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

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: $p(y;\eta ,\tau )=b(a,\tau )exp(({\eta }^{T}T\left(y\right)-a\left(\eta \right))/c\left(\tau \right))$ . 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:

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

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 $h\left(x\right)=\mathrm{E}\left[y\right|x]$ . (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 $h_{\theta }\left(x\right)=p(y=1|x;\theta )=0·p(y=0|x;\theta )+1·p(y=1|x;\theta )=\mathrm{E}\left[y\right|x;\theta ]$ .)
3. The natural parameter $\eta$ and the inputs $x$ are related linearly: $\eta ={\theta }^{T}x$ . (Or, if $\eta$ is vector-valued, then ${\eta }_i={\theta }_i^{T}x$ .)