Multifidelity Approaches to Approximate Bayesian Computation

Transcription

1 Multifidelity Approaches to Approximate Bayesian Computation Thomas P. Prescott Wolfson Centre for Mathematical Biology University of Oxford Banff International Research Station 11th 16th November 2018

2 Approximate Bayesian computation Multifidelity methods Multifidelity modelling example: Repressilator Multifidelity ABC Early accept/reject multifidelity ABC Performance analysis Repressilator revisited Implementation and future extensions 13/11/18 Early Accept/Reject Multifidelity ABC 2

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4 Approximate Bayesian computation Multifidelity methods Multifidelity modelling example: Repressilator Multifidelity ABC Early accept/reject multifidelity ABC Performance analysis Repressilator revisited Implementation and future extensions 13/11/18 Early Accept/Reject Multifidelity ABC 4

5 Bayesian parameter inference Real-world system defined by uncertain parameters θ π( ) belonging to a prior distribution. Summary statistics, yobs, are gathered from observed data. A model p( θ) is a map from θ to a distribution. Distribution p( θ) on an output space containing y obs. Likelihood of observed data is p(yobs θ). Simulation of a model is a draw y p( θ) from the distribution. Cost of simulation is the time taken to simulate y. Model here refers to combination of mathematical model and numerical implementation to generate y. 13/11/18 Early Accept/Reject Multifidelity ABC 5

6 Bayesian parameter inference Posterior distribution for parameters is: p(θ y obs ) = p(y obs θ)π(θ) p(yobs θ)π(θ) dθ. Assume that the likelihood, p(y obs θ), is unavailable. 13/11/18 Early Accept/Reject Multifidelity ABC 6

7 Approximate Bayesian computation Attempt 1: Approximate likelihood, p(y obs θ), using Monte Carlo simulation: Simulate y p( θ) and set ˆp(y obs θ) = I(y = y obs ). Expectation E(ˆp(y obs ) θ) = p(y obs θ). 13/11/18 Early Accept/Reject Multifidelity ABC 7

8 Approximate Bayesian computation Attempt 1: Approximate likelihood, p(y obs θ), using Monte Carlo simulation: Simulate y p( θ) and set ˆp(y obs θ) = I(y = y obs ). Expectation E(ˆp(y obs ) θ) = p(y obs θ). Impractical in large output space. 13/11/18 Early Accept/Reject Multifidelity ABC 7

9 Approximate Bayesian computation Attempt 2: Consider neighbourhood Ω(ɛ) = {y d(y, y obs ) < ɛ} around y obs. Distance d(y, y obs ) in output space. Threshold distance ɛ. 13/11/18 Early Accept/Reject Multifidelity ABC 8

10 Approximate Bayesian computation Attempt 2: Consider neighbourhood Ω(ɛ) = {y d(y, y obs ) < ɛ} around y obs. Distance d(y, y obs ) in output space. Threshold distance ɛ. ABC approximation to the likelihood: for y p( θ), p(y obs θ) p(y Ω(ɛ) θ) =: p ABC (y obs θ) Requires lim ɛ 0 p(y Ω(ɛ) θ) = p(y obs θ). 13/11/18 Early Accept/Reject Multifidelity ABC 8

11 ABC: Monte Carlo rejection sampling For i = 1,..., N: Select θ i π( ) from prior and simulate y i p( θ i ). If y i Ω(ɛ) is in the ɛ-neighbourhood of y obs then accept θ i into sample, else reject. Given θ i, the accept/reject decision is a random weight w(θ i ) = I(y i Ω(ɛ)) with expectation E(w(θ i )) = p ABC (y obs θ i ) p(y obs θ i ). 13/11/18 Early Accept/Reject Multifidelity ABC 9

12 Monte Carlo ABC: Computational bottleneck Simulate y i p( θ i ) for i = 1,..., N for N large. Aim to reduce computational burden. Efficiently exploring parameter space can reduce N: Markov chain Monte Carlo (MCMC); Seqential Monte Carlo (SMC). 13/11/18 Early Accept/Reject Multifidelity ABC 10

13 Monte Carlo ABC: Computational bottleneck Simulate y i p( θ i ) for i = 1,..., N for N large. Aim to reduce computational burden. Efficiently exploring parameter space can reduce N: Markov chain Monte Carlo (MCMC); Seqential Monte Carlo (SMC). Goal: reduce simulation cost within rejection sampling framework. 13/11/18 Early Accept/Reject Multifidelity ABC 10

14 Approximate Bayesian computation Multifidelity methods Multifidelity modelling example: Repressilator Multifidelity ABC Early accept/reject multifidelity ABC Performance analysis Repressilator revisited Implementation and future extensions 13/11/18 Early Accept/Reject Multifidelity ABC 11

15 Alternative models Original model p( θ): distribution on output space containing y obs. Alternative summary statistics ỹ obs induce new output space. Consider new model p( θ)......with new cost, c, of simulating ỹ p( θ). We assume the new model is cheaper: E( c) E(c). 13/11/18 Early Accept/Reject Multifidelity ABC 12

16 Cheaper simulations Early stopping: simulate to t < T ; Coarser discretisations of space/time; Reduce model dimension; Langevin, mean field, or deterministic approximations; Surrogate models and regressions; Steady state analysis. Note: we re assuming that θ well-defines both p and p. 13/11/18 Early Accept/Reject Multifidelity ABC 13

17 Approximate Bayesian computation Using alternative model p: Consider neighbourhood Ω( ɛ) = {ỹ d(ỹ, ỹ obs ) < ɛ} around ỹ obs. Distance d(ỹ, ỹ obs ) in output space. Threshold distance ɛ. 13/11/18 Early Accept/Reject Multifidelity ABC 14

18 Approximate Bayesian computation Using alternative model p: Consider neighbourhood Ω( ɛ) = {ỹ d(ỹ, ỹ obs ) < ɛ} around ỹ obs. Distance d(ỹ, ỹ obs ) in output space. Threshold distance ɛ. ABC approximation to a different likelihood: for ỹ p( θ), p(ỹ obs θ) p ABC (ỹ obs θ) = p(ỹ Ω( ɛ) θ). Requires lim ɛ 0 p(ỹ Ω( ɛ) θ) = p(ỹ obs θ). 13/11/18 Early Accept/Reject Multifidelity ABC 14

19 ABC: Monte Carlo rejection sampling For i = 1,..., N: Select θ i π( ) from prior and simulate ỹ i p( θ i ). If ỹ i Ω( ɛ) is in the ɛ-neighbourhood of ỹ obs then accept θ i into sample, else reject. Given θ i, the accept/reject decision is a random weight w(θ i ) = I(ỹ i Ω( ɛ)) with expectation E( w(θ i )) = p ABC (ỹ obs θ i ) p(ỹ obs θ i ). 13/11/18 Early Accept/Reject Multifidelity ABC 15

20 Multifidelity terminology High-fidelity model p( θ): HF simulation y p( θ) with cost c; HF accept/reject weight w(θ) = I(y Ω(ɛ)). Low-fidelity model p( θ): LF simulation ỹ p( θ) with cost c; LF accept/reject weight w(θ) = I(ỹ Ω( ɛ)). 13/11/18 Early Accept/Reject Multifidelity ABC 16

21 Example: Repressilator Three genes G 1, G 2, and G 3, transcribed and translated into proteins P 1, P 2, and P 3. P 1 represses G 2 transcription; P 2 represses G 3 transcription; Elowitz & Leibler, Nature, P 3 represses G 1 transcription. Goal: identify n and K h in repression f (p) = K n h /(K n h + pn ). 13/11/18 Early Accept/Reject Multifidelity ABC 17

22 Multifidelity models 1000 mrna mrna mrna 3 Molecule Count Protein Protein Protein 3 Molecule Count Time Time Time Simulate network with: High fidelity Stochastic Simulation Algorithm (Gillespie). Low fidelity tau-leap discretisation. 13/11/18 Early Accept/Reject Multifidelity ABC 18

23 Low-fidelity rejection sampling is biased Matching estimator values False positive False negative Threshold , y obs ) (left) and n (right). Each Distance d(y, yobs ) varies with d(y point (N = 104 ) corresponds to parameter sampled from uniform prior. Observed data yobs = y obs is synthetic data using n = 2. 13/11/18 Early Accept/Reject Multifidelity ABC 19

24 Low-fidelity rejection sampling is biased High fidelity: Low fidelity: E(w(θ)) = p(y Ω(ɛ) θ) p(y obs θ). E( w(θ)) = p(ỹ Ω( ɛ) θ) p(ỹ obs θ) p(y obs θ). Can we combine low and high-fidelity models to reduce simulation costs, but without introducing bias? 13/11/18 Early Accept/Reject Multifidelity ABC 20

25 Approximate Bayesian computation Multifidelity methods Multifidelity modelling example: Repressilator Multifidelity ABC Early accept/reject multifidelity ABC Performance analysis Repressilator revisited Implementation and future extensions 13/11/18 Early Accept/Reject Multifidelity ABC 21

26 Early rejection First idea: Use low-fidelity simulation to decide when it s worth simulating the high fidelity model Matching estimator values False positive False negative Threshold /11/18 Early Accept/Reject Multifidelity ABC 22

27 Early rejection First idea: Use low-fidelity simulation to decide when it s worth simulating the high fidelity model Matching estimator values False positive False negative Threshold Simulate ỹ i p( θ i ). Define continuation probability α(ỹ i ) (0, 1]. w.p. 1 α: Reject without simulating y i Else, simulate y i : If y i Ω(ɛ) then accept (with appropriate weight); Else, reject. 13/11/18 Early Accept/Reject Multifidelity ABC 22

28 Early rejection Formulate as early rejection weight: w er (θ) = I(U < α(ỹ)) w(θ) α(ỹ) for unit uniform r.v. U U(0, 1). w er (θ) is an unbiased estimator of p(y Ω(ɛ) θ), and we avoid some high-fidelity simulations. 13/11/18 Early Accept/Reject Multifidelity ABC 23

29 Early rejection Matching estimator values False positive False negative Threshold w er (θ) = I(U < α(ỹ)) w(θ) α(ỹ) Choosing α(ỹ): Make α(ỹ) small when ỹ suggests y / Ω(ɛ) is likely. Converse requires larger α(ỹ) when y Ω(ɛ) more likely. See also: Lazy ABC (Prangle 2016), Delayed Acceptance ABC (Everitt & Rowinska 2017). 13/11/18 Early Accept/Reject Multifidelity ABC 24

30 Re-examine converse w er (θ) = I(U < α(ỹ)) w(θ). α(ỹ) Larger α(ỹ) when y Ω(ɛ) more likely. 13/11/18 Early Accept/Reject Multifidelity ABC 25

31 Re-examine converse w er (θ) = I(U < α(ỹ)) w(θ). α(ỹ) Larger α(ỹ) when y Ω(ɛ) more likely. More likely to simulate high-fidelity model if low-fidelity simulation is close to the data. Why not accept early without simulating? 13/11/18 Early Accept/Reject Multifidelity ABC 25

32 Re-examine converse w er (θ) = I(U < α(ỹ)) w(θ). α(ỹ) Larger α(ỹ) when y Ω(ɛ) more likely. More likely to simulate high-fidelity model if low-fidelity simulation is close to the data. Why not accept early without simulating? Early acceptance requires positive weight without high-fidelity simulation: not possible using w er. 13/11/18 Early Accept/Reject Multifidelity ABC 25

33 Early decision Second idea: Sometimes make an early accept/reject decision based on the low-fidelity simulation alone Matching estimator values False positive False negative Threshold /11/18 Early Accept/Reject Multifidelity ABC 26

34 Early decision Second idea: Sometimes make an early accept/reject decision based on the low-fidelity simulation alone Matching estimator values False positive False negative Threshold Simulate ỹ i p( θ i ). Make an early decision w. Define probability η (0, 1]. w.p. 1 η: Use early decision. Else, simulate y i : Correct early decision, based on w. Correction appropriately weighted. 13/11/18 Early Accept/Reject Multifidelity ABC 26

35 Early decision Formulate as early decision weight: w ed (θ) = w(θ) + for unit uniform r.v. U U(0, 1). I(U < η) η [w(θ) w(θ)] w ed (θ) is an unbiased estimator of p(y Ω(ɛ) θ), and we avoid some high-fidelity simulations. 13/11/18 Early Accept/Reject Multifidelity ABC 27

36 Combining existing approaches Early rejection use ỹ to determine when to generate y: w er = I(U < α(ỹ)) I(y Ω(ɛ)). α(ỹ) Early decision use ỹ to determine the early decision: w ed = I(ỹ Ω( ɛ)) + I(U < η) η Special cases of multifidelity ABC. ( ) I(y Ω(ɛ)) I(ỹ Ω( ɛ)). 13/11/18 Early Accept/Reject Multifidelity ABC 28

37 Early accept/reject multifidelity ABC Combine approaches into a multifidelity weight: w mf (θ) = I(ỹ Ω( ɛ)) + I(U < η(ỹ)) η(ỹ) ( I(y Ω(ɛ)) I(ỹ Ω( ɛ)) ) 13/11/18 Early Accept/Reject Multifidelity ABC 29

38 Early accept/reject multifidelity ABC Combine approaches into a multifidelity weight: w mf (θ) = I(ỹ Ω( ɛ)) + I(U < η(ỹ)) η(ỹ) ( I(y Ω(ɛ)) I(ỹ Ω( ɛ)) ) Natural form of η(ỹ) to give early accept/reject multifidelity ABC: for η 1, η 2 (0, 1]. η(ỹ) = η 1 I(ỹ Ω( ɛ)) + η 2 I(ỹ / Ω( ɛ)) E(w mf (θ)) = p(y Ω(ɛ) θ). 1 η 1 and 1 η 2 are early accept/reject probabilities. 13/11/18 Early Accept/Reject Multifidelity ABC 29

39 Early accept/reject multifidelity ABC There are 6 cases (at most 4 values) of w mf (θ): 1 ỹ Ω( ɛ) & U η 1 0 ỹ / Ω( ɛ) & U η 2 1 ỹ w mf (θ) = Ω( ɛ) & y Ω(ɛ) & U < η 1 0 ỹ / Ω( ɛ) & y / Ω(ɛ) & U < η 2 1 1/η 1 ỹ Ω( ɛ) & y / Ω(ɛ) & U < η /η 2 ỹ / Ω( ɛ) & y Ω(ɛ) & U < η 2 13/11/18 Early Accept/Reject Multifidelity ABC 30

40 Early accept/reject multifidelity ABC There are 6 cases (at most 4 values) of w mf (θ): 1 Predicted positive (unchecked) 0 Predicted negative (unchecked) 1 True positive (checked) w mf (θ) = 0 True negative (checked) 1 1/η 1 False positive (checked) 0 + 1/η 2 False negative (checked) 13/11/18 Early Accept/Reject Multifidelity ABC 30

41 Approximate Bayesian computation Multifidelity methods Multifidelity modelling example: Repressilator Multifidelity ABC Early accept/reject multifidelity ABC Performance analysis Repressilator revisited Implementation and future extensions 13/11/18 Early Accept/Reject Multifidelity ABC 31

42 Early accept/reject multifidelity ABC algorithm For i = 1,..., N: Generate θ i π( ). Simulate (low-fidelity) ỹ i p( θ i ). Calculate w = I(ỹ i Ω( ɛ)). Set w i = w and η = η 1 w + η 2 (1 w). If U < η: Simulate (high fidelity) y i p( θ). Calculate w = I(y i Ω(ɛ)). Update wi = w i + (w w)/η. Return {w i, θ i } N i=1. 13/11/18 Early Accept/Reject Multifidelity ABC 32

43 Weighted sample Input to algorithm is continuation probabilities η 1, η 2. Output of algorithm is weighted sample {w i, θ i } N i=1. Each weight w i incurs simulation cost T i. Use sample in ABC estimator for a function, F (θ), of uncertain parameters: i µ ABC (F ) = w if (θ i ) j w F (θ)p ABC (y obs θ)π(θ) dθ. j 13/11/18 Early Accept/Reject Multifidelity ABC 33

44 Quality of a sample Monte Carlo sample {w i, θ i } N i=1 : Effective Sample Size: ESS = ( i w i) 2 i w i 2 = N ( i w i/n) 2 i w i 2/N Simulation time: T tot = T i 13/11/18 Early Accept/Reject Multifidelity ABC 34

45 Quality of a sample Efficiency is ratio of ESS and simulation time: ESS T tot = ( i w i/n) ( 2 i w i 2/N) ( i T i/n) Approximation is in the limit as N. E(w i)2 E(w 2 i )E(T i) Goal: Tune inputs η 1, η 2 to maximise efficiency (in the limit). 13/11/18 Early Accept/Reject Multifidelity ABC 35

46 Maximising efficiency ESS E(w mf) 2 T tot E(wmf 2 )E(T ) over θ π( ) Numerator: E(w mf ) = p(y Ω(ɛ)) is independent of η 1, η 2. 13/11/18 Early Accept/Reject Multifidelity ABC 36

47 Maximising efficiency ESS E(w mf) 2 T tot E(wmf 2 )E(T ) over θ π( ) Numerator: E(w mf ) = p(y Ω(ɛ)) is independent of η 1, η 2. Maximising efficiency over η 1, η 2 (0, 1] equivalent to minimising product E(w 2 mf )E(T i). 13/11/18 Early Accept/Reject Multifidelity ABC 36

48 Tradeoff: Reduced computational burden... Simulation times: Low-fidelity simulation ỹ p( θ) costs c. High-fidelity simulation y p( θ) costs c. w mf costs T = c + ci(u < η(ỹ)). Expected time to construct w mf (θ) is therefore E(T ) = E( c) + η 1 E[cI(ỹ Ω( ɛ))] + η 2 E[cI(ỹ / Ω( ɛ))]. Depends on simulation cost c relative to c. 13/11/18 Early Accept/Reject Multifidelity ABC 37

49 Tradeoff:...for increased variance. Second moment increases as η i decreases: E(w 2 mf ) = P(y Ω(ɛ)) ( ) P(ỹ Ω( ɛ) y / Ω(ɛ)) η 1 ( ) P(ỹ / Ω( ɛ) y Ω(ɛ)) η 2 Depends on false positive and false negative rates. 13/11/18 Early Accept/Reject Multifidelity ABC 38

50 Optimal early accept/reject continuation probabilities Analytical result: efficiency maximised at ( (η1, η2) = 1 P(y / Ω(ɛ) ỹ Ω( ɛ)) λ E(c ỹ Ω(ɛ))/E( c), P(y Ω(ɛ) ỹ / Ω( ɛ)) ) E(c ỹ / Ω( ɛ))/e( c) If probability P(y / Ω(ɛ) ỹ Ω( ɛ)) of needing to correct a positive prediction is small, then check less often. If the cost E(c ỹ Ω(ɛ)) of checking a positive prediction is large relative to low-fidelity simulation cost E( c), then check less often. Similar for negative predictions. Less effective when λ = P(y Ω(ɛ)) P(w w) is small. 13/11/18 Early Accept/Reject Multifidelity ABC 39

51 Example: Construct realisations of early accept/reject samples Matching estimator values False positive False negative Threshold Calculate w mf (θ i ) for i = 1,..., I(U < η(ỹ i )) is the random decision. Evaluate ESS/T tot for 500 realisations. Optimal (η1, η 2 ) taken from baseline dataset: compare to other continuation probabilities. 13/11/18 Early Accept/Reject Multifidelity ABC 40

52 Example: Efficiency of different continuation probabilities 1.0 Early decision: η 1 = η 2. Early rejection: η 1 = 1. Rejection sampling: η 1 = η 2 = Level sets are 99%, 95%, 90%, 85%, 80%, 75%, 60% of maximum efficiency. 13/11/18 Early Accept/Reject Multifidelity ABC 41

53 Example: Distribution of observed efficiencies /11/18 Early Accept/Reject Multifidelity ABC 42

54 Approximate Bayesian computation Multifidelity methods Multifidelity modelling example: Repressilator Multifidelity ABC Early accept/reject multifidelity ABC Performance analysis Repressilator revisited Implementation and future extensions 13/11/18 Early Accept/Reject Multifidelity ABC 43

55 Implementation of early accept/reject multifidelity ABC Dealt with in recent work: No a priori knowledge of location of (η1, η 2 ): Simulation times and false positive/negative probabilities. Use burn-in approach first, then estimate. Methods to reduce false positive/negative probabilities. Coupling low and high fidelity simulations. Optimising efficiency in terms of estimating specific functions E ABC (F (θ)) i w i F (θ i )/ i w i Minimize E(w 2 mf (F E ABC (F )) 2 )E(T ) instead. 13/11/18 Early Accept/Reject Multifidelity ABC 44

56 Outlook: future work Applicability to MCMC/SMC methods. Deal with wmf < 0. Multiple low-fidelity models. Local hierarchies of accuracy/speed-up. Parameter/estimator-specific low-fidelity model. Alternative forms for η(ỹ) and w. Parameter-dependent η. Analytic/computational bounds on error: ỹ versus y; w versus w. 13/11/18 Early Accept/Reject Multifidelity ABC 45

57 Acknowledgements Project: Next generation approaches to connect models and quantitative data. PIs: Ruth Baker & Michael Stumpf. 13/11/18 Early Accept/Reject Multifidelity ABC 46