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T ( y ) = y a ( η ) = - log ( 1 - Φ ) = log ( 1 + e η ) b ( y ) = 1

This shows that the Bernoulli distribution can be written in the form of Equation  [link] , using an appropriate choice of T , a and b .

Let's now move on to consider the Gaussian distribution. Recall that, when deriving linear regression, the value of σ 2 had no effect on our final choice of θ and h θ ( x ) . Thus, we can choose an arbitrary value for σ 2 without changing anything. To simplify the derivation below, let'sset σ 2 = 1 . If we leave σ 2 as a variable, the Gaussian distribution can also be shown to be in the exponential family, where η R 2 is now a 2-dimension vector that depends on both μ and σ . For the purposes of GLMs, however, the σ 2 parameter can also be treated by considering a more general definition of the exponential family: p ( y ; η , τ ) = b ( a , τ ) exp ( ( η T T ( y ) - a ( η ) ) / c ( τ ) ) . Here, τ is called the dispersion parameter , and for the Gaussian, c ( τ ) = σ 2 ; but given our simplification above, we won't need the more general definition for the examples we will consider here. We then have:

p ( y ; μ ) = 1 2 π exp - 1 2 ( y - μ ) 2 = 1 2 π exp - 1 2 y 2 · exp μ y - 1 2 μ 2

Thus, we see that the Gaussian is in the exponential family, with

η = μ T ( y ) = y a ( η ) = μ 2 / 2 = η 2 / 2 b ( y ) = ( 1 / 2 π ) exp ( - y 2 / 2 ) .

There're many other distributions that are members of the exponential family: The multinomial (which we'll see later), the Poisson (for modelling count-data;also see the problem set); the gamma and the exponential (for modelling continuous, non-negative random variables, such as time-intervals); the beta and the Dirichlet(for distributions over probabilities); and many more. In the next section, we will describe a general “recipe” for constructing models in which y (given x and θ ) comes from any of these distributions.

Constructing glms

Suppose you would like to build a model to estimate the number y of customers arriving in your store (or number of page-views on your website) in any givenhour, based on certain features x such as store promotions, recent advertising, weather, day-of-week, etc. We know that the Poisson distributionusually gives a good model for numbers of visitors. Knowing this, how can we come up with a model for our problem? Fortunately,the Poisson is an exponential family distribution, so we can apply a Generalized Linear Model (GLM).In this section, we will we will describe a method for constructing GLM models for problems such as these.

More generally, consider a classification or regression problem where we would like to predict the value of some random variable y as a function of x . To derive a GLM for this problem, we will make the following threeassumptions about the conditional distribution of y given x and about our model:

  1. y x ; θ ExponentialFamily ( η ) . I.e., given x and θ , the distribution of y follows some exponential family distribution, with parameter η .
  2. Given x , our goal is to predict the expected value of T ( y ) given x . In most of our examples, we will have T ( y ) = y , so this means we would like the prediction h ( x ) output by our learned hypothesis h to satisfy h ( x ) = E [ y | x ] . (Note that this assumption is satisfied in the choices for h θ ( x ) for both logistic regression and linear regression.For instance, in logistic regression, we had h θ ( x ) = p ( y = 1 | x ; θ ) = 0 · p ( y = 0 | x ; θ ) + 1 · p ( y = 1 | x ; θ ) = E [ y | x ; θ ] .)
  3. The natural parameter η and the inputs x are related linearly: η = θ T x . (Or, if η is vector-valued, then η i = θ i T x .)

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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