Log Odds

The log odds is the logarithm applied to the Odds. $f(p)=\log \left(\frac{p}{1-p}\right)$ In a Logistic Regression we try to predict these log odds via a linear combination of the features (just like in Perceptron or Linear Regression we assume this linear relationship). $\log \left(\frac{P(y=1)}{1-P(y=1)}\right)=\log \left(\frac{P(y=1)}{P(y=0)}\right)={\beta }_0+{\beta }_1{x}_1+\dots +{\beta }_p{x}_p$

The Sigmoid Function is the inverse function of the logit function, so we can get back the normal probability $p$. $\sigma (f(p))=p$

Explainability

Increasing a features value $x_j$ by one unit (this is happening in the numerator), the odds ratio changes by a factor of $e^{{\beta }_j}$.

$\frac{odd{s}_{x_j+1}}{odds}=\frac{\exp \left({\beta }_0+{\beta }_1{x}_1+\dots +{\beta }_j\left(x_j+1\right)+\dots +{\beta }_p{x}_p\right)}{\exp \left({\beta }_0+{\beta }_1{x}_1+\dots +{\beta }_j{x}_j+\dots +{\beta }_p{x}_p\right)}=\exp \left({\beta }_j\left(x_j+1\right)-{\beta }_j{x}_j\right)=\exp \left({\beta }_j\right)$

We can rewrite this to ${\text{odds}}_{x_j+1}=\text{odds}\cdot e^{{\beta }_j}.$ Which leads to the conclusion that the odds will increase multiplicative by the exponential of the weight of that feature.