Given a statistical model, a statistic on the model is sufficient if the Fisher metric of the induced model coincides with the original Fisher metric, according to the definition by Ay-Jost-Lê-Schwachhöfer. We introduce and study its quantitative version: for \(0 < \delta \le 1\) , we call a statistic \(\delta \) -almost sufficient if \(\delta ^2 \mathfrak {g}(v,v) \le \mathfrak {g}'(v,v)\) for every tangent vector v of the parameter space, where \(\mathfrak {g}\) and \(\mathfrak {g}'\) are the Fisher metric of the original and the induced model, respectively. By the monotonicity theorem due to Amari-Nagaoka and Ay-Jost-Lê-Schwachhöfer, the Fisher metric \(\mathfrak {g}'\) of the induced model for such a statistic is bi-Lipschitz equivalent to the original one \(\mathfrak {g}\) , which means that the information loss of the statistic is uniformly bounded. We characterize such statistics in terms of the conditional probability or by the existence of a certain decomposition of the density function in a way similar to the characterizations of sufficient statistics due to Ay-Jost-Lê-Schwachhöfer and Fisher-Neyman.
