# Basic elements of statistical decision theory and statistical

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This paper reviews and contrasts the basic elements of statistical decision theory and statistical learning theory. It is not intended to be a comprehensive treatment of either subject, but rather just enough to draw comparisons between the two.

Throughout this module, let $X$ denote the input to a decision-making process and $Y$ denote the correct response or output (e.g., the value of a parameter, the label of a class, the signal of interest). We assume that $X$ and $Y$ are random variables or random vectors with joint distribution ${P}_{X,Y}\left(x,y\right)$ , where $x$ and $y$ denote specific values that may be taken by the random variables $X$ and $Y$ , respectively. The observation $X$ is used to make decisions pertaining to the quantity of interest. For thepurposes of illustration, we will focus on the task of determining the value of the quantity of interest. A decision rule for this task is a function $f$ that takes the observation $X$ as input and outputs a prediction of the quantity $Y$ . We denote a decision rule by $\stackrel{^}{Y}$ or $f\left(X\right)$ , when we wish to indicate explicitly the dependence of the decision rule on the observation. Wewill examine techniques for designing decision rules and for analyzing their performance.

## Measuring decision accuracy: loss and risk functions

The accuracy of a decision is measured with a loss function. For example, if our goal is to determine the value of $Y$ , then a loss function takes as inputs the true value $Y$ and the predicted value (the decision) $\stackrel{^}{Y}=f\left(X\right)$ and outputs a non-negative real number (the “loss”) reflective of theaccuracy of the decision. Two of the most commonly encountered loss functions include:

1. 0/1 loss: ${\ell }_{0/1}\left(\stackrel{^}{Y},Y\right)\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\mathbf{I}}_{\stackrel{^}{Y}\ne Y}$ , which is the indicator function taking the value of 1 when $\stackrel{^}{Y}\ne Y$ and taking the value 0 when $\stackrel{^}{Y}\left(X\right)=Y$ .
2. squared error loss: ${\ell }_{2}\left(\stackrel{^}{Y},Y\right)\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{\parallel \stackrel{^}{Y}-Y\parallel }_{2}^{2}$ , which is simply the sum of squared differences between the elements of $\stackrel{^}{Y}$ and $Y$ .

The 0/1 loss is commonly used in detection and classification problems, and the squared error loss is more appropriate for problemsinvolving the estimation of a continuous parameter. Note that since the inputs to the loss function may be random variables, so is the loss.

A risk $R\left(f\right)$ is a function of the decision rule $f$ , and is defined to be the expectation of a loss with respect to the jointdistribution ${P}_{X,Y}\left(x,y\right)$ . For example, the expected 0/1 loss produces the probability of error risk function; i.e., a simply calculation shows that ${R}_{0/1}\left(f\right)=E\left[\left({\mathbf{I}}_{f\left(X\right)\ne Y}\right]=\text{Pr}\left(f\left(X\right)\ne Y\right)$ . The expected squared error loss produces the mean squared error MSE risk function, ${R}_{2}\left(f\right)={E\left[\parallel f\left(X\right)-Y\parallel }_{2}^{2}\right]$ .

Optimal decisions are obtained by choosing a decision rule $f$ that minimizes the desired risk function. Given complete knowledge of theprobability distributions involved (e.g., ${P}_{X,Y}\left(x,y\right)$ ) one can explicitly or numerically design an optimal decision rule, denoted ${f}^{*}$ , that minimizes the risk function.

## The maximum likelihood principle

The conditional distribution of the observation $X$ given the quantity of interest $Y$ is denoted by ${P}_{X|Y}\left(x|y\right)$ . The conditional distribution ${P}_{X|Y}\left(x|y\right)$ can be viewed as a generative model, probabilistically describing the observations resulting from a givenvalue, $y$ , of the quantity of interest. For example, if $y$ is the value of a parameter, the ${P}_{X|Y}\left(x|y\right)$ is the probability distribution of the observation $X$ when the parameter value is set to $y$ . If $X$ is a continuous random variable with conditional density ${p}_{X|Y}\left(x|y\right)$ or a discrete random variable with conditional probability mass function (pmf) ${p}_{X|Y}\left(x|y\right)$ , then given a value $y$ we can assess the probability of a particular measurment value $y$ by the magnitude of either the conditional density or pmf.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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