Predict Probability Distribution with Neural Network or Monte Carlo

Question

Let's say we would like to predict price of Microsoft Stock. We have historical data and interested in predicting price distribution for future time t+1, like shown on the image.

We can use classical Neural Network approach to predict the price, but it would be a single number, not distribution.

1. Question - is there a way to make Neural Network to output the price distribution (CNN for sequences as a Neural Network)?

2. Question - as far as I know it's possible to do such thing with Markov Chain Monte Carlo. You run it a lots of times, collect the outputs, the predictions and then calculate the distribution. Could Monte Carlo approach be used with Neural Network, instead of Markov Chain (CNN for sequences as a Neural Network)?

Answer 1

MCMC methods are typically used for sampling from a distribution which otherwise would be very difficult to sample from. They do this by defining a markov chain whose equilibrium distribution is the desired distribution.

Neural networks are general function approximators. You can interpret the output as a probability distribution, in which case they can be used to model complex distributions.

So to answer your questions

1. yes

2. not really sure what you're asking.

Answer 2

1. Neural networks can only really give point estimates, not distributions. Bayesian neural networks can output distributions, so you can take a look into those but are harder to work with.

2. MCMC and deep learning are pretty disparate fields. MCMC uses random sampling to approximate a distribution, whereas deep learning approximates a function through minimizing a cost function. You wouldn't use MCMC to train a neural net, if that's what you're asking.