Logistic Regression

In a classification problem, given a list of features values $x$ and their corresponding classes $\{c_i\}$, the posterior for of the classes, aka conditional probability of the classes, is

$$ p(C=c_i\mid X=x). $$


The likelihood of the data is

$$ p(X=x\mid C=c_i). $$

Logistic Regression for Two Classes

For two classes, the simplest model for the posterior is a linear model,

$$ \log \frac{p(C=c_1\mid X=x) }{p(C=c_2\mid X=x)} = \beta_0 + \beta_1 \cdot x, $$

which is equivalent to

$$ p(C=c_1\mid X=x) = \exp\left(\beta_0 + \beta_1 \cdot x\right) p(C=c_2\mid X=x) . $$


The reason that we proposing a linear model for the quantity

$$ \log \frac{p(C=c_1\mid X=x) }{p(C=c_2\mid X=x)}, $$

is that it has a range from $-\infty$ to $\infty$ which matches the range of the linear model $ \beta_0 + \beta_1 \cdot x$.

We can also see in the following results that such relation guarantees that the conditional probabilities are restricted to 0 to 1 after applying the normalization constraint.

Using the normalization condition

$$ p(C=c_1\mid X=x) + p(C=c_2\mid X=x) = 1, $$

we can derive the posterior for each classes

$$ \begin{align} p(C=c_2\mid X=x) &= \frac{1}{1 + \exp\left(\beta_0 + \beta_1 \cdot x\right)} \\ p(C=c_1\mid X=x) &= \frac{\exp\left(\beta_0 + \beta_1 \cdot x\right)}{1 + \exp\left(\beta_0 + \beta_1 \cdot x\right)}. \end{align} $$

For simplicity, we are using $x’=\beta_0 + \beta_1 \cdot x$ in this figure.
The two conditional probabilities

For simplicity, we are using $x’=\beta_0 + \beta_1 \cdot x$ in this figure.

This is the [[sigmoid function]] Uni-Polar Sigmoid Uni-polar sigmoid function and its properties

Limiting behavior

  1. As $\beta_0 + \beta_1 \cdot x \to \infty$, we have $p(C=c_2\mid X=x) \to 0$ and $p(C=c_1\mid X=x)\to 1$.
  2. As $\beta_0 + \beta_1 \cdot x \to 0$, we have $p(C=c_2\mid X=x) \to 0.5$ and $p(C=c_1\mid X=x)\to 0.5$.
  3. As $\beta_0 + \beta_1 \cdot x \to -\infty$, we have $p(C=c_2\mid X=x) \to 1$ and $p(C=c_1\mid X=x)\to 0$.

Relation to Cross Entropy

For two classes, we can write down the likelihood as

$$ \pi_{i=1}^{N} p^{y_i} p^{1-y_i}, $$

where $p$ is the probability of label $y_i=c_1$ and $1-p$ is probability of label $y_i=c_2$.

Taking the neglog, we find that

$$ -l = sum_{i=1}^N ( -y_i \log p - (1-y_i)\log (1-p) ). $$

This is the [[cross entropy]] Cross Entropy Cross entropy is1 $$ H(p, q) = \mathbb E_{p} \left[ -\log q \right]. $$ Cross entropy $H(p, q)$ can also be decomposed, $$ H(p, q) = H(p) + \operatorname{D}_{\mathrm{KL}} \left( p \parallel q \right), $$ where $H(p)$ is the [[entropy of $P$]] Shannon Entropy Shannon entropy $S$ is the expectation of information content $I(X)=-\log \left(p\right)$1, \begin{equation} H(p) = \mathbb E_{p}\left[ -\log \left(p\right) \right]. \end{equation} shannon_entropy_wiki Contributors to Wikimedia projects. …

Logistic Regression for $K$ Classes

It is easily generalized to problems with $K$ classes.

$$ \begin{align} p(C=c_K\mid X=x) &= \frac{1}{1 + \sum_k\exp\left(\beta_{k0} + \beta_k \cdot x\right)} \\ p(C=c_k\mid X=x) &= \frac{\exp\left(\beta_{k0} + \beta_k \cdot x\right)}{1 + \sum_k\exp\left(\beta_{k0} + \beta_k \cdot x\right)} \end{align} $$

Why not non-linear

The log of the posterior ratio can be more complex than linear models. In general, we have1

$$ \log \frac{p(C=c_1\mid X=x) }{p(C=c_2\mid X=x)} = f(x), $$

so that

$$ p(C=c_1\mid X=x) = \frac{\exp(f(x))}{ 1 + \exp(f(x)) }. $$

The logistic regression model we mentioned in the previous sections require

$$ f(x) = \beta_0 + \beta_1 \cdot x. $$

A more general additive model is

$$ f(x) = \sum_i f_i(x), $$

where we can apply algorithms such as local scoring to fit such models1.

Planted: by ;

L Ma (2021). 'Logistic Regression', Datumorphism, 05 April. Available at: https://datumorphism.leima.is/wiki/machine-learning/linear/logistic-regression/.