# Basic elements of statistical decision theory and statistical

 Page 1 / 5
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.

how do I set up the problem?
what is a solution set?
Harshika
hello, I am happy to help!
Abdullahi
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Got questions? Join the online conversation and get instant answers!