Monte Carlo algorithms for Bayesian social network models Alberto Caimo alberto.caimo@usi.ch Social Network Analysis Research Centre InterDisciplinary Institute of Data Science University of Lugano, Switzerland AMS Sectional Meeting, Loyola University Chicago October 2015 Alberto Caimo Bayesian computation for ERGMs
Outline Exponential random graph models (ERGMs) Bayesian exponential random graph models (BERGMs) Computational approaches for BERGMs Examples Alberto Caimo Bayesian computation for ERGMs
Exponential random graph models (ERGMs) The relational structure of an observed network y can be explained by the relative prevalence of a set of overlapping sub-graph configurations s ( y ) called network statistics Alberto Caimo Bayesian computation for ERGMs
Exponential random graph models (ERGMs) The relational structure of an observed network y can be explained by the relative prevalence of a set of overlapping sub-graph configurations s ( y ) called network statistics The likelihood of an ERGM represents the probability distribution of y given a parameter θ : p ( y | θ ) = exp { θ t s ( y ) − γ ( θ ) } Alberto Caimo Bayesian computation for ERGMs
Exponential random graph models (ERGMs) The relational structure of an observed network y can be explained by the relative prevalence of a set of overlapping sub-graph configurations s ( y ) called network statistics The likelihood of an ERGM represents the probability distribution of y given a parameter θ : p ( y | θ ) = exp { θ t s ( y ) − γ ( θ ) } From a computational point of view we have an intractable likelihood problem Alberto Caimo Bayesian computation for ERGMs
Bayesian exponential random graph models (BERGMs) (2) Model specification (3) p ( y | θ , m ) ∝ exp { θ t s ( y ) } Doubly-intractable posterior (1) p ( θ , m | y ) ∝ p ( y | θ , m ) p ( θ | m ) p ( m ) Observed network y (5) (4) Model choice Parameter inference across-model p ( θ | y , m ) p ( m | θ , y ) within-model p ( y | m ) (6) Goodness of fit Alberto Caimo Bayesian computation for ERGMs
. . Bayesian exponential random graph models (BERGMs) .parameter .network statistics .normalising constant exp { θ t likelihood: . . . s ( y ) − γ ( θ ) . . } .. prior . . p ( y | θ ) . . p ( θ ) . posterior: p ( θ | y ) = . p ( y ) . model evidence . . . . . . . . Alberto Caimo Bayesian computation for ERGMs
Computational approaches Let’s define: q θ ( y ) = exp { θ t s ( y ) } unnormalised likelihood z ( θ ) = exp { γ ( θ ) } normalising constant Metropolis-Hastings 1 Gibbs update of ( θ ′ ) i Draw θ ′ ∼ h ( ·| θ ) 2 Accept move from θ to θ ′ with probability p ( θ ′ ) h ( θ | θ ′ ) 1 ∧ q θ ′ ( y ) h ( θ ′ | θ ) × z ( θ ) z ( θ ′ ) q θ ( y ) p ( θ ) Alberto Caimo Bayesian computation for ERGMs
Computational approaches Approximate exchange algorithm (AEA) (Murray et al., 2006; Caimo and Friel, 2011) 1 Gibbs update of ( θ ′ , y ′ ) (i) Draw θ ′ ∼ h ( ·| θ ) (ii) Draw y ′ ∼ p ( ·| θ ′ ) via MCMC Alberto Caimo Bayesian computation for ERGMs
Computational approaches Approximate exchange algorithm (AEA) (Murray et al., 2006; Caimo and Friel, 2011) 1 Gibbs update of ( θ ′ , y ′ ) (i) Draw θ ′ ∼ h ( ·| θ ) (ii) Draw y ′ ∼ p ( ·| θ ′ ) via MCMC 2 Exchange move from θ to θ ′ with probability p ( θ ′ ) h ( θ | θ ′ ) q θ ′ ( y ′ ) × z ( θ ) z ( θ ′ ) q θ ( y ′ ) 1 ∧ q θ ′ ( y ) h ( θ ′ | θ ) z ( θ ) z ( θ ′ ) q θ ( y ) p ( θ ) � �� � 1 Alberto Caimo Bayesian computation for ERGMs
Computational approaches Computational challenges Posterior distribution p ( θ | y ) is difficult to sample efficiently from as ERGM parameters is typically very thin and highly correlated Alberto Caimo Bayesian computation for ERGMs
Computational approaches Improving chain mixing and convergence (Caimo and Friel, 2011) 1(i) Parallel adaptive direction sampling (ADS) for Gibbs update of θ ′ 1(ii) Tie/no tie (TNT) sampler (as in the ergm package for R ) Alberto Caimo Bayesian computation for ERGMs
Computational approaches θ h 2 θ h θ h 1 θ h 1 ADS sampler: “snooker move” Alberto Caimo Bayesian computation for ERGMs
Bayesian exponential random graph models (BERGMs) 10 5 0 − 5 − 10 − 15 − 10 − 5 0 5 10 Alberto Caimo Bayesian computation for ERGMs
Computational approaches Adaptive approximate exchange algorithm with delayed rejection (Caimo and Mira, 2015) 1(i) Adaptive strategies Alberto Caimo Bayesian computation for ERGMs
Computational approaches Adaptive approximate exchange algorithm with delayed rejection (Caimo and Mira, 2015) 1(i) Adaptive strategies 2 Approximate exchange algorithm with delayed rejection Alberto Caimo Bayesian computation for ERGMs
Computational approaches Adaptive strategies for Gibbs update of θ ′ (Roberts and Rosenthal, 2007; Haario et al., 2001) vertical adaptation: all past particles along the same chain (AAEA-1+DR) horizontal adaptation: all particles at the current time for all chains (AAEA-2+DR) rectangular adaptation: particles from all chains and all past simulations (AAEA-3+DR) Alberto Caimo Bayesian computation for ERGMs
Computational approaches Adaptive exchange algorithm with delayed rejection First stage move: p ( θ ′ ) h 1 ( θ | θ ′ ) q θ ( y ′ ) α 1 ( θ , θ ′ ) = 1 ∧ q θ ′ ( y ) h 1 ( θ ′ | θ ) q θ ′ ( y ′ ) q θ ( y ) p ( θ ) Alberto Caimo Bayesian computation for ERGMs
Computational approaches Adaptive exchange algorithm with delayed rejection First stage move: p ( θ ′ ) h 1 ( θ | θ ′ ) q θ ( y ′ ) α 1 ( θ , θ ′ ) = 1 ∧ q θ ′ ( y ) h 1 ( θ ′ | θ ) q θ ′ ( y ′ ) q θ ( y ) p ( θ ) If θ ′ rejected, try a second stage move to θ ′′ with probability α 2 ( θ , θ ′ , θ ′′ ) = 1 ∧ q θ ( y ′′ ) p ( θ ′′ ) h 1 ( θ ′ | θ ′′ ) [1 − α 1 ( θ ′′ , θ ′ )] h 2 ( θ | θ ′′ , θ ′ ) q θ ′′ ( y ) q θ ( y ) p ( θ ) h 1 ( θ ′ | θ ) [1 − α 1 ( θ , θ ′ )] h 2 ( θ ′′ | θ , θ ′ ) q θ ′′ ( y ′ ) Alberto Caimo Bayesian computation for ERGMs
Computational approaches A hierarchy of proposal distributions can be exploited Moves (tennis-service strategy): “first bold” h 1 ( · ) proposal versus “second timid” h 2 ( · ) Alberto Caimo Bayesian computation for ERGMs
Examples 26 24 27 28 25 30 29 19 32 16 34 33 21 3 8 13 23 4 14 9 15 6 10 17 2 31 1 7 20 11 22 5 12 18 Zachary Karate Club Network : Friendship relations between 34 members of a karate club at a US university in the 1970 Alberto Caimo Bayesian computation for ERGMs
Examples ERGM: p ( y | θ ) ∝ exp { θ 1 s 1 ( y )+ θ 2 s 2 ( y , φ u )+ θ 3 s 3 ( y , φ v ) } Alberto Caimo Bayesian computation for ERGMs
Examples ERGM: p ( y | θ ) ∝ exp { θ 1 s 1 ( y )+ θ 2 s 2 ( y , φ u )+ θ 3 s 3 ( y , φ v ) } where: s 1 ( y ) = ∑ i < j y ij = number of edges � 1 − e − φ v � i � � s 2 ( y , φ v ) = e φ v ∑ n − 2 1 − EP i ( y ) i =1 geometrically weighted edgewise shared partners (gwesp) � 1 − e − φ u � i � � s 3 ( y , φ u ) = e φ u ∑ n − 1 1 − D i ( y ) i =1 geometrically weighted degrees (gwdegree) EP i ( y ) = distribution of the number of unordered pairs of connected nodes having exactly k common neighbours D i ( y ) = degree distribution Alberto Caimo Bayesian computation for ERGMs
Examples for model 14 θ ( 1 ) (edges) θ ( 2 ) (gwesp) θ ( 3 ) (gwdegree) ADS-AEA Post. mean − 3.51 0.74 1.18 Post. sd 0.62 0.21 1.12 AAEA-2+DR (horizontal adaptation + DR) Post. mean − 3.44 0.72 1.01 Post. sd 0.59 0.21 1.07 Estimated posterior means and standard deviations Alberto Caimo Bayesian computation for ERGMs
Examples 1.0 1.0 0.5 0.6 0.5 0.5 0.4 0.3 0.4 0.0 0.0 0.2 -0.5 0.2 -0.5 0.1 -1.0 -1.0 0.0 0.0 -6 -5 -4 -3 -2 -1 0 0 10 20 30 40 50 -6 -5 -4 -3 -2 0 10 20 30 40 50 θ 1 ( edges ) θ 1 ( edges ) Lag Lag 1.0 1.0 2.0 1.5 0.5 0.5 1.5 1.0 0.0 1.0 0.0 0.5 -0.5 -0.5 0.5 0.0 -1.0 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 0 10 20 30 40 50 0.0 0.5 1.0 1.5 0 10 20 30 40 50 θ 2 ( gwesp.fixed.0.693147180559945 ) Lag θ 2 ( gwesp.fixed.0.693147180559945 ) Lag 0.30 1.0 0.4 1.0 0.5 0.3 0.5 0.20 0.0 0.2 0.0 0.10 -0.5 -0.5 0.1 0.00 -1.0 0.0 -1.0 -4 -2 0 2 4 6 8 0 10 20 30 40 50 -2 0 2 4 6 0 10 20 30 40 50 θ 3 ( gwdegree ) θ 3 ( gwdegree ) Lag Lag Posterior density estimates for AEA (left) and AAEA+DR (right) Alberto Caimo Bayesian computation for ERGMs
Examples Effective sample size (ESS): AAEA-2+DR +70% Performance (= ESS/CPU time): AAEA-2+DR +40% Alberto Caimo Bayesian computation for ERGMs
Examples 0.7 0.8 0.6 0.3 0.6 0.5 proportion of dyads proportion of edges 0.4 0.2 0.4 0.3 0.2 0.1 0.2 0.1 0.0 0.0 0.0 0 2 4 6 8 10 12 14 16 18 1 2 3 4 5 6 7 8 9 NR 0 2 4 6 8 10 12 14 degree minimum geodesic distance edge-wise shared partners Bayesian goodness-of-fit diagnostics: the observed network y is compared with a set of networks simulated from independent realisations of p ( θ | y ) in terms of high-level network statistics Alberto Caimo Bayesian computation for ERGMs
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