# 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 ${\eta }_{i}={\theta }_{i}^{T}x$ .)

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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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research.net
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what is the actual application of fullerenes nowadays?
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