Is it possible to prove that a set of exponential family distributions are linearly independent?
I want to show that $\sum_k c_{k} F_{k}(\pmb{x},\pmb{\theta})=0$ has only the trivial solution $c_k$'s are $0$, where $F_{k}(\pmb{x},\pmb{\theta})$ is the CDF of an exponential family.
can we prove it for dirichlet distributions?
Take for example the beta-distributions defined for $x\in [0,1]$
$$f(x,\alpha,\beta) = x^{\alpha-1}(1-x)^{\beta-1}$$
like
$$f(x,1,1) = 1$$ $$f(x,2,1) = x$$ $$f(x,1,2) = (1-x)$$
and these are linearly dependent
$$f(x,2,1)+f(x,2,1)-f(x,1,1) = 0$$
This counterexample shows that it can not be proven for beta-distributions, and so it won't be either for Dirichlet distributions of any categories number. (You can consider extending the beta binomial by adding categories. If you do this with the three functions above than the dependence remains)
