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.