Using an information point of view, we discuss deterministic versus stochastic tools for residual generation and evaluation for fault detection and isolation (FDI) in linear time-invariant (LTI) state-space systems. In both types of approaches to off-line FDI, residual generation can be viewed as the design of a linear transformation of a Gaussian vector (the finite-window input-adjusted observations). Several statistical isolation methods are revisited, using both a linear-transform formulation and the information content of the corresponding residuals. We formally state several multiple-fault cases, with or without causality assumptions, and discuss an optimality criterion for the most general one. New information criteria are proposed for investigating the residual optimization problem.