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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

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Eke Reply
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Missy Reply
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Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
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Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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da
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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da
Application of nanotechnology in medicine
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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.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
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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
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Rafiq
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
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
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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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