Model Assessment and Adaptive
Design Group (MAAD)
Carlos M. Carvalho
Summary Report
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Model Assessment and Adaptive Design
• General Goal: Enhance, explore and demonstrate the potential of
particle based methods to represent sequential model uncertainty.
• Specifics...
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Explore the ability of SeqMC methods to effectively explore
very large discrete model spaces.
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Parallel computation issues... connections to Shotgun
methods of Jones et. al (2005) and Hans, Dobra and West
(2006)
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Reinvent chapter 11 of West and Harrison (1997). Model
monitoring and intervention.
• Models and Applications...
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Different specification of the economy... again, DSGE models
in Macroeconomics.
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Finance models...
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Dynamic variable selection
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...
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Summary of Discussion: Week 1
Three main topics were discussed:
1. Marginal Likelihood Computation via SMC
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Many examples in Finance and Economics...
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Astrophysics...
2. SMC Exploration of Large Model Spaces
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Variable selection problems
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DLM model specification
3. Adaptive Design and Control
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Astrophysics context
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Dynamic control and decision making
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Five Months Later...
1. Marginal Likelihood Computation via SMC
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Astrophysics (Tom Loredo et al.)
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Non-parametric regression (Dunson, Das and Shi)
2. SMC Exploration of Large Model Spaces
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Variable selection problems (Dunson, Das and Shi)
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DLM model specification (Hao, Reeson and Carvalho)
3. Adaptive Design and Control
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Dynamic Control and Sequential Decision Making (Argon,
Rodriguez and Bain)
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Adaptive Sampling (Liu, Li and Dunson)
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Papers in preparation...
1. “Bayesian Distribution Regression via Augmented Particle
Learning” (Dunson and Das, 09)
2. “Particle Stochastic Search for High-dimensional Variable
Selection” (Shi and Dunson, 09)
2. “Sequential Learning in Dynamic Graphical Models” (Wang,
Reeson and Carvalho, 09)
2. “Adaptive Sampling for Bayesian Variable Selection” (Liu, Li
and Dunson, 09)
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Sequential Learning in Dynamic
Graphical Models
Hao Wang
Craig Reeson
Carlos M. Carvalho
Application Context
High-dimensional asset allocation problems
9 “Structured” models for the time-varying covariance
matrix of a large number of assets
9 Tractable models in very many dimensions
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Gaussian Graphical Models
9 A graph G=(V,E) defines conditional independence relationships for
a set of random variables
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Hyper-Inverse Wishart
9 Conjugate prior (local) distribution for
9 Unique hyper-Markov law with consistent componentmarginals
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Exchange Rate Example
Explore distribution of weights
Carvalho, Massam and West (2007) – Biometrika
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Exchange Rate Example
Smaller variability of portfolio weights
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Portfolios and Graphs
Stevens (1998)
“… the amount of money invested in asset i depends on the
ratio of the expected return that cannot be explained by the
linear combination of assets to the nondiversifiable risk.”
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DLMs and Graphs
Conditional independence in multivariate time series models
Common components defining each individual DLMs
General, fully-conjugate framework
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Extension to Graphs
Closed-form sequential update
9 Forecasting
9 Filtering – retrospective analysis
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Sequential Update
Posterior at time t-1:
Prior at time t:
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Sequential Update
One-step forecast:
Posterior at time t:
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Time-Varying Covariance
9 Treat the covariance matrix as a state in the model, allowing it to
vary stochastically in time
9 Define an evolution equation -- transition distribution
Stochastic map established through multiplicative beta shocks to the
diagonal elements of the Cholesky decomposition of the precision
9 Generalizes Uhlig(1994) and extends Quintana(2003) to graphs
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Time-Varying Covariance
“Locally Smooth” – Dynamic Discounting
Information loss – discount factor
Prior has the same “location” of the previous posterior
Implies a random walk in the transition equation of log-variances
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Sequential Update
Posterior at t-1:
Prior at t:
Posterior at t:
Approximate EWMA for the harmonic mean
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EX Example Revisited
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EX Example Revisited
δ = 0.99
δ = 0.90
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Dynamic Portfolios (EX)
Variance of Portfolio Weights
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EX Example Revisited
9 Higher realized cumulative returns – structure
helps in practical terms
9 Lower risk portfolios in terms of one-step ahead
predicted variances
9 Lower volatility of the optimal portfolio weights –
more stable portfolios
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S&P 500 – Model Selection
9 346 stocks in the index from 1999 until 2004
Challenge: uncertainty about the graph.
Efficient exploration of model space
Metropolis Search
Shotgun Stochastic Search
Jones, Carvalho et. al. (2005) – Stat. Sci.
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Marginal Likelihood for Graphs
Computations facilitated by one-edge moves
Small changes to allow for time dependence
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S&P 500
Top graph with 29,181 edges
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Modelling time-varying Gt
Recursively conditional sufficient statistic for (Gt , Σt ):
St = δSt−1 + yt yt0
Information set Dt = {y1 , · · · , yt } and update Gt as follows
i Posterior at time t:
p(Gt | Dt ) = p(Gt | St ) ∝ π(Gt )p(Y1:t |Gt , St )
ii Prior at time t + 1 : p(Gt+1 | Dt ) ∝ π(Gt+1 )p(Y1:t |Gt+1 , δSt )
iii Posterior at time t + 1 :
p(Gt+1 | Dt+1 ) = p(Gt+1 | St+1 ) ∝ π(Gt+1 )p(Y1:t+1 |Gt+1 St+1 )
Wang, Reeson and Carvalho
Sequential Graph DLM
Sequential Learning of Gt
Particle Learning (Carvalho et al., 2008) + Particle
Stochastic Search (Shi & Dunson, 2009)
Suppose at time t, most probable graphs
(i)
{Gt } ∼ p(Gt | Dt )
Re-sample: ∝ p(Y1:t+1 |Gt , St+1 )
Propagate:
i Update estimation of edge inclusion probabilities of
p(Gt+1 | Dt+1 )
ii Sample new particles using edge inclusion probabilities
iii Repeat until “some” stop criteria is satisfied.
Wang, Reeson and Carvalho
Sequential Graph DLM
Exchange Rate Data
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Figure:
Sequential Graph DLM
Wang, Reeson and Carvalho
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References
n Carvalho and West (2007) “Dynamic Matrix-Variate Graphical
Models”
o Wang, Reeson and Carvalho (2009) “Sequential Learning in
Dynamic Graphical Models”
p Carvalho and Scott (2009) “Objective Bayesian Model Selection in
Gaussian Graphical Models”
q Reeson, Carvalho and West (2009) “Financial Time Series Graphical
Modeling and Portfolio Analysis”
r Quintana, Carvalho, Scott and Costigliola (2009)“Futures Markets,
Bayesian Forecasting and Risk Modeling”
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