When a model for a binary outcomes returns 0.9 for a given data point, that implies a 90% probably that the value is true.
Evaluating the quality of this estimates (often called measuring the calibration of the model) is even very common.
(There are some exception of course. Max margin models aren't probabilistic. And sometimes people use fixed variance parameters for their normal models, etc).
[0] https://www.sciencedirect.com/science/article/pii/S156625352...
See my other comment for more detail.
You can actually recover from this a bit. I saw a paper once where they used the Hessian to approximate the posterior as a gaussian distribution around the maximum likelihood. Can't remember what this paper was called unfortunately.
Consider tossing a coin. If I see 2 heads and 2 tails, I might report "the probability of heads is 50%". If you see 2000 heads and 2000 tails you'd also report the SAME probability estimate -- but you'd be more certain than me.
Neural networks give probability estimates. Bayesian methods (and also frequentist methods) give us probability estimates AND uncertainty.
The literature on neural network calibration seems to me to have missed this distinction.
[0] https://jeffreyling.github.io/2018/01/09/vaes-are-bayesian.h...
I'm just trying to point out that there is a dichotomy between the Bayesian and the non-Bayesian, and that the standard neural network models are non-Bayesian, and that we need Bayesianism (or something like it) to talk about (epistemic) uncertainty.
Standard neural networks are non-Bayesian, because they do not treat the neural network parameters as random variables. This includes most of the examples that have been mentioned in this thread: classifiers (which output a probability distribution over labels), networks that estimate mean and variance, and VAEs (which use Bayes's rule for the latent variable but not for the model parameters). These networks all deal with probability distributions, but that's not enough for us to call them Bayesian.
Bayesian neural networks are easy, in principle -- if we treat the edge weights of a neural network as having a distribution, then the entire neural network is Bayesian. And as you say these can be approximated, e.g. by using dropout at inference time [0], or by careful use of ensemble methods [1].
[0] https://arxiv.org/abs/1506.02142
Quote: "Deep learning tools have gained tremendous attention in applied machine learning. However such tools for regression and classification do not capture model uncertainty."
[1] https://arxiv.org/abs/1810.05546
Quote: "Ensembling NNs provides an easily implementable, scalable method for uncertainty quantification, however, it has been criticised for not being Bayesian."
Bayesians use the terms 'aleatoric' and 'epistemic' uncertainty. Aleatoric uncertainty is the part of uncertainty that says "I don't know the outcome, and I wouldn't know it even if I knew the exact model parameters", and epistemic uncertainty says "I don't even know the model".
Your example (outputting a mean and variance) is reporting a probability distribution, and it captures aleatoric uncertainty. When Bayesians talk about uncertainty or confidence, they're referring to model uncertainty -- how confident are you about the mean and the variance that you're reporting?
I’d hazard a guess that analytical solutions are intractable and numerical solutions would be infeasible.