I read on Wikipedia that all exponential family distributions have conjugate priors. I have not, however, been able to find a reference that describes how to find it. So given $$f_X(x\mid\theta) = h(x) \exp \left (\theta^T \cdot T(x) -A(\theta)\right )$$
how do I find its conjugate prior?
(In particular, I am trying to find the conjugate prior for a distribution that is the product of many binomials, which is also in the exponential family: \begin{align} P(\Phi=\{\phi_1, \cdots, \phi_k\}|X=\{(d_1, v_1), \cdots, ... (d_k, v_k)\}) =\prod\limits_{j=1}^k\binom{d_{j}}{v_{j}} \left(\phi_{j}\right)^{v_{j}} \left(1 - {\phi}_{j}\right)^{{d}_{j} -{v}_{j}} \end{align} ).