In estimation problems, a widespread way of assessing the quality of a
given estimator is to compare its risk to the minimax risk. It is
however typically impossible (especially in nonparametric problems) to
determine the minimax risk exactly. Consequently, one attempts to
obtain good lower bounds on the minimax risk and the risk of the
estimator is then compared to these lower bounds. Minimax lower bounds
are hence important and applicable in any estimation problem where the
minimax criterion is used. 

A popular method for obtaining minimax lower bounds is to reduce the
general estimation problem to a multiple hypothesis testing problem
and then employ Fano's inequality from information theory. Fano's
inequality involves logarithms and Kullback-Leibler divergences and
provides a lower bound for the uniform Bayes risk in a  
multiple hypothesis testing problem. 

In this paper, I show that logarithms and Kullback-Leibler divergences
are not really necessary to obtain lower bounds for the Bayes risk. In
fact, the logarithm can be replaced by any convex function f and the
Kullback-Leibler divergence can be replaced by the corresponding
f-divergence (f-divergences are a general class of divergences between
probability measures which include many common distances/divergences
like the Kullback-Leibler divergence, chi-squared divergence, total
variation distance, Hellinger distance etc). This results in a more
general class of minimax lower bounds, one for each convex function f,
and includes the classical Fano method as a special case. This
flexibility provided by the various convex functions f can be useful
in applications. There also turn out to be qualitative differences
between the bounds for different functions f.  

The lower bounds developed here are applicable in many estimation
problems. I have included two specific applications in the paper. The
first application is to nonparametric regression where the regression
function is the support function of an unknown high-dimensional convex
body. The second application is to covariance matrix estimation.