Tutorial on ABC rejection and ABC SMC for parameter estimation and model selection

T utorial on ABC rejection and ABC SMC for parameter estimation and mo del selection Tina T oni, Mic hael P . H. Stumpf ttoni@imp erial.ac.uk, m.stumpf@imp erial.ac.uk In this tutorial w e sc hematically illustrate four algorithms: 1. ABC rejection for parameter estimation [1, 2], 2. ABC SMC for parameter estimation [3, 4, 5], 3. ABC rejection for mo del selection on the join t space [6], 4. ABC SMC for mo del selection on the join t space [7]. W e suggest to read this tutorial from the b eginning. W e start with a detailed explanation of the ABC rejection algorithm, which later helps to understand ABC SMC as it is based on the same concepts. Also, b oth mo del selection algorithms are closely related to parameter estimation algorithms and it is therefore helpful to understand those ﬁrst. This tutorial forms a part of the supplemen tary material of the pap er ”Simulation- based mo del selection for dynamical systems in systems and population biology , Bioinformatics, 26 (1), 104-110, 2010” (T. T oni, M. P . H. Stumpf ). 1 ABC rejection (a) W e deﬁne a prior distribution P ( θ ) and we w ould like to approximate the p osterior distribution P ( θ | D 0 ). W e start by sampling a parameter θ ∗ from the prior distribution. W e call this sampled parameter a p article . (b) W e simulate a data set D ∗ according to some sim ulation framework f ( D | θ ∗ ). In our examples w e use diﬀerent sim ulation framew orks. If we sim ulate a deterministic dynamical mo del, we add some noise at the time points of interest. If w e sim ulate a sto c hastic dynamical mo del, we do not add an y additional noise to the tra jectories. W e compare the simulated data set D ∗ (circles) to the exp erimen tal data D 0 (crosses) using a distance function, d , and tolerance  ; if d ( D 0 , D ∗ ) ≤  , we accept θ ∗ . The tolerance  ≥ 0 is the desired lev el of agreemen t b et ween D 0 and D ∗ . (c) The particle θ ∗ is accepted b ecause D ∗ and D 0 are suﬃcien tly close. (d) W e sample another parameter θ ∗ from the prior distribution and sim ulate a corresp onding dataset D ∗ . In this case D ∗ and D 0 are very diﬀeren t and we reject the particle (we ”throw it a wa y”). (e) W e rep eat the whole pro cedure until N particles hav e b een accepted. They repre- sen t a sample from P ( θ | d ( D 0 , D ∗ ) ≤  ), which appro ximates the posterior distribution. If  is suﬃcien tly small then the distribution P ( θ | d ( D 0 , D ∗ ) ≤  ) will b e a go od approxi- mation for the “true” p osterior distribution, P ( θ | D 0 ). (f ) Man y particles w ere rejected in the pro cedure, for which we hav e sp en t a lot of computational eﬀort for sim ulation. ABC rejection is therefore computationally ineﬃ- cien t. W e can use ABC SMC to reduce the computational cost. ABC fo r dynamical systems ABC framew o rk fo r dynamical systems Prior Posterio r P ( θ ) P ( θ | D 0 ) x x x x time o o o o o o o o Tina T oni ABC SMC 26/06/2009 3 / 3 (a) ABC fo r dynamical systems ABC framew o rk fo r dynamical systems Prio r P osterio r P ( θ ) P ( θ | D ) x x x x time o o o o o o o o Tina T oni ABC SMC 26/06/2009 4 / 24 (b) ABC fo r dynamical systems ABC framew o rk fo r dynamical systems Prior Posterio r P ( θ ) P ( θ | D 0 ) x x x x time o o o o o o o o Tina T oni ABC SMC 26/06/2009 3 / 3 (c) ABC fo r dynamical systems ABC framew o rk fo r dynamical systems Prior Posterio r P ( θ ) P ( θ | D 0 ) x x x x time o o o o o o o o Tina T oni ABC SMC 26/06/2009 3 / 3 (d) ABC framew o rk fo r dynamical systems Prior P osterior P ( θ ) P ( θ | D 0 ) x x x x time o o o o o o o o August 2, 2009 1 / 1 (e) ABC fo r dynamical systems ABC framew o rk fo r dynamical systems Prio r P osterio r P ( θ ) P ( θ | D ) x x x x time o o o o o o o o Tina T oni ABC SMC 26/06/2009 4 / 24 (f ) Figure 1: Schematic r epresentation of ABC re- jection. 2 ABC SMC (T oni et al. , 2009) (a) As in ABC rejection, we deﬁne a prior distribution P ( θ ) and w e w ould like to appro xi- mate a p osterior distribution P ( θ | D 0 ). In ABC SMC we do this sequentially by constructing in termediate distributions, whic h con v erge to the posterior distribution. W e deﬁne a tolerance sc hedule  1 >  2 > . . .  T ≥ 0. (b) W e sample particles from a prior distribu- tion until N particles hav e b een accepted (hav e reac hed the distance smaller than  1 ). F or all accepted particles w e calculate w eigh ts (see [4] for formulas and deriv ation). W e call the sample of all accepted particles ”P opulation 1”. (c) W e then sample a particle θ ∗ from popu- lation 1 and p erturb it to obtain a p erturb ed particle θ ∗∗ ∼ K ( θ | θ ∗ ), where K is a p er- turbation kernel (for example a Gaussian random walk). W e then simulate a dataset D ∗ ∼ f ( D | θ ∗∗ ) and accept the particle θ ∗∗ if d ( D 0 , D ∗∗ ) ≤  2 . W e rep eat this un til we ha ve accepted N particles in p opulation 2. W e calculate w eights for all accepted particles. (d) W e rep eat the same procedure for the follo wing p opulations, until we hav e accepted N particles of the last p opulation T and calculated their weigh ts. P opulation T is a sample of particles that approximates the p osterior distribution. ABC SMC is computationally muc h more eﬃcien t than ABC rejection (see [4] for comparison). ABC SMC (Sequential Monte Ca rlo) Intermediate Distributions Prior Posterio r  1  2 . . .  T − 1  T P opulation 1 P opulation 2 P opulation T Tina T oni, Michael Stumpf ABC dynamical sy stems 03/07/08 1 / 1 (a) ABC SMC (Sequential Monte Ca rlo) Intermediate Distributions Prior Posterio r  1  2 . . .  T − 1  T Population 1 P opulation 2 P opulation T Tina T oni, Michael Stumpf ABC dynamical sy stems 03/07/08 1 / 1 (b) ABC SMC (Sequential Monte Ca rlo) Intermediate Distributions Prior Posterio r  1  2 . . .  T − 1  T Population 1 Tina T oni, Michael Stumpf ABC dynamical sy stems 03/07/08 1 / 1 (c) ABC SMC (Sequential Monte Ca rlo) Intermediate Distributions Prior Posterio r  1  2 . . .  T − 1  T Population 1 Population 2 Population T Tina T oni, Michael Stumpf ABC dynamical sy stems 03/07/08 1 / 1 (d) Figure 2: Schematic represen tation of ABC SMC. 3 ABC rejection for mo del selection ABC framew o rk fo r dynamical systems Prio r P osterio r P ( m , θ ) P ( m , θ | D 0 ) x x x x time o o o o o o o o August 2, 2009 1 / 1 (a) !"#$%&'(")*+%",-("&*.+*"/.#,*.012+3"45-., (6"$3/"$77(&,(/"&$*27'(#"40'1(6 " % 8" % 9 " % : " θ 8""""" θ :""""" θ 9 """ θ ; """ θ < """"" θ = """ θ > """" θ ?""""" θ @ """ θ 8A """ θ 88 "" θ 8: "" θ 89 "" θ 8; "" θ 8< " (b) Approximation of P(m|D 0 ) M d l 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 Model (c) Figure 3: (a) Prior and posterior distributions, P ( m, θ ) and P ( m, θ | D 0 ), are now deﬁned on a join t model and parameter space. (b) P articles ( m, θ ) are sampled from the prior distribution and accepted/rejected according to the distance b et ween the simulated and exp erimen tal datasets. The accepted particles are shown in dark blue. (c) Six particles hav e b een accepted: one from mo del 1 and ﬁv e from mo del 2. The approximated marginal p osterior probability of the mo del can b e calculated as P ( m = 1 | D 0 ) = 1 6 , P ( m = 2 | D 0 ) = 5 6 , P ( m = 3 | D 0 ) = 0 6 . F or illustrative purp oses we ha ve chosen a small num b er of particles. In principle this algorithm will yield consisten t marginal p osterior model distributions for N → ∞ . 4 ABC SMC for mo del selection !"#$%&'(")*+%",-("&*.+*"/.#,*.012+3"$3/"$44(&, (/"&$*24'(#"+)"5+&6 " % 6" % 7 " % 8 " θ 6""""" θ 8""""" θ 7 """ θ 9 """ θ : """"" θ ; """ θ < """" θ =""""" θ > """ θ 6? """ θ 66 "" θ 68 "" θ 67 "" θ 69 "" θ 6: " (a) M d l Population 1 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 (b) !"#$%&'"()* +, ) ))))))) θ ,))))) ) ))))))) θ -))))) )) θ . ))))) θ / ))))))))) θ 0)))) θ 1 )))))) 23)4)2 -) 233)5)67 * 82923: ) 2 ,) 2 . ) 2 - ) 233)4)2 .) ;&2#%<)8233=) θ 3:)>?"2)@8233=) θ :A *+, B ) θ 33)5)6! *=233) 8 θ 9 θ 3:) CDD<#*E?)&DD<#*

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