Efficient Likelihood-Free Inference
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1 Efficient Likelihood-Free Inference Michael Gutmann Institute for Adaptive and Neural Computation School of Informatics, University of Edinburgh 8th November 2017
2 Problem statement Likelihood-free inference: Perform statistical inference for models where 1. the likelihood function is too costly to evaluate 2. sampling simulating data from the model is possible Michael Gutmann Efficient Likelihood-Free Inference 2 / 32
3 Strain Importance Such models and inference problems occur widely Neuroscience: Simulating neural circuits Evolutionary biology: Simulating evolution Computer vision: Simulating naturalistic scenes Health science: Simulating the spread of an infectious disease 5 Strain Individual 30 Individual Time Sampled individuals Michael Gutmann Efficient Likelihood-Free Inference 3 / 32
4 Program Background Previous work Our approach Michael Gutmann Efficient Likelihood-Free Inference 4 / 32
5 Program Background Previous work Our approach Michael Gutmann Efficient Likelihood-Free Inference 5 / 32
6 Assumptions on the models Only assumption: sampling simulating data from the model is possible Models specified by a data generating mechanism e.g. stochastic nonlinear dynamical systems e.g. computer models / simulators of some complex physical or biological process Different communities use different names: Simulator-based models Stochastic simulation models Implicit models Generative (latent-variable) models Probabilistic programs Michael Gutmann Efficient Likelihood-Free Inference 6 / 32
7 Definition of simulator-based models (SBMs) Let (Ω, F, P) be a probability space. A simulator-based model is a collection of (measurable) functions g(., θ) parametrised by θ, ω Ω x θ = g(ω, θ) X (1) For any fixed θ, x θ = g(., θ) is a random variable. g(., θ) typically not available in closed form Simulation / Sampling Michael Gutmann Efficient Likelihood-Free Inference 7 / 32
8 Strengths of SBMs Direct implementation of hypotheses of how the observed data were generated. Neat interface with scientific models (e.g. from physics or biology). Modelling by replicating the mechanisms of nature that produced the observed/measured data. ( Analysis by synthesis ) Possibility to perform experiments in silico. Michael Gutmann Efficient Likelihood-Free Inference 8 / 32
9 Weaknesses of SBMs Generally elude analytical treatment. Can be easily made more complicated than necessary. Statistical inference is difficult. Michael Gutmann Efficient Likelihood-Free Inference 9 / 32
10 Weaknesses of SBMs Generally elude analytical treatment. Can be easily made more complicated than necessary. Statistical inference is difficult. Main reason: Likelihood function is too expensive to evaluate Michael Gutmann Efficient Likelihood-Free Inference 9 / 32
11 The likelihood function L(θ) Well defined but generally intractable for SBMs Probability that the model generates data like x o when using parameter value θ Data source Observation Data space Unknown properties M(θ) Model Data generation Michael Gutmann Efficient Likelihood-Free Inference 10 / 32
12 Three foundational issues 1. How should we assess whether x θ x o? 2. How should we compute the probability of the event x θ x o? 3. For which values of θ should we compute it? Data source Observation Data space Unknown properties M(θ) Model Data generation Likelihood: Probability that the model generates data like x o for parameter value θ Michael Gutmann Efficient Likelihood-Free Inference 11 / 32
13 Program Background Previous work Our approach Michael Gutmann Efficient Likelihood-Free Inference 12 / 32
14 Approximate Bayesian computation For recent review, see: Lintusaari et al (2017) Fundamentals and recent developments in approximate Bayesian computation, Systematic Biology 1. How should we assess whether x θ x o? Check whether T (x θ ) T (x o ) ɛ 2. How should we compute the proba of the event x θ x o? By counting 3. For which values of θ should we compute it? Sample from the prior (or other proposal distributions) Michael Gutmann Efficient Likelihood-Free Inference 13 / 32
15 Approximate Bayesian computation For recent review, see: Lintusaari et al (2017) Fundamentals and recent developments in approximate Bayesian computation, Systematic Biology 1. How should we assess whether x θ x o? Check whether T (x θ ) T (x o ) ɛ 2. How should we compute the proba of the event x θ x o? By counting 3. For which values of θ should we compute it? Sample from the prior (or other proposal distributions) Difficulties: Choice of T () and ɛ Typically high computational cost Michael Gutmann Efficient Likelihood-Free Inference 13 / 32
16 Implicit likelihood approximation Likelihood: Probability to generate data like x o for parameter value θ Model M(θ) Likelihood L(θ) proportion of green outcomes (1) x θ (2) x θ (3) x θ (4) x θ (5) x θ (6) x θ x o ε Data space L(θ) P N (d(x θ, x o ) ɛ) = 1 N ) N (d(x 1 (i) i=1 θ, xo ) ɛ Michael Gutmann Efficient Likelihood-Free Inference 14 / 32
17 Synthetic likelihood (Simon Wood, Nature, 2010) 1. How should we assess whether x θ x o? 2. How should we compute the proba of the event x θ x o? Compute summary statistics t θ = T (x θ ) Model their distribution as a Gaussian Compute likelihood function with T (x o ) as observed data 3. For which values of θ should we compute it? Use obtained synthetic likelihood function as part of a Monte Carlo method Michael Gutmann Efficient Likelihood-Free Inference 15 / 32
18 Synthetic likelihood (Simon Wood, Nature, 2010) 1. How should we assess whether x θ x o? 2. How should we compute the proba of the event x θ x o? Compute summary statistics t θ = T (x θ ) Model their distribution as a Gaussian Compute likelihood function with T (x o ) as observed data 3. For which values of θ should we compute it? Use obtained synthetic likelihood function as part of a Monte Carlo method Difficulties: Choice of T () Gaussianity assumption may not hold Typically high computational cost Michael Gutmann Efficient Likelihood-Free Inference 15 / 32
19 Program Background Previous work Our approach Michael Gutmann Efficient Likelihood-Free Inference 16 / 32
20 Overview of some of my work 1. How should we assess whether x θ x o? Use classification (Gutmann et al, 2014, 2017) 1. How should we assess whether x θ x o? 2. How should we compute the proba of the event x θ x o? Use density ratio estimation (Dutta et al, 2016, arxiv: ) 2. How should we compute the proba of the event x θ x o? 3. For which values of θ should we compute it? Use Bayesian optimisation (Gutmann and Corander, ) Michael Gutmann Efficient Likelihood-Free Inference 17 / 32
21 Overview of some of my work 1. How should we assess whether x θ x o? Use classification (Gutmann et al, 2014, 2017) Basic idea: Classification accuracy (discriminability) serves as distance measure Value of 1: far; Value of 1/2: close 3 2 Observed data Simulated data, mean (6,0) 3 2 Observed data Simulated data, mean (1/2,0) 1 1 y coordinate 0 y coordinate x coordinate x coordinate Michael Gutmann Efficient Likelihood-Free Inference 18 / 32
22 Overview of some of my work 1. How should we assess whether x θ x o? 2. How should we compute the proba of the event x θ x o? Use density ratio estimation (Dutta et al, 2016, arxiv: ) Basic idea: frame posterior estimation as ratio estimation problem p(θ x) = p(x θ)p(θ) p(x) = r(x, θ)p(θ) (2) Estimate ˆr(x, θ) yields estimate of the likelihood function and posterior ˆL(θ) ˆr(x o, θ), ˆp(θ x o ) = ˆr(x o, θ)p(θ). (3) Michael Gutmann Efficient Likelihood-Free Inference 19 / 32
23 Overview of some of my work 1. How should we assess whether x θ x o? Use classification (Gutmann et al, 2014, 2017) 1. How should we assess whether x θ x o? 2. How should we compute the proba of the event x θ x o? Use density ratio estimation (Dutta et al, 2016, arxiv: ) 2. How should we compute the proba of the event x θ x o? 3. For which values of θ should we compute it? Use Bayesian optimisation (Gutmann and Corander, ) Michael Gutmann Efficient Likelihood-Free Inference 20 / 32
24 Overview of some of my work 1. How should we assess whether x θ x o? Use classification (Gutmann et al, 2014, 2017) 1. How should we assess whether x θ x o? 2. How should we compute the proba of the event x θ x o? Use density ratio estimation (Dutta et al, 2016, arxiv: ) 2. How should we compute the proba of the event x θ x o? 3. For which values of θ should we compute it? Use Bayesian optimisation (Gutmann and Corander, ) Michael Gutmann Efficient Likelihood-Free Inference 21 / 32
25 Why is the ABC algorithm so expensive? 1. It rejects most samples when ɛ is small 2. It does not make assumptions about the shape of L(θ) 3. It does not use all information available 4. It aims at equal accuracy for all parameters Model M(θ) Likelihood L(θ) proportion of green outcomes (1) x θ (2) x θ (3) x θ (4) x θ (5) x θ (6) x θ x o ε Data space L(θ) P N (d(x θ, x o ) ɛ) = 1 N ( N ) 1 d(x (i) i=1 θ, xo ) ɛ Michael Gutmann Efficient Likelihood-Free Inference 22 / 32
26 Proposed solution (Gutmann and Corander, JMLR, 2016) 1. It rejects most samples when ɛ is small Don t reject samples learn from them 2. It does not make assumptions about the shape of L(θ) Model the distances, assume average distance is smooth 3. It does not use all information available Use Bayes theorem to update the model 4. It aims at equal accuracy for all parameters Prioritize parameter regions with small distances equivalent strategy applies to inference with synthetic likelihood Michael Gutmann Efficient Likelihood-Free Inference 23 / 32
27 Modelling (points 1 & 2) Data are tuples (θ i, d i ), where d i = d(x (i) θ, xo ) Model the conditional distribution of d given θ Estimated model yields approximation ˆL(θ) for any choice of ɛ ˆL(θ) P (d ɛ θ) P is probability under the estimated model. Here: Use (log) Gaussian process with squared exponential covariance function as model Approach not restricted to this model or Gaussian processes (comparison of different GP models: Järvenpää et al, 2016, arxiv: ) Michael Gutmann Efficient Likelihood-Free Inference 24 / 32
28 Data acquisition (points 3 & 4) Samples of θ could be obtained by sampling from the prior or some adaptively constructed proposal distribution Give priority to regions in the parameter space where distance d tends to be small. Use Bayesian optimization to find such regions Here: Use lower confidence bound acquisition function (e.g. Cox and John, 1992; Srinivas et al, 2012) A t (θ) = µ t (θ) }{{} η2 t v t (θ) }{{}}{{} post mean weight post var t: number of samples acquired so far Approach not restricted to this acquisition function. (new acquisition function: Järvenpää et al, 2017, arxiv: ) (4) Michael Gutmann Efficient Likelihood-Free Inference 25 / 32
29 Bayesian optimization for likelihood-free inference 5 Model based on 2 data points 95% 6 Model based on 3 data points 0 mean 90% 80% 50% distance -5 20% 10% Acquisition function 5% Competition parameter Model based on 4 data points Next parameter to try Competition parameter 5 4 Exploration vs exploitation distance 3 2 Model Data 1 0 Bayes' theorem Competition parameter Michael Gutmann Efficient Likelihood-Free Inference 26 / 32
30 Strain Example: Bacterial infections in child care centers Likelihood intractable for cross-sectional data But generating data from the model is possible Strain Parameters of interest: - rate of infections within a center - rate of infections from outside - competition between the strains Individual Individual 5 10 Strain Individual 10 Strain Strain (Numminen et al, 2013) Time Individual Individual Michael Gutmann Efficient Likelihood-Free Inference 27 / 32
31 Example: Bacterial infections in child care centers Comparison of the proposed approach with a standard population Monte Carlo ABC approach. Roughly equal results using 1000 times fewer simulations. 4.5 days with 200 cores 90 minutes with seven cores Posterior means: solid lines, credibility intervals: shaded areas or dashed lines. Competition parameter Developed Fast Method Standard Method Computational cost (log10) (Gutmann and Corander, JMLR, 2016) Michael Gutmann Efficient Likelihood-Free Inference 28 / 32
32 Example: Bacterial infections in child care centers Comparison of the proposed approach with a standard population Monte Carlo ABC approach. Roughly equal results using 1000 times fewer simulations Developed Fast Method Standard Method Developed Fast Method Standard Method Internal infection parameter External infection parameter Computational cost (log10) Computational cost (log10) Posterior means are shown as solid lines, credibility intervals as shaded areas or dashed lines. Michael Gutmann Efficient Likelihood-Free Inference 29 / 32
33 Benefits The proposed method makes the inference more efficient. allowed us to perform far more comprehensive data analysis than with standard approach (Numminen et al, 2016) Enables inference for models which were out of reach till now model of evolution where simulating a single data set took us hours (Marttinen et al, 2015) Enables easier assessment of parameter identifiability for complex models model about transmission dynamics of tuberculosis (Lintusaari et al, 2016) Michael Gutmann Efficient Likelihood-Free Inference 30 / 32
34 Open questions Model: How to best model the distance between simulated and observed data? Acquisition function: Can we find strategies which are optimal for parameter inference? Efficient high-dimensional inference: Can we use the approach to infer the joint distribution of 1000 variables? see Gutmann and Corander, JMLR, 2016 for a discussion for first answers: Michael Gutmann Efficient Likelihood-Free Inference 31 / 32
35 Summary Topic: Inference for models where the likelihood is intractable but sampling is possible Inference principle: Find parameter values for which the distance between simulated and observed data is small Problem considered: Computational cost Proposed approach: Combine statistical modeling of the distance with decision making under uncertainty (Bayesian optimization) Outcome: Approach increases the efficiency of the inference by several orders of magnitude Michael Gutmann Efficient Likelihood-Free Inference 32 / 32
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