Porbabilistic Machine Learning

 Stochastic Variational Inference

https://www.it.uu.se/research/systems_and_control/education/2018/pml/lectures/

Such a good source for probabilistic ML.

Integration: p(D) = INT( p(D |w)p(w)dt ) 

Optimization: D=arg max p(D|w) over w parameter

The three cornerstones: 

1. (Data) The observed data becomes useful when we have extracted knowledge from it. 

2. (Mathematical model) A mathematical model is a compact representation of the data that in precise mathematical form captures the key properties of the underlying situation. 

3. (Learning algorithm) Used to compute the unknown variables from the observed data using the model.

Key probabilistic objects (notation: D - measured data and w - unknown model variables): 

The full probabilistic model (joint distribution of all known and unknown variables present in the model) is given by

 p(D,w) = p(D |w) | {z } data distribution p(D|w) and {z } prior

 In the Bayesian setting learning amounts to computing the posterior distribution 

p(w | D) = [p(D |w) |  p(w)] /p(D) (i.e., p(D|w) likelihood , p(w) prior, p(D|w) marginal likelihood)

1)p(x, z) = p(x | z) | {z } likelihood p(z) |{z} prior

2)Marginal likelihood, p(x) = INT[ p(x, z)dz],  is highly intractable for many models of interest.

1)SVI is also called Stochastic: Markov chain Monte Carlo (MCMC), sequential Monte Carlo (SMC), stochastic variational inference 

2) Deterministic: variational inference, expectation propagation