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Overview

  • Motivation for likelihood-free inference
  • Basics of ABC
  • pyABC: Features and usage

Why?

Parameter inference

(Likelihood-free) Bayesian inference

  • goal: infer parameters $\theta$ given data $y_\text{obs}$, i.e. analyze the posterior distribution
  • common optimization and sampling methods require the (unnormalized) likelihood
  • can happen: numerical evaluation infeasible
  • ... but possible to simulate data $y\sim\pi(y|\theta)$

Multi-scale models

Multiscale models based on Hasenauer et al., J. Coup. Sys. and Mult. Dyn. 2015

Example: Modeling tumor growth

based on Jagiella et al., Cell Systems 2017 Tumor gif
  • cells modeled as interacting stochastic agents, dynamics of extracellular substances by PDEs
  • simulate up to 106 cells
  • 10s - 1h for one forward simulation
  • 7-18 parameters

What we tried
  • multi-start local methods
    • deterministic gradient descent
      • Levenberg-Marquardt
      • interior-point
      • trust-region
    • stochastic gradient descent
    • Bayesian optimization
  • global methods
    • simulated annealing
    • > 20 particle methods
    • enhanced scatter search

Failed

Worked
  • multi-start local methods
    • deterministic gradient descent
      • Levenberg-Marquardt
      • interior-point
      • trust-region
    • stochastic gradient descent
    • Bayesian optimization
  • global methods
    • simulated annealing
    • > 20 particle methods
    • enhanced scatter search
 ---

How to infer parameters for complex stochastic models?

ABC

Approximate Bayesian Computation

Rejection ABC

With distance $d$, threshold $\varepsilon>0$, summary statistics $s$:
until $N$ acceptances:
  1. sample $\theta\sim g(\theta)$
  2. simulate data $y\sim\pi(y|\theta)$
  3. accept $\theta$ if $d(s(y), s(y_\text{obs}))\leq\varepsilon$

A "derivation"

Rejection sampling

Background: Want to sample from $f$, but can only sample from $g \gg f$.
until $N$ acceptances:
  1. sample $\theta\sim g(\theta)$
  2. accept $\theta$ with probability $\propto \frac{f(\theta)}{g(\theta)}$
Accepted $\theta$ are independent samples from $f(\theta)$.
Let $f=\pi(\theta|y_\text{obs}), g=\pi(\theta) \Rightarrow \frac{\pi(\theta|y_\text{obs})}{\pi(\theta)} \propto \pi(y_\text{obs}|\theta)$
  • not available
  • idea: circumvent likelihood evaluation by simulating data and matching them to the observed data

Likelihood-free rejection sampling

until $N$ acceptances:
  1. sample $\theta\sim \pi(\theta)$
  2. simulate data $y\sim\pi(y|\theta)$
  3. accept $\theta$ if $y=y_\text{obs}$
  • Acceptance probability: $\mathbb{P}[y_\text{obs}]$
  • can be small in particular for continuous data
  • idea: accept simulations that are similar to $y_\text{obs}$

Rejection ABC

With distance $d$, threshold $\varepsilon>0$:
until $N$ acceptances:
  1. sample $\theta\sim \pi(\theta)$
  2. simulate data $y\sim\pi(y|\theta)$
  3. accept $\theta$ if $d(y, y_\text{obs})\leq\varepsilon$
  • curse of dimensionality: if the data are too high-dimensional, the probability of simulating similar data sets is small
  • idea: create an informative lower-dimensional representation via summary statistics
  • ideally minimal sufficient statistics

Toy example

$y\sim\mathcal{N}(2(\theta-2)\theta(\theta+2), 1+\theta^2)$, $y_\text{obs}=2$
$\pi(\theta) = U[-3,3]$
$d=|{\cdot}|_1$
$N=1000$ acceptances

Approximate Bayesian Posterior

We want: \[\pi(\theta|y_\text{obs}) \propto \color{red}{p(y_\text{obs}|\theta)}\pi(\theta)\]

We get: \[\pi_{ABC}(\theta|s(y_\text{obs})) \propto \color{red}{\int I(\{d(s(y), s(y_\text{obs})) \leq \varepsilon\})p(y|\theta)\operatorname{dy}}\pi(\theta) \approx \frac{1}{N} \sum_{i=1}^N\delta_{\theta^{(i)}}(\theta)\] with distance $d$, threshold $\varepsilon>0$, and summary statistics $s$

Sources of approximation errors in ABC

  • model error (as for every model of reality)
  • Monte-Carlo error (as for sampling in general)
  • summary statistics
  • epsilon threshold
Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.
John Tukey 1962

Efficient samplers

  • Rejection ABC, the basic ABC algorithm, can be inefficient due to repeatedly sampling from the prior
  • smaller $\varepsilon$ leads to lower acceptance rates
  • many (likelihood-based) algorithms like IS, MCMC, SMC, SA have ABC-fied versions
pyABC logo white
github.com/icb-dcm/pyabc
Klinger et al., Bioinformatics 2018
pyABC logo black Points

Easy to use

                
# define problem
abc = pyabc.ABCSMC(model, prior, distance)

# pass data
abc.new(database, observation)

# run it
abc.run()

Parallel backends: 1 to 1,000s cores

Parallelization backends

Parallelization strategies

Klinger et al., CMSB Proceedings 2017 Parallelization strategies

Example: Tumor growth model

based on Jagiella et al., Cell Systems 2017

Define summary statistics

Tumor summary statistics
  • 400 cores
  • 2 days
  • 1.8e6 simulations
Tumor KDEs
ABC worked where many other methods had failed.
Tumor final KDE
ABC worked where many other methods had failed.

Uncertainty-aware predictions, easy data integration.

Algorithmic details

Adaptive population sizes

Klinger et al., CMSB Proceedings 2017 Adaptive population sizes idea: adapt population size trying to match a target accuracy

Self-tuning distance functions

based on Prangle, Bayesian Analysis 2015
Adaptive distances

Measurement noise

Schälte et al., Bioinformatics 2020
How to efficiently account for measurement noise in ABC? Measurement noise methods

And ...

And

...


Joint initiative to perform inference for multi-cellular models Morpheus
Morpheus toolbox: Staruß et al., Bioinformatics 2014

Summary

Summary

  • parameter estimation when we cannot evaluate the likelihood is challenging
  • ABC allows for reliable statistical inference
  • pyABC provides an easy-to-use framework
Not everything is a nail.

Literature

  • Sisson, Scott A., Yanan Fan, and Mark Beaumont, eds. "Handbook of approximate Bayesian computation." CRC Press, 2018.
  • Beaumont, Mark A. "Approximate Bayesian computation in evolution and ecology." Annual review of ecology, evolution, and systematics 41 (2010): 379-406.
  • Martin, Gael M., David T. Frazier, and Christian P. Robert. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century." arXiv preprint arXiv:2004.06425 (2020).

Acknowledgments

Thanks to: Jan Hasenauer (and the whole lab), Elba Raimúndez-Álvarez, Emad Alamoudi, the FitMultiCell team, ...

Thanks! Questions?