Minimization/maximization of Sum-Product cost functions

Typically in digital detection/decoding one encounters a sum-product cost function. Mathematically, this may represented by
J(x)=\sum_i\prod_j f(x_{ij}), where f(x) is a continuous function (typically a pdf). If f(x) is a univariate-pdf, then j signifies the dimension of the joint-pdf \prod_j f(x_{ij}). The case for j=1 reduces to \sum_if(x_{i}) and is easily solvable. For j\neq1, would expressions of this nature be concave and implementable in cvx? In accordance to the DCP rule set products are not entertained. If J(x) is concave, how would one implement it in cvx?

For the purpose of discussion, let us start of by trying to implement the form \sum_i\ f(x_{i1})f(x_{i2}). We may assume f(x) to be Gaussian or Cauchy, i.e., f(x)=\frac{1}{\sqrt(2\pi)}\exp(-\frac{x^2}{2\sigma^2}) or f(x)=\frac{\delta}{\pi(x^2+\delta^2)}, respectively, where \delta and \sigma are positive constants.

Any input will be greatly appreciated, thanks!