On the Chi square and higher-order Chi distances for approximating f-divergences

We report closed-form formula for calculating the Chi square and higher-order Chi distances between statistical distributions belonging to the same exponential family with affine natural space, and instantiate those formula for the Poisson and isotropic Gaussian families. We then describe an analytic formula for the $f$-divergences based on Taylor expansions and relying on an extended class of Chi-type distances.