Consider the problem of classifyingD dimensional inputs xRD. Suppose we have 2 classes and the output is denoted y{0,1}. If we use a binary logistic regression classifier then the model is:

pb(yx)=Binomial(y(wTx+b))

where(a)=1+ea1 and wRD,bR are the parameters of the model.

Multiclass logistic regression can also be used when there are only two classes. In that case, the model is:

pm(yx)=Categoricial(yS(Ax+c))

whereS(Ax+c)=iexp(Axi+c)exp(Ax+c) is the softmax function andAR2d,cR2 are the parameters of the model.

1. Prove that this multiclass logistic regression model is equivalent to the binary logistic regression model. In particular show how, given the parametersw,b of any binary logistic regression model, you could construct parameters A,cof a multiclass logistic regression model which would always give the same predictions.
2. Also, show how, given A,c, you can compute parametersw,b which would always give the same predictions.
3. Finally, are these transformations unique? Given the valuesA,c (or w,b), is the value ofw,b (or A,c) unique? If a direction is unique, give an argument why. If it's not unique, give at least one example of a different transformation which would be equivalent.