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Gaussian processes are ubiquitous in nature and engineering. A case in pointis a class of neural networks in the infinite-width limit, whose priorscorrespond to Gaussian processes. Here we perturbatively extend thiscorrespondence to finite-width neural networks, yielding non-Gaussian processesas priors. The methodology developed herein allows us to track the flow ofpreactivation distributions by progressively integrating out random variablesfrom lower to higher layers, reminiscent of renormalization-group flow. Wefurther develop a perturbative procedure to perform Bayesian inference withweakly non-Gaussian priors.

Non-Gaussian processes and neural networks at finite widths