# 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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