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10 Generalized Linear Regression In the problems of this section $x^T \beta = \beta_0 + \sum_{i=1}^p \beta_i x_i$ Problem 10.6. This is a multiclass generalization of logistic regression known as multinomial logistic regression. Here Y is an r.v. that assumes K values in a discrete set S = {1,... K}. Note that the elements in S are symbols, whose meaning depends on the context of the data analysis at hand. The training set is $D_{tr} = \{x_i, Y_i\}_{i=1}^n$ where $x_i$ is a predictor with values in (some subset of) $R^d$. We model the symbol probabilities by $\ln P(Y_i = k | x_i) = x_i^T \beta_k - \ln Z$. Z is a normalization factor, the logarithm of the partition function. Z is determined by the constraint $\sum_{k=1}^K P(Y_i = k | x_i) = 1$. Show that $(P(Y_i = 1 | x_i),... P(Y_i = K | x_i)) = \text{softmax} (x_i^T \beta_1,...,x_i^T \beta_k)$.

          10 Generalized Linear Regression
In the problems of this section
$x^T \beta = \beta_0 + \sum_{i=1}^p \beta_i x_i$
Problem 10.6.
This is a multiclass generalization of logistic regression known as multinomial logistic
regression. Here Y is an r.v. that assumes K values in a discrete set S = {1,... K}. Note that
the elements in S are symbols, whose meaning depends on the context of the data analysis at
hand. The training set is
$D_{tr} = \{x_i, Y_i\}_{i=1}^n$
where $x_i$ is a predictor with values in (some subset of) $R^d$. We model the symbol probabilities
by
$\ln P(Y_i = k | x_i) = x_i^T \beta_k - \ln Z$.
Z is a normalization factor, the logarithm of the partition function. Z is determined by the
constraint
$\sum_{k=1}^K P(Y_i = k | x_i) = 1$.
Show that
$(P(Y_i = 1 | x_i),... P(Y_i = K | x_i)) = \text{softmax} (x_i^T \beta_1,...,x_i^T \beta_k)$.

$10 Generalized Linear Regression In the problems of this section x^T β = β0 + ∑i=1^p xi Problem 10.6. This is a multiclass generalization of logistic regression known as multinomial logistic regression. Here Y is an r.v. that assumes K values in a discrete set S = 1,... K. Note that the elements in S are symbols, whose meaning depends on the context of the data analysis at hand. The training set is Dtr = {xi, Yi}i=1^n where xi is a predictor with values in (some subset of) R^d. We model the symbol probabilities by ln P(Yi = k | xi) = xi^T - ln Z. Z is a normalization factor, the logarithm of the partition function. Z is determined by the constraint ∑k=1^K P(Yi = k | xi) = 1. Show that (P(Yi = 1 | xi),... P(Yi = K | xi)) = softmax (xi^T β1,...,xi^T ).$

Added by Nicholas G.

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