Thanks for writing back! I appreciate the plan to integrate the two architectures. On that front, it might be interesting to have a future research section - like what would be uniquely good about this architecture if scaled up?

On ‘usefulness’ I think I’m still at my original question - it seems like an open theoretical q to say that the combination of a tripled-or-greater training budget, data size budget of the NN, and probably a close to triple or greater inference budget, the costs of the architecture you described, cannot be closely approximated by the “fair equivalent”-ly sized MLP.

I hear you that the architecture can do more, but can you talk about this fair size question I have? That is, if a PGM of the same size as your original network in terms of weights and depth is as effective, then we’d still have a space savings to just have the two networks (MLP and PGM) side by side.

Thanks again for publishing!

That’s a fair question. You’re right that on paper a uGMM neuron looks like it “costs” ~3× an MLP weight. But there are levers to balance that. For example, the paper discusses parameter tying, where the Gaussian component means are tied directly to the input activations. In that setup, each neuron only learns the mixture weights and variances, which cuts parameters significantly while still preserving probabilistic inference. The tradeoff may be reduced expressiveness, but it shows the model doesn’t have to be 3x heavier.

More broadly: traditional graphical models were largely intractable at deep learning scale until probabilistic circuits, which introduced tractable probabilistic semantics without exploding parameter counts. Circuits do this by constraining model structure. uGMM-NN sits differently: it brings probabilistic reasoning inside dense architectures.

So while compute cost is real, the “fair comparison” isn’t just params-per-weight, it’s also about what kinds of inference the model can do at all, and the added interpretability of mixture-based neurons, which traditional MLP neurons don’t provide - it shares some spirit with recent work like KAN, but tackles the problem through probabilistic modeling rather than spline-based function fitting.