Introduction to Probabilistic Machine Learning with PyMC3

Daniel Emaasit

Data Scientist

Haystax Technology

Bayesian Data Science DC Meetup

April 26, 2018

Data Science & Cybersecurity Meetup


Download slides & code:

Application (1/3)

  • Optimizing expensive functions in autonomous vehicles (using Bayesian optimization).

Application (2/3)

  • Probabilistic approach to ranking & matching gamers.

Application (3/3)

  • Supplying internet to remote areas (using Gaussian processes).

Media Attention

Academics flocking to Industry

Intro to Probabilistic Machine Learning

ML: A Probabilistic Perspective (1/3)

  • Probabilistic ML:
  • An Interdisciplinary field that develops both the mathematical foundations and practical applications of systems that learn models of data. (Ghahramani, 2018)

ML: A Probabilistic Perspective (2/3)

  • A Model:
  • A model describes data that one could observe from a system (Ghahramani, 2014)
  • Use the mathematics of probability theory to express all forms of uncertainty

Generative Process


ML: A Probabilistic Perspective (3/3)

P(\theta \mid y) = \frac{P(y \mid \theta) \, P(\theta)}{P(y)}
P(θy)=P(yθ)P(θ)P(y)P(\theta \mid y) = \frac{P(y \mid \theta) \, P(\theta)}{P(y)}
  • \(\mathbf{\theta}\) = parameters e.g. coefficients

\(p(\mathbf{\theta})\) = prior over the parameters

\(p(\textbf{y} \mid \mathbf{\theta})\) = likelihood given the covariates & parameters

\(p(\textbf{y})\) = data distribution to ensure normalization

\(p(\mathbf{\theta} \mid \textbf{y})\) = posterior over the parameters, given observed data


Probabilistic ML Vs Traditional ML

Algorithmic ML Probabilistic ML
Examples K-Means, Random Forest GMM, Gaussian Process
Specification Model + Algorithm combined Model & Inference separate
Unknowns Parameters Random variables
Inference Optimization (MLE) Bayes (MCMC, VI)
Regularization Penalty terms Priors
Solution Best fitting parameter Full posterior distribution

Limitations of deep learning

  • Deep learning systems give amazing performance on many benchmark tasks but they are generally:
  • very data hungry (e.g. often millions of examples)
  • very compute-intensive to train and deploy (cloud GPU resources)
  • poor at representing uncertainty
  • easily fooled by adversarial examples
  • finicky to optimize: choice of architecture, learning procedure, etc, require expert knowledge and experimentation
  • uninterpretable black-boxes, lacking in transparency, difficult to trust

Probabilistic Programming

Probabilistic Programming (1/2)

  • Probabilisic Programming (PP) Languages:
  • Software packages that take a model and then automatically generate inference routines (even source code!) e.g Pyro, Stan, Infer.Net, PyMC3, TensorFlow Probability, etc.

Probabilistic Programming (2/2)

  • Steps in Probabilisic ML:
  • Build the model (Joint probability distribution of all the relevant variables)
  • Incorporate the observed data
  • Perform inference (to learn distributions of the latent variables)

Resources to get started

Thank You!



  • Ghahramani, Z. (2015). Probabilistic machine learning and artificial intelligence. Nature, 521(7553), 452.
  • Bishop, C. M. (2013). Model-based machine learning. Phil. Trans. R. Soc. A, 371(1984), 20120222.
  • Murphy, K. P. (2012). Machine learning: a probabilistic perspective. MIT Press.
  • Barber, D. (2012). Bayesian reasoning and machine learning. Cambridge University Press.