Is there a difference between the two models
- Multinomial logit regression
- Multinomial logistic regression?
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They are the same. Logit and logistic are inverse functions of each other (i.e: $f^{-1}(x)$ and $f(x)$). Therefore in statistical softwares and usage, the terms are actually referring to the same thing.
See https://stats.stackexchange.com/a/120364/132847 for difference between logit and logistic.