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\title{Stochastic Gradient Descent: Lecture Notes} 

\author{William W. Cohen}

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\section{A Warmup}

If you have a continuous mathematical expression for a probability,
you can often learn the parameter values by simply maximizing the
probability of the data with respect to the parameter values.

For example, suppose we have a binomial---a coin with some unknown
probability $\theta$ of heads---and $n$ draws from it,
$x_1,\ldots,x_n$, of which $k$ came up heads.  We can write the
probability of a single $x_i$ as
\[ P(X=x_i) = \left{\begin{array}{ll} \theta & \mbox{if $x_i=1$} \\ 1-\theta & \mbox{if $x_i=0$} \end{array}
\]

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