Human-centered Machine Learning

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Presentation transcript:

Human-centered Machine Learning Sampling and parameter fitting with Hawkes Processes Human-centered Machine Learning http://courses.mpi-sws.org/hcml-ws18/

We will first sample and then fit. Recap: How to fit and why sample? Infer parameters (Learning) Sampling (Predicting) Raw Data Parametrize intensity Event times drawn from: Derive Log-likelihood: Helps with: Prediction Model Checking Sanity check Gaining Intuition Simulator Summary statistics Maximum likelihood estimation: We will first sample and then fit.

Recap: What are Hawkes processes? time History, Intensity of self-exciting (or Hawkes) process: Observations: Clustered (or bursty) occurrence of events Intensity is stochastic and history dependent

Recap: How to fit a Hawkes process? time Maximum likelihood The max. likelihood is jointly convex in and (use CVX!)

Recap: How to sample from a Hawkes process time Thinning procedure (similar to rejection sampling): Sample from Poisson process with intensity Inversion sampling Generate Keep sample with prob. Keep the sample if

Coding assignment overview Parameter fitting (Assignment 2) Sampler (Assignment 1) Outline of tutorial Sanity check (provided)

Sampling: Ogata’s algorithm Ogata’s method of thinning Drawing 1 sample with intensity Accepting it with prob

Careful about exploding Sampling: Achtung I Ogata’s method of thinning Careful about exploding intensities! Include a check on i. Ensure

Sampling: Achtung II Ogata’s method of thinning History up to but not including t.

Sampling: Achtung III Ogata’s method of thinning is not an actual sample!

Sampling: Evaluation python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png Theoretical value Empirical average Juxtaposed events from sequences

Sampling: Evaluation python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png Theoretical value Empirical average Poisson (incorrect) sampler Juxtaposed events from sequences

Sampling: Evaluation python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png Theoretical value Empirical average Deterministic (incorrect) sampler Juxtaposed events from sequences

Live coding Show baseline + sanity check Show Poisson + sanity check

Parameter fitting: Problem setting (Assignment 2) Sampler (Assignment 1) samples.txt samples-test.txt

Can we do it faster for exponential kernels? Parameter fitting: Method Maximum likelihood estimation: Can we do it faster for exponential kernels? Nested sum

Parameter fitting: Using cvxpy

Parameter fitting: Using cvxpy Change this to maximize the likelihood

Live coding Show baseline + desired output Evaluation via sampling

Happy coding and holidays! Questions? Drop me an e-mail at utkarshu@mpi-sws.org Skype: utkarsh.upadhyay