Lets say we have a dependent variable $Y$ with few categories and set of independent variables.
What are the advantages of multinomial logistic regression over set of binary logistic regressions (i.e. one-vs-rest scheme)? By set of binary logistic regression I mean that for each category $y_{i} \in Y$ we build separate binary logistic regression model with target=1 when $Y=y_{i}$ and 0 otherwise.
Best Answer
If $Y$ has more than two categories your question about "advantage" of one regression over the other is probably meaningless if you aim to compare the models' parameters, because the models will be fundamentally different:
$\bf log \frac{P(i)}{P(not~i)}=logit_i=linear~combination$ for each $i$ binary logistic regression, and
$\bf log \frac{P(i)}{P(r)}=logit_i=linear~combination$ for each $i$ category in multiple logistic regression, $r$ being the chosen reference category ($i \ne r$).
However, if your aim is only to predict probability of each category $i$ either approach is justified, albeit they may give different probability estimates. The formula to estimate a probability is generic:
$\bf P'(i)= \frac{exp(logit_i)}{exp(logit_i)+exp(logit_j)+\dots+exp(logit_r)}$, where $i,j,\dots,r$ are all the categories, and if $r$ was chosen to be the reference one its $\bf exp(logit)=1$. So, for binary logistic that same formula becomes $\bf P'(i)= \frac{exp(logit_i)}{exp(logit_i)+1}$. Multinomial logistic relies on the (not always realistic) assumption of independence of irrelevant alternatives whereas a series of binary logistic predictions does not.
A separate theme is what are technical differences between multinomial and binary logistic regressions in case when $Y$ is dichotomous. Will there be any difference in results? Most of the time in the absence of covariates the results will be the same, still, there are differences in the algorithms and in output options. Let me just quote SPSS Help about that issue in SPSS: