Partial specification of risk models

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Partial specification of risk models

• Tim Bedford
• Strathclyde Business School
• Glasgow, Scotland

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Contents...

• Modelling considerations
• Information or KL divergence
• Vines
• Copula assessment
• Conclusions

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Modelling goals

• Decision Theory perspective slightly different
  to Statistics
• Modelling aims
• How do we judge what model is appropriate?
• Type
• Outputs
• Level of detail for cost-effective modelling
• Bottom-up or top-down

Caveat convenient labels for discussion only.
Not intended to describe the views of any actual
person.
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What is a model?

• Device for making predictions
• Creative activity giving understanding
• A statement of beliefs and assumptions

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Nominal versus non-nominal

• What behaviour are we trying to capture in risk
  models?
• Nominal / Non-nominal
• EVT is trying to model the extremes of nominal
  behaviour
• Technical risk models try to find non-nominal
  behaviour
• Discrete or continuous nature of departures from
  nominal state is important from modelling
  perspective

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Model as a statement of beliefs

• Often many different plausible statistical models
  consistent with the data
• May be identifiability problems
• Non-statistical validation is required
• Providing finer grain structure to model arising
  from knowledge of context can be a way of
  providing validation

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Model dimensions

• George Mitchell describes 7 dimensions on which
  models can be compared
• Actuality abstract
• Black box structural
• Off the shelf purpose built
• Absolute relative
• Passive behavioural
• Private public
• Subsystem whole system

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Model dimensions - 2
Black box
Structural
FT
ISRD
Statistical Regression
Predictive
Explanatory
Instrumental
Realistic
Macro
Micro
Back of envelope
Large computer
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Role of EJ
Black box
Structural
model

• Decision variables introduced in the model
• Causal connections differ from correlated
  connections

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Example common cause

• Modelling dependent failure in NPPs etc
• Existing models used for risk assessment use
  historical data but give no sights
• Alpha factor model
• Multiple greek letter model
• Etc etc

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Common cause model
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Role of experts

• Structuring providing framework, specifying
  important variables, conditional independencies
• Quantifying providing assessments on quantities
  that they can reasonably assess

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• Essentially, all models are wrong, but some are
  useful
• George Box

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General research theme

• Develop good ways to determine a decision model
  when only a partial specification is possible
• Mainly working in technical/engineering
  applications

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Expert assessment methods

• Many methods for expert assessment of
  distributions - for applications in
  reliability/risk often non-parametric
• Experts provide input by
• Means, covariances..
• Marginal quantiles, product-moment correlations
• Marginal quantiles, rank correlations
• Here look at methods for building up a subjective
  joint distribution

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Building a joint distribution

• Assume experts have given us information on
  marginals
• How do we build a joint distribution with
  information from experts?
• Iman-Conover method
• Markov trees
• Vines

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Copula

• Joint distribution on unit square with uniform
  marginals
• Copula plus marginals specify joint distribution
• If XF and YG then (F(X),G(Y)) is a copula
• Any (Spearman) rank correlation is possible
  between 1, 1. But range of PM correlation
  depends on F and G

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Information

• Also known as
• Relative entropy
• Kullback Leibler divergence
• Coordinate free measure
• Requires specification of background distribution
• Another role for expert?
• Other things being equal....

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Minimum information copulae

• Partially specify the copula, eg by (rank)
  correlation
• Find most independent copula given information
  specified
• Minimize relative information to independent
  copula uniform distribution
• Equivalent to min inf in original space

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Markov trees and Vines

• (Minimum information) copulae used to couple
  random variables
• Marginals specified plus certain (conditional)
  rank correlations
• Main advantage is no algebraic restrictions on
  correlations
• Disadvantage is difficulty of assessing
  correlations

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Markov tree example

• Decomposition Theorem

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Vine example
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Vine example
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Vine example
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Vine example
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Vine example
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Information decomposition
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Information decomposition
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Information calculation
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So

• You can build up multivariate distributions from
  bivariate pieces
• Minimum information pieces give global minimum
  information
• But is it realistic to elicit correlations?

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Observable quantities and expert assessment

• Experts are best able to judge observable
  functions of data
• Distributions of such functions are not free,
  restrictions depend on
• marginals
• other functions being assessed
• REMM project feeds back contradictory
  information real-time to allow experts to reflect
  on inconsistencies
• PARFUM method allows probabilistic inversion of
  random quantities to build joint

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Formulation as minimal information problem

• Define domain specific functions of the variables
  to obtain quantiles

Expectations of regions are fixed by quantiles
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Examples

• Product Moment correlation
• If marginals are known then can just consider
  range of E(XY)
• Equivalent to looking at E(F-1(X)G-1(Y))
• All possible values of this can be reached by the
  min information distributions
• Can map out possible values using information
  function as measure of distance to infeasibility
• Differences quantiles for X-Y

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Example - exp marginals FR 1, 2
E(XY) as a function of lambda
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Example - exp marginals FR 1, 2
Information as a function of lambda
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Copula density for Lambda2
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Copula density for Lambda-2
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Example constraints on X-Y
Marginals as before Expert assesses P(X-Y P(X-Y 39
Sequential Step 1
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Sequential Step 2
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Sequential Step 3
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Conclusions

• General problem of eliciting dependencies from
  experts
• Copulae are a good tool, but how do we select the
  copula?
• Can use vines to get simple parameterisation of
  covariance matrix
• Can tie together observables and copulae using
  interactive computer based methods

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Questions

• Eliciting dependence from experts some work
  done but more needed
• General guidance to use Expert Judgement to add
  structure and variables to existing models, or to
  link models
• Find ways to incorporate features specified by
  experts
• Methods must recognize limitations of experts
• All the usual biases
• Poor in the tails
• Insight not much deeper than the data

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Contributors/Collaborators

• Roger Cooke
• Dorota Kurowicza
• Anca Hanea
• Daniel Lewandowski
• Lesley Walls
• John Quigley
• Athena Zitrou

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D1AD2 algorithm - 1

• If specify expectations of functions
• or equivalently of
• then minimally informative density has form

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D1AD2 algorithm - 2

• View discretised density as matrix product D1AD2
  where Di are diagonal
• Iterative algorithm generates D1 and D2

normalising by
Iteration is contraction in hyperbolic metric