Exchangeability and Hierarchical Modeling

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Exchangeability and Hierarchical Modeling

• David Madigan
• Chapter 5 of Gelman et al.

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Borrowing Strength
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Exchangeability

• In the absence of any other information to
  distinguish the qis, one must assume symmetry a
  priori
• The simplest form of exchangeable distribution
  says the qis are iid from some distribution with
  parameter f
• Since f is unknown

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deFinettis Theorem

• As any exchangeable distribution on
  can be written as

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PGM for Hospitals
stochastic
plate
constant
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Conditional Exchangeability

• Suppose we had a covariate xj for each hospital
  that measures typical illness severity for
  incoming patients
• Now could use this model

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Computing the Posterior

• the hospital model is as follows

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Simulation strategy - draw from - draw q from

Prior for ?
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Normal Hierarchical Model
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rats