# multinomial and ordinal regression

Given a set of objects $\{x_1,x_2,\dots,x_n\}$ we can define an order to them and re-arrange them in that order. For example, if $n$ objects in the set and we define the object $x_i$ to be the integer value $n-i+1$, then the ordering of that set from low to high is $\{x_{\{1\}},\dots,x_{\{n\}}\}=\{x_n,\dots,x_1\}$

If each member of the set has a probability, that ordering can be used within the definition of cumulative distribution

$\Pr(X\le K)=\Pr(X = x_{\{1\}}) + \dots +\Pr(X = x_{\{k\}})$

(So here in this example $x_{\{1\}}=x_n$, etc.). This is not just necessarily integers, but any abstract set with a user-defined ordering.

My question:

Why is Ordered Logit, or proportional odds model, defined so differently than just taking the cumulative of user-provided ordering within the members of Multinomial logistic?

Multinomial logistic solves the probability of being in a particular category

$$P(Y=K) \propto \exp(\boldsymbol\beta_K \cdot \mathbf{X}_i)$$

If we define the ordering as $\{1,2,\dots,K\}$ (say) the cumulative probability, given that ordering, is defined as

$\Pr(Y \le K)=\Pr(Y = 1) + \dots +\Pr(X = K)$

I see multinomial models applied to categorical data, but then when the categories have an order the model switches to an ordered logit with different assumptions and different structure. I don't see why the model has to switch, other than making a proportional odds assumption which may not be valid. The user defines the ordering, which then enables the cumulative distribution function to be used directly when individual membership probabilities can be solved by multinomial logistic. (Alternatively, ordered logistic should be able to determine member probabilities by simple subtraction of the cumulative)

Given a set and a defined order, the probability of being in member ${j}$ , $\Pr(X = j)$, should equal the difference between the cumulative probability of $\Pr(X \le j)$ and $\Pr(X \le (j-1))$