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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 ( x , y ) , 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 Y ^ or f ( X ) , 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) Y ^ = f ( X ) 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: 0 / 1 ( Y ^ , Y ) = I Y ^ Y , which is the indicator function taking the value of 1 when Y ^ Y and taking the value 0 when Y ^ ( X ) = Y .
  2. squared error loss: 2 ( Y ^ , Y ) = Y ^ - Y 2 2 , which is simply the sum of squared differences between the elements of 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 ( f ) 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 ( x , y ) . For example, the expected 0/1 loss produces the probability of error risk function; i.e., a simply calculation shows that R 0 / 1 ( f ) = E [ ( I f ( X ) Y ] = Pr ( f ( X ) Y ) . The expected squared error loss produces the mean squared error MSE risk function, R 2 ( f ) = E [ f ( X ) - Y 2 2 ] .

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 ( x , y ) ) 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 ( x | y ) . The conditional distribution P X | Y ( x | y ) 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 ( x | y ) 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 ( x | y ) or a discrete random variable with conditional probability mass function (pmf) p X | Y ( x | y ) , 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

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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