# Multinomial Logistic Regression likelihood

Suppose we have the following parametric model for logistic regression: $$\phi_{i} = \frac{\exp{(a^{T}x_{[i]}})}{\sum_{k = 1} ^{M} \exp{(a^{T}x_{[k]})}}$$for $$i = 1, \dots, M$$ and that the parameter set $$X = \{x_{[i]}: i = 1, \dots, M \}$$ consists of $$M$$ vectors $$x_{[I]} \in \mathbb{R}^{n}$$ we want to calibrate.

I want to find the log-likelihood for this model and the associated maximisation problem. Further, I want to show that this is a concave function.

I am however very confused as to how I should proceed. I believe I should introduce some vectors $$y_{j}$$ that should represent the class the observation should be in, e.g. $$y_{j} = e_{i}$$ representing that $$a_{j}$$ should be in class $$C_{i}$$. I am though not sure how to introduce these in the calculations.

you need to maximize the product of the probability function $$\prod_{i=1}^M \phi_i$$
Maximize with the log as log function will have the same optimal points $$x_i$$ as original function. $$\log L = \sum_i \log \phi_i$$
Now $$\log \phi_i = \log (\exp{a^{T}x_{[i]}}) = a^{T}x_{[i]} - \log \sum_k \exp (a^Tx_k)$$\
So max Log L = $$\sum_i a^{T}x_{[i]}-M{\sum_{k = 1} ^{M} (\log \exp(a^{T}x_{[k]})})$$
but I think parametric model will be $$\phi_i = {{e^{-a^Tx_i}}\over{e^{\sum_k -a^Tx_k}} }$$
So $$\log \phi_i = \sum_k^M a^Tx_k- a^Tx_i$$