Evidence estimation for Markov random fields: a triply intractable - PowerPoint PPT Presentation

Markov random fields A doubly intractable problem A triply intractable problem Evidence estimation for Markov random fields: a triply intractable problem Richard Everitt University of Reading January 7th, 2014 beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem 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. beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem Markov random fields Image analysis The log expression of 72 genes on a particular chromosome beamer-icsi-logo over 46 hours (from Friel et al. 2009). Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem Markov random fields Pairwise Markov random fields beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem 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 ) . beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem 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 � � 1 , q ( θ | θ ∗ ) p ( θ ∗ ) γ ( y | θ ∗ ) 1 Z ( θ ) min . q ( θ ∗ | θ ) Z ( θ ∗ ) p ( θ ) γ ( y | θ ) 1 beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem 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 � � 1 , q ( θ | θ ∗ ) p ( θ ∗ ) γ ( y | θ ∗ ) 1 Z ( θ ) min . q ( θ ∗ | θ ) Z ( θ ∗ ) p ( θ ) γ ( y | θ ) 1 beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem ABC-MCMC Approximate an intractable likelihood at θ with: R 1 ∑ π ε ( S ( x r ) | S ( y )) R 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 beamer-icsi-logo approximate posterior. Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem ABC-MCMC Approximate an intractable likelihood at θ with: R 1 ∑ π ε ( S ( x r ) | S ( y )) R 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 beamer-icsi-logo approximate posterior. Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem ABC on ERGMs ”True” ABC beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem 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 : � � µ θ , � S ( y ) | � L ( S ( y ) | θ ) = N Σ θ , where R µ θ = 1 ∑ � S ( x r ) , R r = 1 Σ θ = ss T R − 1 , beamer-icsi-logo with s = ( S ( x 1 ) − � µ θ ,..., S ( x R ) − � µ θ ) . Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem 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 � � 1 , q ( θ | θ ∗ ) p ( θ ∗ ) γ ( y | θ ∗ ) q u ( u ∗ | θ ∗ , y ) γ ( u | θ ) min . beamer-icsi-logo q ( θ ∗ | θ ) p ( θ ) γ ( y | θ ) γ ( u ∗ | θ ∗ ) q u ( u | θ , y ) Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem Exact approximations Note that q u ( u ∗ | θ ∗ , y ) is an unbiased importance sampling γ ( u ∗ | θ ∗ ) 1 estimator of Z ( θ ∗ ) . 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”). beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem Exact approximations Note that q u ( u ∗ | θ ∗ , y ) is an unbiased importance sampling γ ( u ∗ | θ ∗ ) 1 estimator of Z ( θ ∗ ) . 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”). beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem Exact approximations Note that q u ( u ∗ | θ ∗ , y ) is an unbiased importance sampling γ ( u ∗ | θ ∗ ) 1 estimator of Z ( θ ∗ ) . 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”). beamer-icsi-logo Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem Estimating the marginal likelihood The marginal likelihood (also known as the evidence) is � p ( y ) = θ p ( θ ) f ( y | θ ) d θ . Used in Bayesian model comparison 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 beamer-icsi-logo “a triply intractable problem” - Friel (2013). Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem