Approximate Bayesian computation for spatial extremes via open-faced sandwich adjustment

Approximate Bayesian computation for spatial extremes via open-faced sandwich adjustment Ben Shaby SAMSI August 3, 2010 Ben Shaby (SAMSI) OFS adjustment August 3, 2010 1 / 29

Outline 1 Introduction 2 Spatial Extremes 3 The OFS adjustment 4 Simulation Ben Shaby (SAMSI) OFS adjustment August 3, 2010 2 / 29

Outline 1 Introduction 2 Spatial Extremes 3 The OFS adjustment 4 Simulation Ben Shaby (SAMSI) OFS adjustment August 3, 2010 3 / 29

Posterior distributions I will describe how to draw from a posterior distribution. Note the finger quotes. What do we want out of a posterior distribution? Ben Shaby (SAMSI) OFS adjustment August 3, 2010 4 / 29

Posterior distributions I will describe how to draw from a posterior distribution. Note the finger quotes. What do we want out of a posterior distribution? For our purposes, we will want a distribution that 1 Describes our state of knowledge (uncertainty) about a parameter. 2 Produces equi-tailed credible intervals that have nominal frequentist coverage rates. This is not a very Bayesian view! Ben Shaby (SAMSI) OFS adjustment August 3, 2010 4 / 29

A good posterior An ideal ''posterior'' empirical density of θ^ π(θ x) Ben Shaby (SAMSI) OFS adjustment August 3, 2010 5 / 29

Outline 1 Introduction 2 Spatial Extremes 3 The OFS adjustment 4 Simulation Ben Shaby (SAMSI) OFS adjustment August 3, 2010 6 / 29

Extreme values Of the environmental variables we care about, usually what we really care about are the extremes. heat waves storms sea levels Why do we care? manage risk (insurance, etc.) emergency preparedness Ben Shaby (SAMSI) OFS adjustment August 3, 2010 7 / 29

Floods Ben Shaby (SAMSI) OFS adjustment August 3, 2010 8 / 29

Heat waves Ben Shaby (SAMSI) OFS adjustment August 3, 2010 9 / 29

Extreme values Extreme values can mean many things We consider only block maxima (block minima). Asymptotically follow generalized extreme value (GEV) distribution { [ ( x η )] } 1/ξ G(x) = exp 1 ξ τ + where z + = max(z, 0). η is a location and τ a scale parameter. ξ is a shape parameter, and determines the tail behavior. Then any maximal process should have GEV marginal distributions! Ben Shaby (SAMSI) OFS adjustment August 3, 2010 10 / 29

Spatial extremes It is possible to construct processes with spatial structure and GEV marginals. This leads us to max stable processes. The Smith model is one example. Z(x) 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8 10 x Ben Shaby (SAMSI) OFS adjustment August 3, 2010 11 / 29

Smith process in 2 dimensions Ben Shaby (SAMSI) OFS adjustment August 3, 2010 12 / 29

GP margins Lest you fear that this process is unrealistic, the margins don t have to be the same everywhere. Unit Frechet margins Gaussian process GEV parameters Ben Shaby (SAMSI) OFS adjustment August 3, 2010 13 / 29

Pairwise likelihoods Unfortunately, joint likelihoods for the Smith process are not known for n 2. But we can write the pairwise likelihood, a form of composite likelihood. L p (θ; y) = i j f(y i, y j ; θ) Ben Shaby (SAMSI) OFS adjustment August 3, 2010 14 / 29

Pairwise likelihoods Unfortunately, joint likelihoods for the Smith process are not known for n 2. But we can write the pairwise likelihood, a form of composite likelihood. L p (θ; y) = i j f(y i, y j ; θ) It turns out that L p (θ; y) that behaves similarly to the likelihood. Can we trick MCMC into doing something useful with L p (θ; y)? Ben Shaby (SAMSI) OFS adjustment August 3, 2010 14 / 29

The quasi-posterior Yes! We define the quasi-posterior distribution as π p,n (θ y n ) = L p,n (θ; y n )π(θ) Θ L p,n(θ; y n )π(θ) dθ, We will assume, for convenience, that π(θ) proper. L p,n is not necessarily a density, so π p,n (θ y n ) is not a true posterior. L p,n is integrable, so as long as the prior π(θ) is proper, then π p,n (θ Z n ) will be a proper density. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 15 / 29

More definitions Now we can write down a quasi-bayes estimator Define loss in the usual way. Define quasi-posterior risk R n (θ) as the quasi-posterior expectation of loss. The pairwise quasi-bayes estimator is then ˆθ QB = argmin R n (θ). θ Θ Ben Shaby (SAMSI) OFS adjustment August 3, 2010 16 / 29

Outline 1 Introduction 2 Spatial Extremes 3 The OFS adjustment 4 Simulation Ben Shaby (SAMSI) OFS adjustment August 3, 2010 17 / 29

The sandwich matrix P n = E 0 [ 0 l p,n 0 l p,n] B n = E 0 [ 2 0l p,n ] S n = B n P 1 n B n Bread Ben Shaby (SAMSI) OFS adjustment August 3, 2010 18 / 29

The sandwich matrix P n = E 0 [ 0 l p,n 0 l p,n] B n = E 0 [ 2 0l p,n ] S n = B n P 1 n B n Bread Peanut butter Ben Shaby (SAMSI) OFS adjustment August 3, 2010 18 / 29

The sandwich matrix P n = E 0 [ 0 l p,n 0 l p,n] B n = E 0 [ 2 0l p,n ] S n = B n P 1 n B n Bread Peanut butter Bread Ben Shaby (SAMSI) OFS adjustment August 3, 2010 18 / 29

Asymptotic normality of quasi-bayes estimators Then as long as we don t use a crazy prior, Chernozhukov and Hong (2003) says that: Theorem S 1/2 n (ˆθ QB θ 0 ) D N(0, I) When we use pairwise likelihoods for MCMC, the sandwich matrix describes the (asymptotic) sampling variability of the estimator. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 19 / 29

Convergence of the quasi-posterior Furthermore, (also from Chernozhukov and Hong, 2003) Theorem Asymptotically, π p,n (θ y n ) N(θ 0, B 1 n ). This has important consequences for inference from the MCMC sample! B 1 B 1 PB 1! Equi-tailed credible intervals based on MCMC quantiles will NOT have the correct frequentist coverage probabilities Ben Shaby (SAMSI) OFS adjustment August 3, 2010 20 / 29

Distortion of the posterior The two curves are very different! Density 0 2 4 6 8 empirical density of θ^ π p (θ x) 0.5 1.0 1.5 2.0 θ Ben Shaby (SAMSI) OFS adjustment August 3, 2010 21 / 29

The OFS adjustment The main idea: Whereas ˆθ QB is distributed like a sandwich normal (S 1 n ), the quasi-posterior looks like a single slice of bread normal (B 1 n ). We want to complete the sandwich by joining the slice of bread B 1 n to the open-faced sandwich B n P 1 n to get S 1 n. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 22 / 29

The OFS adjustment The main idea: Whereas ˆθ QB is distributed like a sandwich normal (S 1 n ), the quasi-posterior looks like a single slice of bread normal (B 1 n ). We want to complete the sandwich by joining the slice of bread B 1 n to the open-faced sandwich B n P 1 n to get S 1 n. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 22 / 29

The OFS adjustment The trick: Let Ω = B 1 P 1/2 B 1/2, the (OFS) adjustment matrix. Take samples from π p (θ y) obtained via MCMC and pre-multiply them (after centering) by an estimator ˆΩ of Ω If everything goes according to plan, if you squint a bit, each (centered) sample Z N(0, B 1 ), making the transformed sample Z = ΩZ N(0, S 1 ). So we should end up with a sample that has the right frequentist properties. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 23 / 29

Outline 1 Introduction 2 Spatial Extremes 3 The OFS adjustment 4 Simulation Ben Shaby (SAMSI) OFS adjustment August 3, 2010 24 / 29

Simulated data I simulated 1000 datasets, each with y Smith process(σ) Unit Frechet margins [ ] 0.75 0.5 Σ = 0.5 1.25 100 spatial locations 100 blocks Ben Shaby (SAMSI) OFS adjustment August 3, 2010 25 / 29

MCMC with OFS For each realization of y, we run MCMC using the pairwise likelihood. The OFS matrix is constructed via the four combinations of: 1 ˆP a Monte Carlo estimate of the expected information at θ0 2 ˆP a moment estimate of the expected information at ˆθ 3 ˆB the sample covariance of the MCMC sample 4 ˆB the observed information at ˆθ Intervals are constructed as equi-tailed quantiles of the adjusted MCMC sample, and coverage rates computed. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 26 / 29

Coverage rates Σ σ 11 Σ σ 12 Σ σ 22 coverage 0.0 0.2 0.4 0.6 0.8 1.0 coverage 0.0 0.2 0.4 0.6 0.8 1.0 coverage 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 nominal coverage 0.0 0.2 0.4 0.6 0.8 1.0 nominal coverage 0.0 0.2 0.4 0.6 0.8 1.0 nominal coverage Dashed lines are OFS-adjusted samples, solid line is un-adjusted. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 27 / 29

Summary In summary: Max stable processes are useful for modeling spatial extremes, but their corresponding joint densities are unavailable. One can construct a quasi-posterior using pairwise likelihoods, but The quasi posterior does not reflect parameter uncertainty. Using OFS, we can adjust MCMC samples of the quasi posterior to have the properties we want. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 28 / 29

Summary In summary: Max stable processes are useful for modeling spatial extremes, but their corresponding joint densities are unavailable. One can construct a quasi-posterior using pairwise likelihoods, but The quasi posterior does not reflect parameter uncertainty. Using OFS, we can adjust MCMC samples of the quasi posterior to have the properties we want. A few caveats: The OFS matrix can be difficult to estimate (in particular, the peanut butter center). This approach would really shine in hierarchical models, which I have not shown you. It s not really Bayesian. Ribatet et al. (2010) have a different approach to the same problem. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 28 / 29

References Victor Chernozhukov and Han Hong. An MCMC approach to classical estimation. J. Econometrics, 115(2):293 346, 2003. ISSN 0304-4076. Mathieu Ribatet, Daniel Cooley, and Anthony Davison. Bayesian inference from composite likelihoods, with an application to spatial extremes. Extremes, 2010. to appear. Ben Shaby (SAMSI) OFS adjustment August 3, 2010 29 / 29