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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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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