The problem of determining the achievable sensitivity with digitization exhibiting minimal complexity is addressed. In this case, measurements are exclusively available in hard-limited form. Assessing the achievable sensitivity via the Cramér-Rao lower bound requires characterization of the likelihood function, which is intractable for multivariate binary distributions. In this context, the Fisher matrix of the exponential family and a lower bound for arbitrary probabilistic models are discussed. The conservative approximation for Fisher's information matrix rests on a surrogate exponential family distribution connected to the actual data-generating system by two compact equivalences. Without characterizing the likelihood and its support, this probabilistic notion enables designing estimators that consistently achieve the sensitivity as defined by the inverse of the conservative information matrix. For parameter estimation with multivariate binary samples, a quadratic exponential surrogate distribution tames statistical complexity such that a quantitative assessment of an achievable sensitivity level becomes tractable. This fact is exploited for the performance analysis concerning parameter estimation with an array of low-complexity binary sensors in comparison to an ideal system featuring infinite amplitude resolution. Additionally, data-driven assessment by estimating a conservative approximation for the Fisher matrix under recursive binary sampling as implemented in $ΣΔ$-modulating analog-to-digital converters is demonstrated.