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.

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yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
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narayan
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I only see partial conversation and what's the question here!
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Damian
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Professor
I think
Professor
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LITNING
scanning tunneling microscope
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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