We had used Bayesian learning for house price prediction project, notebook is available at [0]. Purpose of this blog is to have quick summary of concept involved.

• Bayesian learning allows us to have distribution for parameters rather than point estimate.
• We keep sampling values of parameters say 10000 times. Histogram of last 6000 sample represent approximate posterior of parameter.
• Every parameters, latent variables, output have distribution associated with them.
• Top parameters in the hierarchy will have prior associated with them.
  • We sample for these top parameters
  • Calculate corresponding value for all latent variable coming down the tree
  • Finally at output calculate likelihood against observed data
• Different MCMC algorithms differs in two steps:
  • How to jump to next to next sample
  • How to decide whether next sample is acceptable or not
• Metropolis algorithm:
  • Jumps considering normal distribution at previous parameter value and some fixed standard deviation
  • Acceptance test : random(0,1) < ratio of (Pnew/Pold)
    • When Pnew is higher we will definitely accept
    • Else probability of acceptance depends on how large Pold is
    • If Pold = 2 * Pnew it is 50 %
  • Pnew = liklihood_new * prior_new
  • Pold = liklihood_old * prior_old
• Example of other MCMC algorithms:
  • Metropolis-Hastings
  • The Gibbs Sampler
  • Hamiltonian MCMC
  • The No-U-Turn Sampler (NUTS)

Reference

[0] : https://github.com/arcarchit/House-Prices-Advanced-Regression-Techniques/blob/master/notebooks/Bayesian%2BMaster.ipynb

[1] : http://twiecki.github.io/blog/2015/11/10/mcmc-sampling/

[2] : https://www.quantstart.com/articles/Bayesian-Inference-of-a-Binomial-Proportion-The-Analytical-Approach