# Machine learning lecture 1 course notes  (Page 11/13)

 Page 11 / 13
$\begin{array}{ccc}\hfill T\left(y\right)& =& y\hfill \\ \hfill a\left(\eta \right)& =& -log\left(1-\Phi \right)\hfill \\ & =& log\left(1+{e}^{\eta }\right)\hfill \\ \hfill b\left(y\right)& =& 1\hfill \end{array}$

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 ${\sigma }^{2}$ had no effect on our final choice of $\theta$ and ${h}_{\theta }\left(x\right)$ . Thus, we can choose an arbitrary value for ${\sigma }^{2}$ without changing anything. To simplify the derivation below, let'sset ${\sigma }^{2}=1$ . If we leave ${\sigma }^{2}$ as a variable, the Gaussian distribution can also be shown to be in the exponential family, where $\eta \in {\mathbb{R}}^{2}$ is now a 2-dimension vector that depends on both $\mu$ and $\sigma$ . For the purposes of GLMs, however, the ${\sigma }^{2}$ parameter can also be treated by considering a more general definition of the exponential family: $p\left(y;\eta ,\tau \right)=b\left(a,\tau \right)exp\left(\left({\eta }^{T}T\left(y\right)-a\left(\eta \right)\right)/c\left(\tau \right)\right)$ . Here, $\tau$ is called the dispersion parameter , and for the Gaussian, $c\left(\tau \right)={\sigma }^{2}$ ; but given our simplification above, we won't need the more general definition for the examples we will consider here. We then have:

$\begin{array}{ccc}\hfill p\left(y;\mu \right)& =& \frac{1}{\sqrt{2\pi }}exp\left(-,\frac{1}{2},{\left(y-\mu \right)}^{2}\right)\hfill \\ & =& \frac{1}{\sqrt{2\pi }}exp\left(-,\frac{1}{2},{y}^{2}\right)·exp\left(\mu ,y,-,\frac{1}{2},{\mu }^{2}\right)\hfill \end{array}$

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

$\begin{array}{ccc}\hfill \eta & =& \mu \hfill \\ \hfill T\left(y\right)& =& y\hfill \\ \hfill a\left(\eta \right)& =& {\mu }^{2}/2\hfill \\ & =& {\eta }^{2}/2\hfill \\ \hfill b\left(y\right)& =& \left(1/\sqrt{2\pi }\right)exp\left(-{y}^{2}/2\right).\hfill \end{array}$

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

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Machine learning' conversation and receive update notifications? By OpenStax By OpenStax By Yasser Ibrahim By OpenStax By Brooke Delaney By By OpenStax By Briana Hamilton By Janet Forrester By OpenStax