1. Human-centered Machine Learning
Sampling and parameter fitting with Hawkes Processes
Human-centered Machine Learning

2. 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.

3. 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

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

5. 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

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

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

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

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

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

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

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

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

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

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

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

17. Parameter fitting: Using cvxpy

18. Parameter fitting: Using cvxpy
Change this to maximize the likelihood

19. Live coding
Show baseline + desired output
Evaluation via sampling

20. Happy coding and holidays!
Questions?
Drop me an at
Skype: utkarsh.upadhyay