Approximate Bayesian computation for spatial extremes via...

Approximate Bayesian computation for spatial extremesvia 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


Outline

1 Introduction

2 Spatial Extremes


4 Simulation


“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 that1 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!


A good “posterior”’

An ideal ''posterior''

empirical density of θπ(θ|x)


Outline

1 Introduction

2 Spatial Extremes


4 Simulation


Extreme values

Of the environmental variables we care about, usually what we really careabout are the extremes.

heat waves

storms

sea levels

Why do we care?

manage risk (insurance, etc.)

emergency preparedness


Floods


Heat waves


Extreme values

“Extreme values” can mean many things

We consider only “block maxima” (block minima).

Asymptotically follow generalized extreme value (GEV) distribution

G(x) = exp

−[1− ξ

(x− ητ

)]−1/ξ

+

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!


Spatial extremes

It is possible to construct processes with spatial structure and GEVmarginals. This leads us to max stable processes.

The “Smith model” is one example.

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

x

Z(x

)


Smith process in 2 dimensions


GP margins

Lest you fear that this process is unrealistic, the margins don’t have to bethe same everywhere.

Unit Frechet margins Gaussian process GEV parameters


Pairwise likelihoods

Unfortunately, joint likelihoods for the Smith process are not knownfor n ≥ 2.

But we can write the pairwise likelihood, a form of compositelikelihood.

Lp(θ;y) =∏i 6=j

f(yi, yj ;θ)

It turns out that Lp(θ;y) that behaves “similarly” to the likelihood.

Can we trick MCMC into doing something useful with Lp(θ;y)?


The quasi-posterior

Yes!

We define the quasi-posterior distribution as

πp,n(θ|yn) =Lp,n(θ;yn)π(θ)∫

Θ Lp,n(θ;yn)π(θ) dθ,

We will assume, for convenience, that π(θ) proper.

Lp,n is not necessarily a density, so πp,n(θ|yn) is not a true posterior.

Lp,n is integrable, so as long as the prior π(θ) is proper, thenπp,n(θ|Zn) will be a proper density.


More definitions

Now we can write down a quasi-Bayes estimator

Define loss in the usual way.

Define quasi-posterior risk Rn(θ) as the quasi-posterior expectationof loss.

The pairwise quasi-Bayes estimator is then

θQB = argminθ∈Θ

Rn(θ).


Outline

1 Introduction

2 Spatial Extremes


4 Simulation


The sandwich matrix

Pn = E0[∇0`p,n∇0`′p,n]

Bn = −E0[∇20`p,n]

Sn = Bn P−1n Bn

Bread

Peanut butter

Bread


Asymptotic normality of quasi-Bayes estimators

Then as long as we don’t use a crazy prior, Chernozhukov and Hong(2003) says that:

Theorem

S1/2n (θQB − θ0)

D−→ N(0, I)

When we use pairwise likelihoods for MCMC, the sandwich matrixdescribes the (asymptotic) sampling variability of the estimator.


Convergence of the quasi-posterior

Furthermore, (also from Chernozhukov and Hong, 2003)

Theorem

Asymptotically, πp,n(θ|yn) ∼ N(θ0,B−1n ).

This has important consequences for inference from the MCMC sample!

B−1 6= B−1PB−1!

⇒ Equi-tailed credible intervals based on MCMC quantiles will NOThave the correct frequentist coverage probabilities


“Distortion” of the posterior

The two curves are very different!

0.5 1.0 1.5 2.0

02

46

8

θ

Den

sity

empirical density of θπp (θ|x)


The OFS adjustment

The main idea:

Whereas θQB is distributed like a sandwich normal (S−1n ), the

quasi-posterior looks like a single slice of bread normal (B−1n ).

We want to complete the sandwich by joining the slice of bread B−1n

to the open-faced sandwich BnP−1n to get S−1

n .

−→


The OFS adjustment

The trick:

Let Ω = B−1P1/2B1/2, the (OFS) adjustment matrix.

Take samples from πp(θ|y) obtained via MCMC and pre-multiply

them (after centering) by an estimator Ω of Ω

If everything goes according to plan, if you squint a bit, each(centered) sample Z ∼ N(0,B−1), making the transformed sampleZ∗ = ΩZ ∼ N(0,S−1).

So we should end up with a sample that has the right frequentistproperties.


Outline

1 Introduction

2 Spatial Extremes


4 Simulation


Simulated data

I simulated 1000 datasets, eachwith

y ∼ Smith process(Σ)

Unit Frechet margins

Σ =

[0.75 −0.5−0.5 1.25

]100 spatial locations

100 blocks


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 P a Monte Carlo estimate of the expected information at θ02 P a moment estimate of the expected information at θ3 B the sample covariance of the MCMC sample4 B the observed information at θ

Intervals are constructed as equi-tailed quantiles of the adjustedMCMC sample, and coverage rates computed.


Coverage rates

σ11

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Σ11

nominal coverage

cove

rage

σ12

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Σ12

nominal coverage

cove

rage

σ22

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Σ22

nominal coverage

cove

rage

Dashed lines are OFS-adjusted samples, solid line is un-adjusted.


Summary

In summary:

Max stable processes are useful for modeling spatial extremes, buttheir 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 tohave 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 havenot shown you.

It’s not really Bayesian.

Ribatet et al. (2010) have a different approach to the same problem.


References

Victor Chernozhukov and Han Hong. An MCMC approach to classicalestimation. J. Econometrics, 115(2):293–346, 2003. ISSN 0304-4076.

Mathieu Ribatet, Daniel Cooley, and Anthony Davison. Bayesian inferencefrom composite likelihoods, with an application to spatial extremes.Extremes, 2010. to appear.


Approximate Bayesian computation for spatial extremes via...

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