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<!-- navigation toc: --> <li><a href="._week38-bs011.html#___sec10" style="font-size: 80%;">The Ising model</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs012.html#___sec11" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs013.html#___sec12" style="font-size: 80%;">Linear regression</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs014.html#___sec13" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#___sec28" style="font-size: 80%;">The logistic function</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs030.html#___sec29" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
<!-- navigation toc: --> <li><a href="._week38-bs031.html#___sec30" style="font-size: 80%;">Two parameters</a></li>
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<h2 id="___sec37" class="anchor">More classes </h2>
<p>
In our discussion of neural networks we will encounter the above again
in terms of a slightly modified function, the so-called <b>Softmax</b> function.
<p>
The softmax function is used in various multiclass classification
methods, such as multinomial logistic regression (also known as
softmax regression), multiclass linear discriminant analysis, naive
Bayes classifiers, and artificial neural networks. Specifically, in
multinomial logistic regression and linear discriminant analysis, the
input to the function is the result of \( K \) distinct linear functions,
and the predicted probability for the \( k \)-th class given a sample
vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two
predictors):
$$
p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
$$
It is easy to extend to more predictors. The final class is
$$
p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
$$
<p>
and they sum to one. Our earlier discussions were all specialized to
the case with two classes only. It is easy to see from the above that
what we derived earlier is compatible with these equations.
<p>
To find the optimal parameters we would typically use a gradient
descent method. Newton's method and gradient descent methods are
discussed in the material on <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_self">optimization
methods</a>.
<p>
This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that <b>Scikit-learn</b> offers.
<p>
<p>
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