In a multinomial logistic regression, the predicted probability $\pi$ of each outcome $j$ (in a total of $J$ possible outcomes) is given by:
$ \pi_j = \frac{e^{A_j}}{1+\sum_{g \neq j}^Je^{A_j}} $
where the value $A_j$ is predicted by a series of predictor variables. For instance, here it is predicted by two covariates ($x_1$ and $x_2$), with their associated regression slopes $\beta_1$ and $\beta_2$, and the interaction between the two covariates (with associated regression slope $\beta_{12}$):
$ A_j = e^{(\alpha_j+\beta_{1,j}x_1+\beta_{2,j}x_2+\beta_{12,j}x_1x_2)} $
The model needs to be fitted to real data, and we will want to know how well it fits. For instance, perhaps out of a draw of 10 balls from a sack, 5 were red, 2 were green, and 3 were yellow. We are interested in whether the variables $x_1$ and $x_2$ associated with the sack allow us to predict this result if we plug in the values for $\beta_1$, $\beta_2$, and $\beta_{12}$ from the fitted model.
To obtain a measure of the goodness-of-fit of the model, we need to calculate the log-likelihood formula for a multinomial logistic regression. I am unsure how to go about this. What is the formula for log-likelihood in a multinomial logistic regression of the kind described above?