That's where I have you fooled. I have had enough experience with "nifty" results that now I think everything is wrong. Care to post your slides? YouTube?
What most here refer to 'Neural Networks' are actually quite antiquated machine learning approaches. Those approaches have not really pushed forward "intelligent" relationship building by machines in the areas of Natural Language Processing (NLP), image processing/recognition, sequence learning, and the likes. Only with the advent of deep learning networks have many problems, that today are not even considered problems anymore, been solved. I strongly recommend to those, interested, to take a close look at the following summary of deep learning. Otherwise, the introductory course, taught by Andrew Ng, is an excellent course aimed at total beginners. https://www.coursera.org/specializations/deep-learning
You inherently still make very strong assumptions about the distribution. You might get a better idea about the parameterization of the distribution you make an assumption about but your approach does not really get close to estimating the type of distribution of data because you make an apriori assumption about the distribution. I am not making a case that knowing the distributional nature of financial data is useless, in fact, it is a very valuable property in financial trading, just saying that your described approach does not really do what you set out to achieve.
In your example, you assume a normal distribution and derive the parameterization. To my knowledge, there is currently no machine learning or neural network approach that, for example, derives the type of distribution without making apriori assumptions. Otherwise, the entire stochastic approach to derivatives pricing would be a headache-inducing detour of a much simpler approach. Such an approach does not exist hence the use my market practitioners of the log normal distribution assumptions because we currently do not know better and because the log normal lends itself well to mathematical function transformations.
A gaussian mixture model isn't really making a strong gaussian assumption, in my opinion (I'm often lose with my definitions...) Another option (not a great one, it's inefficient) is to pose your regression problem as a classification problem using cross entropy loss. The outputs will be a probability mass function. Obviously, you'd need a ton of outputs to do anything interesting.
Check out AlphaPy here: https://github.com/ScottfreeLLC/AlphaPy The key for trading is to use the probabilities generated by your classifier as trading signals. The higher the probability, the higher the confidence in the signal. For neural networks, I would focus on trading regimes (e.g., volatility) to select the right system rather than trying to apply vanilla LSTMs based on price action.
A very neat project. Many useful features some of which I plan on adopting to add to my architecture. Thanks for sharing this. One of the very few gems on this site.
I'll dig out my notes and see if they make sense without the verbage -- regardless, it's (I think) an interesting collection. About a dozen slides, IIRC...