The large diversity of behaviors even among closely related species indicates the evolvability of the underlying neural circuit. At the same time, the behaviors must be functionally robust, but how systems can be both robust and evolvable is still an open question in neuroscience and evolution. Studying robustness and evolvability requires the mapping between genotypes and phenotypes, which is challenging to obtain experimentally. However, models of neural circuits that generate behavior can be used as a proxy of the biological system, and the mapping between model parameters and model output can be used as a proxy for the genotype-to-phenotype map. Here, we combine Bayesian inference and information theory to quantify robustness and evolvability in circuit models. We test this method using a model of the acoustic pattern recognition circuit in crickets. This circuit consists of linear filters and nonlinearities and can reproduce the full behavioral diversity of song recognition found in crickets. We demonstrate that the method correctly obtains the mapping from the model parameters to the recognition behavior and quantifies the model's evolvability and robustness. The method also identifies directions of sloppiness and stiffness and illustrates how the properties of the parameter map could shape circuit evolution. This approach of characterizing the evolvability and robustness in neural circuit models is applicable to a wide variety of circuits and systems.