Github user sethah commented on a diff in the pull request:
https://github.com/apache/spark/pull/13139#discussion_r63890670
--- Diff: docs/ml-classification-regression.md ---
@@ -374,6 +374,197 @@ regression model and extracting model summary
statistics.
</div>
+## Generalized linear regression
+
+When working with data that has a relatively small number of features (<
4096), Spark's GeneralizedLinearRegression interface
+allows for flexible specification of [generalized linear
models](https://en.wikipedia.org/wiki/Generalized_linear_model) (GLMs) which
can be used for various types of
+prediction problems including linear regression, Poisson regression,
logistic regression, and others.
+
+Contrasted with linear regression where the output is assumed to follow a
Gaussian
+distribution, GLMs are specifications of linear models where the response
variable $Y_i$ may take on _any_
+distribution from the [exponential family of
distributions](https://en.wikipedia.org/wiki/Exponential_family).
+
+$$
+Y_i \sim f\left(\cdot|\theta_i, \phi, w_i\right)
+$$
+
+An exponential family distribution is any probability distribution of the
form
+
+$$
+f\left(y|\theta, \phi, w\right) = \exp{\left(\frac{y\theta -
b(\theta)}{\phi/w} - c(y, \phi)\right)}
+$$
+
+where the parameter of interest $\theta_i$ is related to the expected
value of the response variable
+$\mu_i$ by
+
+$$
+\theta_i = h(\mu_i)
+$$
+
+Here, $h(\mu_i)$ is defined by the form of the exponential family
distribution used. GLMs also allow specification
+of a link function, which defines the relationship between the expected
value of the response variable $\mu_i$
+and the so called _linear predictor_ $\eta_i$:
+
+$$
+g(\mu_i) = \eta_i = \vec{x_i}^T \cdot \vec{\beta}
+$$
+
+Often, the link function is chosen such that $h(\mu) = g(\mu)$, which
yields a simplified relationship
+between the parameter of interest $\theta$ and the linear predictor
$\eta$. In this case, the link
+function $g(\mu)$ is said to be the "canonical" link function.
+
+$$
+\theta_i = h(g^{-1}(\eta_i)) = \eta_i
+$$
+
+A GLM finds the regression coefficients $\vec{\beta}$ which maximize the
likelihood function.
+
+$$
+\min_{\vec{\beta}} \mathcal{L}(\vec{\theta}|\vec{y},X) =
+\prod_{i=1}^{N} \exp{\left(\frac{y_i\theta_i - b(\theta_i)}{\phi/w_i} -
c(y_i, \phi)\right)}
+$$
+
+where the parameter of interest $\theta_i$ is related to the regression
coefficients $\vec{\beta}$
+by
+
+$$
+\theta_i = h(g^{-1}(\vec{x_i} \cdot \vec{\beta}))
+$$
+
+Spark's generalized linear regression interface also provides summary
statistics for diagnosing the
+fit of GLM models, including residuals, p-values, deviances, the Akaike
information criterion, and
--- End diff --
I would prefer not to. Summary statistics are relatively easy to add and so
this could change rather frequently. We should let the examples document them.
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