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beamer-icsi-logo 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

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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

• ver 46 hours (from Friel et al. 2009).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem Markov random fields

Pairwise Markov random fields

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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) =

∏

(i,j)∈Nei(Y) φ(Yi,Yj|θ). We also need to specify the normalising constant Z(θ) =

• Y

∏

(i,j)∈Nei(Y) φ(Yi,Yj|θ)dY In general we are interested in models that take the form f (Y |θ) = γ(Y |θ) Z(θ) .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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 min

• 1, q(θ|θ ∗)

q(θ ∗|θ) p(θ ∗) p(θ) γ(y|θ ∗) γ(y|θ) 1 Z(θ ∗) Z(θ) 1

• .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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 min

• 1, q(θ|θ ∗)

q(θ ∗|θ) p(θ ∗) p(θ) γ(y|θ ∗) γ(y|θ) 1 Z(θ ∗) Z(θ) 1

• .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem

ABC-MCMC

Approximate an intractable likelihood at θ with: 1 R

R

∑

r=1

πε (S(xr)|S(y)) where the xr ∼ f (.|θ) are R simulations from f (originally in Ratmann et al. (2009)). Often R = 1 and πε (.|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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem

ABC-MCMC

Approximate an intractable likelihood at θ with: 1 R

R

∑

r=1

πε (S(xr)|S(y)) where the xr ∼ f (.|θ) are R simulations from f (originally in Ratmann et al. (2009)). Often R = 1 and πε (.|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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem

ABC on ERGMs

”True” ABC

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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 , xr ∼ 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)|

µθ, Σθ

• ,

where

• µθ = 1

R

R

∑

r=1

S (xr), Σθ = ssT R −1, with s = (S (x1)− µθ,...,S (xR)− µθ).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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) ∝ qu(u|θ,y)f (y|θ)p(θ) where qu is some (normalised) arbitrary distribution and u is

• n 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 ∗)

q(θ ∗|θ) p(θ ∗) p(θ) γ(y|θ ∗) γ(y|θ) qu(u∗|θ ∗,y) γ(u∗|θ ∗) γ(u|θ) qu(u|θ,y)

• .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem

Exact approximations

Note that qu(u∗|θ ∗,y)

γ(u∗|θ ∗)

is an unbiased importance sampling estimator of

1 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”).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem

Exact approximations

Note that qu(u∗|θ ∗,y)

γ(u∗|θ ∗)

is an unbiased importance sampling estimator of

1 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”).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A doubly intractable problem

Exact approximations

Note that qu(u∗|θ ∗,y)

γ(u∗|θ ∗)

is an unbiased importance sampling estimator of

1 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”).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo 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)dθ.

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|M1) p(y|M2). 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).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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 πε(S(x(p) r

)|S(y)) q(θ (p)) where

• x(p)

r

R

r=1 ∼ f (.|θ (p)).

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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Exponential family models

Didelot et al. (2011): when comparing two exponential family models, if

S1(y) is sufficient for the parameters in model 1 S2(y) is sufficient for the parameters in model 2

Then using the vector S(y) = (S1(y),S2(y)) for both models gives p(y|M1) p(y|M2) = p(S(y)|M1) p(S(y)|M2). Marin et al. (2014) has much more general guidance.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Synthetic likelihood IS

We could also use the SL approximation within IS. The weight update is then

• w(p) =

p(θ (p))N

• S(y)|

µθ, Σθ

• q(θ (p))

, where µθ, Σθ are based on

• x(p)

r

R

r=1 ∼ f (.|θ (p)).

Does not require choosing ε, but relies on normality assumption.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Exact methods?

Importance sampling: p(y) =

• θ

f (y|θ)p(θ) q(θ) q(θ)dθ ≈ 1 P

P

∑

p=1

f (y|θ (p))p(θ (p)) q(θ (p)) = 1 P

P

∑

p=1

γ(y|θ (p))p(θ (p)) q(θ (p)) 1 Z(θ (p)). Intractable...

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Exact methods?

Importance sampling: p(y) =

• θ

f (y|θ)p(θ) q(θ) q(θ)dθ ≈ 1 P

P

∑

p=1

f (y|θ (p))p(θ (p)) q(θ (p)) = 1 P

P

∑

p=1

γ(y|θ (p))p(θ (p)) q(θ (p)) 1 Z(θ (p)). Intractable...

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

SAV importance sampling

Consider the SAV target p(θ,u|y) ∝ qu(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

P

∑

p=1

qu(u|θ (p),y)γ(y|θ (p))p(θ (p)) γ(u|θ (p))q(θ (p)) Z(θ (p)) Z(θ (p)) = 1 P

P

∑

p=1

γ(y|θ (p))p(θ (p)) q(θ (p)) qu(u|θ (p),y) γ(u|θ (p)) .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

SAV importance sampling

Consider the SAV target p(θ,u|y) ∝ qu(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

P

∑

p=1

qu(u|θ (p),y)γ(y|θ (p))p(θ (p)) γ(u|θ (p))q(θ (p)) Z(θ (p)) Z(θ (p)) = 1 P

P

∑

p=1

γ(y|θ (p))p(θ (p)) q(θ (p)) qu(u|θ (p),y) γ(u|θ (p)) .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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 M

M

∑

m=1

qu(u(m)|θ,y) γ(u(m)|θ) . For u the proposal is pre-determined, but we need to choose qu(u|θ,y). Møller et al. (2006):

• ne possible choice is qu(u|

θ,y) = γ(u| θ)/Z( θ) where θ is an ML estimate (or some other appropriate estimate) of θ.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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 M

M

∑

m=1

qu(u(m)|θ,y) γ(u(m)|θ) . For u the proposal is pre-determined, but we need to choose qu(u|θ,y). Møller et al. (2006):

• ne possible choice is qu(u|

θ,y) = γ(u| θ)/Z( θ) where θ is an ML estimate (or some other appropriate estimate) of θ.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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 M

M

∑

m=1

qu(u(m)|θ,y) γ(u(m)|θ) . For u the proposal is pre-determined, but we need to choose qu(u|θ,y). Møller et al. (2006):

• ne possible choice is qu(u|

θ,y) = γ(u| θ)/Z( θ) where θ is an ML estimate (or some other appropriate estimate) of θ.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

SAVIS / MAVIS

Using the suggested qu 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 fk(.|θ, θ,y) ∝ γk(.|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

γk+1(u(m)

k

|θ ∗,θ,y) γk(u(m)

k

|θ ∗,θ,y) .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

SAVIS / MAVIS

Using the suggested qu 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 fk(.|θ, θ,y) ∝ γk(.|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

γk+1(u(m)

k

|θ ∗,θ,y) γk(u(m)

k

|θ ∗,θ,y) .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Toy example: Poisson vs geometric

Consider i.i.d. observations {yi}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)

f1 ({yi}n

i=1 |θ)

= ∏

i

λ xi exp(−λ) xi! = 1 exp(nλ) ∏

i

λ xi xi!

2 Y |θ ∼ Geometric(θ), θ ∼ Unif(0,1)

f2 ({yi}n

i=1 |θ)

= ∏

i

p(1−p)x = 1 p−n ∏

i

(1−p)xi .

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Results: box plots

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Results: ABC-IS

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Results: SL-IS

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Results: MAVIS

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

Application to social networks

Compare the evidence for two alternative exponential random graph models p(y|θ) ∝ exp(θ TS(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).

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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|M1)/ p(y|M2) ≈ 4; ε = 0.05 gives p(y|M1)/ p(y|M2) ≈ 20, but has only 5 points with non-zero weight!

Synthetic likelihood obtains p(y|M1)/ p(y|M2) ≈ 40. MAVIS finds

log[ p(y|M1)] = −69.62304, log[ p(y|M2)] = −73.33692 giving p(y|M1)/ p(y|M2) ≈ 41.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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|M1)/ p(y|M2) ≈ 4; ε = 0.05 gives p(y|M1)/ p(y|M2) ≈ 20, but has only 5 points with non-zero weight!

Synthetic likelihood obtains p(y|M1)/ p(y|M2) ≈ 40. MAVIS finds

log[ p(y|M1)] = −69.62304, log[ p(y|M2)] = −73.33692 giving p(y|M1)/ p(y|M2) ≈ 41.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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|M1)/ p(y|M2) ≈ 4; ε = 0.05 gives p(y|M1)/ p(y|M2) ≈ 20, but has only 5 points with non-zero weight!

Synthetic likelihood obtains p(y|M1)/ p(y|M2) ≈ 40. MAVIS finds

log[ p(y|M1)] = −69.62304, log[ p(y|M2)] = −73.33692 giving p(y|M1)/ p(y|M2) ≈ 41.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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|M1)/ p(y|M2) ≈ 4; ε = 0.05 gives p(y|M1)/ p(y|M2) ≈ 20, but has only 5 points with non-zero weight!

Synthetic likelihood obtains p(y|M1)/ p(y|M2) ≈ 40. MAVIS finds

log[ p(y|M1)] = −69.62304, log[ p(y|M2)] = −73.33692 giving p(y|M1)/ p(y|M2) ≈ 41.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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|M1)/ p(y|M2) ≈ 4; ε = 0.05 gives p(y|M1)/ p(y|M2) ≈ 20, but has only 5 points with non-zero weight!

Synthetic likelihood obtains p(y|M1)/ p(y|M2) ≈ 40. MAVIS finds

log[ p(y|M1)] = −69.62304, log[ p(y|M2)] = −73.33692 giving p(y|M1)/ p(y|M2) ≈ 41.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem

beamer-icsi-logo Markov random fields A doubly intractable problem A triply intractable problem A triply intractable problem

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.

Richard Everitt University of Reading Evidence estimation for Markov random fields: a triply intractable problem