If you want to estimate probabilities using the multinomial logistic regression it doesn't really matter which value (A,B,C,D) is the reference.

So the "problem" is only with the interpretation.

After defining A as the reference you get all the results as a comparison to A, but you can easily translate it to any other relation like the odds of B related to C or the odd of C relate to D.

See the following example:

In the following example if you run A as a reference and when to calculate C compare to B:

C to B = C to A / B to A = 2.7723/3.9961=0.6938

C to B = C to A / B to A = 0.9668/0.8780=1.1012 (rounding)

**Example with A, B, C when A is the reference**
Interpretation

When all the values of the predictors (Xj) are zero:

The odds of B in comparison to A is: 3.9961

The odds of C in comparison to A is: 2.7723

One unit increase in X1:

Will decrease the odds of B in comparison to A by 12.2% (a.k.a. the odds will be multiplied by 0.8780).

Will decrease the odds of C in comparison to A by 3.3% (a.k.a. the odds will be multiplied by 0.9668).

**Example with A, B, C when B is the reference**
When all the values of the predictors (Xj) are zero:

The odds of C in comparison to B is: 0.2502

The odds of A in comparison to B is: 0.6938

One unit increase in X1:

Will increase the odds of C in comparison to B by 13.9% (a.k.a. the odds will be multiplied by 1.1389).

Will increase the odds of A in comparison to B by 10.1% (a.k.a. the odds will be multiplied by 1.1012).

Since it is a bit confusing to interpret the multinomial regression, I used the following interpretation calculator:

http://www.statskingdom.com/430logistic_regression.html