MNIST with a TWIST, no labels given, only probabilities Let's say we have basic MNIST dataset, and we have the same goal to predict the digit, BUT we're swapping all the labels by RED and BLUE. The label becomes BLUE with probability DIGIT/10. So if the digit is FIVE, then probability of label BLUE is 50%, so we have equal number of RED and BLUE labels. And if that digit is TWO, then probability of BLUE is only 20% and we have mostly RED.
This on one hand seems like an extremely trivial task, but on the other hand I can't see any way of approaching it, despite many years in the fields of ML and programming.

My question: what architecture or methodology could be used to approach this kind of a model? I understand that traditional feed-forward neural networks won't be able to deal with probabilistic nature of this problem. While Probabilistic Programming tools aren't suitable either. I would appreciate links to similar problems solved or any proposals on how can this be conceptually approached. Right now it doesn't seem that traditional neural network will do any good on this problem.

By the way you’ve set up the problem, perfect accuracy is impossible, and we must accept that, even if we know a digit to be a $1$, there’s only a $90\%$ chance of being red, and that’s the best we can do.
The setup is quite simple, arguable more so than the original MNIST. We have input data (features) in the pixels that we will analyze with some model (perhaps a convolutional neural network). Then, as our $Y$, we have the binary red/blue labels (discrete labels, not probabilities).
This winds up being a pretty interesting problem, since image recognition is almost synonymous with using convolutional neural networks, yet neural networks are known to be poor predictors of the probability values that we seek (“neural networks lack calibration and are overconfident”). Thus, while a CNN might be the go-to for getting accurate categorical predictions in a high signal-to-noise ratio problem like in the original MNIST digit recognition, we have a lower signal-to-noise ratio (more uncertainty, inherently) that might warrant a simpler model that gives better-calibrated probability values, such as a logistic regression. After all, for an input image of a $5$, the correct answer is that there is a $0.5$ probability of being red and $0.5$ probability of being blue. There is no definite color label.
