Evidence estimation for Markov random fields: a triply intractable problem

Evidence estimation for Markov random fields: a triply intractable problem January 7th, 2014

Markov random fields Interacting objects Markov random fields (MRFs) are used for modelling (often large numbers of) interacting objects usually modelling symmetrical interactions. Used widely in statistics, physics and computer science, e.g. image analysis; ferromagnetism; geostatistics; point processes; social networks.

Markov random fields Image analysis The log expression of 72 genes on a particular chromosome over 46 hours (from Friel et al. 2009).

Markov random fields Pairwise Markov random fields

Markov random fields Intractable normalising constants Pairwise MRFs correspond to the factorisation f (Y θ) γ(y θ) = φ(y i,y j θ). (i,j) Nei(Y) We also need to specify the normalising constant Z(θ) = φ(y i,y j θ)dy Y (i,j) Nei(Y) In general we are interested in models that take the form f (Y θ) = γ(y θ) Z(θ).

A doubly intractable problem Doubly intractable Suppose we want to estimate parameters θ after observing Y = y. Use Bayesian inference to find p(θ y) p(y θ)p(θ). Could use MCMC, but the acceptance probability in MH is { min 1, q(θ θ ) p(θ ) γ(y θ } ) 1 Z(θ) q(θ θ) p(θ) γ(y θ) Z(θ. ) 1

A doubly intractable problem ABC-MCMC Approximate an intractable likelihood at θ with: R 1 R π ε (S(x r ) S(y)) r=1 where the x r f (. θ) are R simulations from f (originally in Ratmann et al. (2009)). Often R = 1 and π ε (. S(y)) = U (. (S(y) ε,s(y) + ε)). Essentially a nonparametric kernel estimator to the conditional distribution of the statistics given θ, based on simulations from f. ABC-MCMC is an MCMC algorithm that targets this approximate posterior.

A doubly intractable problem ABC on ERGMs True ABC

A doubly intractable problem Synthetic likelihood An alternative approximation proposed in Wood (2010). Again take R simulations from f, x r f (. θ), and take the summary statistics of each. But instead use a multivariate normal approximation to the distribution of the summary statistics given θ: L(S(y) θ) = N (S(y) µ θ, Σ ) θ, where µ θ = 1 R R S (x r ), r=1 Σ θ = sst R 1, with s = (S (x 1 ) µ θ,...,s (x R ) µ θ ).

A doubly intractable problem The single auxiliary variable method Møller et al. (2006) augment the target distribution with an extra variable u and use p(θ,u y) q u (u θ,y)f (y θ)p(θ) where q u is some (normalised) arbitrary distribution and u is on the same space as y. As the MH proposal in (θ,u)-space they use (θ,u ) f (u θ )q(θ θ). This gives an acceptance probability of { min 1, q(θ θ ) p(θ ) γ(y θ ) q u (u θ,y) q(θ θ) p(θ) γ(y θ) γ(u θ ) γ(u θ) q u (u θ,y) }.

A doubly intractable problem Exact approximations Note that q u(u θ,y) γ(u θ ) 1 estimator of Z(θ ). is an unbiased importance sampling still targets the correct distribution! first seen in the pseudo-marginal methods of Beaumont (2003) and Andrieu and Roberts (2009). Relies on being able to simulate exactly from f (. θ ), which is usually not possible or computationally expensive. Girolami et al. (2013) introduce an approach that does not require exact simulation ( Russian Roulette ).

Estimating the marginal likelihood The marginal likelihood (also known as the evidence) is p(y) = p(θ)f (y θ)dθ. Used in Bayesian model comparison θ p(m y) = p(m)p(y M), most commonly seen in the Bayes factor, for comparing models p(y M 1 ) p(y M 2 ). All commonly used methods require f (y θ) to be tractable in θ, and usually can t be estimated from MCMC output a triply intractable problem - Friel (2013).

Using importance sampling (IS) Importance sampling Returns a weighted sample {(θ (p),w (p) ) 1 p P} from p(θ y). For p = 1 : P Simulate θ (p) q(.) Weight w (p) = p(θ (p) )f (y θ (p) ). q(θ (p) ) Then p(y) = 1 P P p=1 w (p).

Using ABC-IS Didelot, Everitt, Johansen and Lawson (2011) investigate the use of the ABC approximation when using IS for marginal likelihoods. The weights are w (p) = p(θ (p) ) 1 R R r=1 π ε (S(x r (p) ) S(y)) q(θ (p) ) } R where { x r (p) f (. θ (p) ). r=1 This method gives p(s(y)) p(y). Didelot et al. (2011), Grelaud et al. (2009), Robert et al. (2011), Marin et al. (2014), discuss the choice of summary statistics.

Exponential family models Didelot et al. (2011): when comparing two exponential family models, if S 1 (y) is sufficient for the parameters in model 1 S 2 (y) is sufficient for the parameters in model 2 Then using the vector S(y) = (S 1 (y),s 2 (y)) for both models gives p(y M 1 ) p(y M 2 ) = p(s(y) M 1) p(s(y) M 2 ). Marin et al. (2014) has much more general guidance.

Synthetic likelihood IS We could also use the SL approximation within IS. The weight update is then p(θ (p) )N (S(y) µ θ, Σ ) θ w (p) = q(θ (p), ) where µ θ, Σ { θ are based on x (p) r } R f (. θ (p) ). r=1 Does not require choosing ε, but relies on normality assumption.

Exact methods? Importance sampling: p(y) = θ 1 P = 1 P f (y θ)p(θ) q(θ)dθ q(θ) P f (y θ (p) )p(θ (p) ) p=1 q(θ (p) ) P γ(y θ (p) )p(θ (p) ) 1 p=1 q(θ (p) ) Z(θ (p) ). Intractable...

SAV importance sampling Consider the SAV target p(θ,u y) q u (u θ,y)f (y θ)p(θ), noting that it has the same marginal likelihood as p(θ y). Suppose we do importance sampling on this SAV target, and choose the proposal to be q(θ,u) = f (u θ)q(θ). We obtain p(y) = 1 P = 1 P P p=1 P p=1 q u (u θ (p),y)γ(y θ (p) )p(θ (p) ) Z(θ (p) ) γ(u θ (p) )q(θ (p) ) Z(θ (p) ) γ(y θ (p) )p(θ (p) ) q(θ (p) ) q u (u θ (p),y) γ(u θ (p). )

Exact approximations revisited Using unbiased weight estimates within importance sampling: (IS) 2 (Tran et al., 2013); random weight particle filters (Fearnhead et al. 2010); (SMC) 2 (Chopin et al. 2011). For each θ, we could use multiple u variables and use the estimate 1 Z(θ) = 1 q u (u (m) θ,y) M γ(u (m). θ) M m=1 For u the proposal is pre-determined, but we need to choose q u (u θ,y). Møller et al. (2006): one possible choice is q u (u θ,y) = γ(u θ)/z( θ) where θ is an ML estimate (or some other appropriate estimate) of θ.

SAVIS / MAVIS Using the suggested q u gives the following importance sampling estimate of 1/Z(θ) 1 Z(θ) = 1 Z( θ) 1 M M m=1 γ(u (m) θ) γ(u (m) θ). Or, using annealed importance sampling (Neal, 2001) with the sequence of targets f k (. θ, θ,y) γ k (. θ, θ) = γ(. θ) (K+1 k)/(k+1) +γ(. θ) k/(k+1), we obtain 1 Z(θ) = 1 Z( θ) 1 M M m=1 K k=0 γ k+1 (u (m) k θ,θ,y) γ k (u (m) k θ,θ,y).

Non-exact approximations... MAVIS is exact only if exact sampling from f (. θ) is possible (also applies to ABC and synthetic likelihood); 1/Z( θ) is known. In practice use MCMC to simulate from f (. θ); estimate 1/Z( θ) offline in advance of running the IS. In the context of MCMC, one can show that these approximations do not introduce large errors see MCMW approach in Andrieu and Roberts (2009) (also Everitt (2012)), and Nial Friel s talk tomorrow ( Monte Carlo methods in network analysis session).

Toy example: Poisson vs geometric Consider i.i.d. observations {y i } n i=1 of a discrete random variable that takes values in N. We find the Bayes factor for the models 1 Y θ Poisson(θ), θ Exp(1) f 1 ({y i } n i=1 θ) = λ x i exp( λ) i x i! 1 = exp(nλ) λ x i i x i! 2 Y θ Geometric(θ), θ Unif(0,1) f 2 ({y i } n i=1 θ) = p(1 p) x i = 1 p n (1 p) x i. i

Results: box plots

Results: ABC-IS

Results: SL-IS

Results: MAVIS

Application to social networks Compare the evidence for two alternative exponential random graph models p(y θ) exp(θ T S(y)). in model 1 S(y) = number of edges in model 2 S(y) = (number of edges, number of two stars) (so now θ is 2-d). Use prior p(θ) = N (0,25I ), as in Friel (2013).

Results: social network Friel (2013) finds that the evidence for model 1 is 37.499 that for model 2. Using 1000 importance points (with 100 simulations from the likelihood for each point)... ABC: ε = 0.1 gives p(y M 1 )/ p(y M 2 ) 4; ε = 0.05 gives p(y M 1 )/ p(y M 2 ) 20, but has only 5 points with non-zero weight! Synthetic likelihood obtains p(y M 1 )/ p(y M 2 ) 40. MAVIS finds log[ p(y M 1 )] = 69.62304, log[ p(y M 2 )] = 73.33692 giving p(y M 1 )/ p(y M 2 ) 41.

Comparison of methods ABC vs MAVIS both require the simulation of auxiliary variables, but in ABC/SL the use of summary statistics dramatically reduces the dimension of the space; but MAVIS only requires the auxiliary variable to look like it is a good simulation from f (. θ), not (the different requirement) that it is a good match to y. Plus the standard drawbacks of ABC remain choice of tolerance ε not able to estimate the evidence, only Bayes factors. SL vs ABC SL fails when Gaussian assumption is not appropriate...... but it is surprisingly robust and there is no need to choose an ε.

Summary We can tackle doubly intractable problems using: ABC; synthetic likelihood; auxiliary variable methods; Russian Roulette. Used in importance sampling, we can also estimate marginal likelihoods and Bayes factors. For high-dimensional θ, SMC algorithms can be employed in some cases the Bayes factor can be estimated directly. Thanks to Nial Friel, Melina Evdemon-Hogan and Ellen Rowing.