The exponential function is prone to overflow if the scale of the input features are “large”. It’s good practice to always standardize (subtract mean, divide by std) all inputs and targets before you run a GLM. I believe Bambi does this for you automatically. See this discussion, for example. MH probably overflows more than NUTS because, prior to scale tuning, the sampler is drunkenly taking large steps into idiotic regions of the parameter space where theta ends up being too big.
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