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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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 By By Danielrosenberger