Sensor Network Applications for Environmental Monitoring Carla Ellis SAMSI 11-Sept-07.
Approximate Bayesian computation for spatial extremes via...
Transcript of Approximate Bayesian computation for spatial extremes via...
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Approximate Bayesian computation for spatial extremesvia open-faced sandwich adjustment
Ben Shaby
SAMSI
August 3, 2010
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Outline
1 Introduction
2 Spatial Extremes
3 The OFS adjustment
4 Simulation
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Outline
1 Introduction
2 Spatial Extremes
3 The OFS adjustment
4 Simulation
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“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!
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“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!
Ben Shaby (SAMSI) OFS adjustment August 3, 2010 4 / 29
![Page 6: Approximate Bayesian computation for spatial extremes via ...stat.duke.edu/~bs128/papers/shaby_jsm_2010.pdfBen Shaby (SAMSI) OFS adjustment August 3, 2010 23 / 29. Outline 1 Introduction](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b002895cc6a598b6fe329/html5/thumbnails/6.jpg)
A good “posterior”’
An ideal ''posterior''
empirical density of θπ(θ|x)
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Outline
1 Introduction
2 Spatial Extremes
3 The OFS adjustment
4 Simulation
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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
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Floods
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Heat waves
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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!
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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
)
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Smith process in 2 dimensions
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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
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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)?
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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)?
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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.
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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).
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Outline
1 Introduction
2 Spatial Extremes
3 The OFS adjustment
4 Simulation
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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
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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
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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
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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.
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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
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“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)
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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 .
−→
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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 .
−→
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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.
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Outline
1 Introduction
2 Spatial Extremes
3 The OFS adjustment
4 Simulation
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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
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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.
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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.
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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.
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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.
Ben Shaby (SAMSI) OFS adjustment August 3, 2010 28 / 29
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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.
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