss function.

Now let us consider the expectation of this loss, with respect to the conditional distribution P X | Y ( X | y * ) :

- E [ log p X | Y ( X | y ) ] = log 1 p X | Y ( x | y ) p X | Y ( x | y * ) d x

The true value y * minimizes the expected negative log-likelihood (or, equivalently, maximizes the expected log-likelihood ). To seethis, compare the expected log-likelihood of y * with that of any other value y :

E [ log p X | Y ( X | y * ) - log p X | Y ( X | y ) ] = E log p X | Y ( X | y * ) p X | Y ( X | y ) = log p X | Y ( x | y * ) p X | Y ( x | y ) p X | Y ( x | y * ) d x = KL ( p X | Y ( x | y * ) , p X | Y ( x | y ) ) .

The quantity KL ( p X | Y ( x | y * ) , p X | Y ( x | y ) ) is called the Kullback-Leibler (KL) divergence between the conditional densityfunction p X | Y ( x | y * ) and p X | Y ( x | y ) . The KL divergence is non-negative, and zero if and only if the two densities are equal [link] . So, we see that the KL divergence acts as a sort of risk function in the context of Maximum Likelihood Estimation.

The cramer-rao lower bound

The MLE is based on finding the value for Y that maximizes the likelihood function. Intuitively, if the maximum point is verydistinct, say a well isolated peak in the likelihood function, then the easier it will be to distinguish the MLE from alternativedecisions. Consider the case in which Y is a scalar quantity. The “peakiness” of the log-likelihood function can be gauged byexamining its curvature, - 2 log p X | Y ( x | y ) y 2 , at the point of maximum likelihood. The higher the curvature, the more peaky is the behavior of the likelihood functionat the maximum point. Of course, we hope that the MLE will be a good predictor (decision) for the unknown true value y * . So, rather than looking at the curvature of the log-likelihood function at themaximum likelihood point, a more appropriate measure of how easily it will be to distinguish y * from the alternatives is the expected curvature of the log-likelihood function evaluated at the value y * . The expectation taken over all possible observations with respect tothe conditional density p X | Y ( x | y * ) . This quantity, denoted I ( y * ) = E [ - 2 log p X | Y ( x | y ) y 2 ] | y = y * , is called the Fisher Information (FI). In fact, the FI provides us with an important performance bound known asthe Cramer-Rao Lower Bound (CRLB).

Questions & Answers

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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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