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Generalized Linear Model (GLMs) analysis is a popular tool in psychophysics and neuroscience for inferring the relative influence of various experimental factors onto choices, reaction times, neural activity and other observables. However, GLMs are intrinsically limited by their linearity assumption, and can lead to severe misattribution errors when (correlated) regressors contribute nonlinearly to the observed response. We show how this framework can be expanded to capture nonlinear functions. First, Generalized Additive Models (GAMs) allow to capture a nonlinear contribution for each regressor. A Gaussian Processes (GP) treatment of GAMs allows to recover the posterior distribution for each nonlinear mapping. Second, as neuroscience is often interested in the interaction of cognitive factors, we present a Bayesian treatment of Generalized Multilinear Models (GMMs) that capture multilinear interactions between different sets of regressors. GMMs can be applied eg when inferring the modulation of sensory processing by additional factors such as attention factors. Merging the frameworks of GAMs and GMMs yield Generalized Unrestricted Models (GUMs), a highly versatile and interpretable environment to capture cognitive determinants of behavior and neural activity. Crucially, these models can be efficiently estimated, even with limited dataset; Bayesian techniques can be applied to test which model is best supported by data.