Quant Toolbox - 24. Bayesian Statistics - Conditional likelihood prior posterior predictive
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Transcript of Quant Toolbox - 24. Bayesian Statistics - Conditional likelihood prior posterior predictive
Quant Toolbox > 23. Bayesian statistics > Analytical resultsConjugate distribution
Conjugate distribution
A conjugate distribution with respect to a Bayesian model (24.3) is aparametric distribution fη with a set of hyperparameters η such that
fpri(θ) = fηpri(θ) ⇒ fpos(θ) = fηpos
(θ) (24.16)
where fpri is the prior distribution (24.4) and fpos is the posteriordistribution (24.6).
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Feb-21-2017 - Last update
Quant Toolbox > 23. Bayesian statistics > Analytical resultsConjugate distribution
Conjugate distribution
Consider the exponential family
X ∼ Exp(θ, φ(·), h(·)) ⇔ f(x|θ) = h(x)eθ′φ(x)−ψ(θ) (23.16)
where θ ≡ (θ1, . . . , θl̄)′ are the natural parameters; h(x) > 0 is the base
measure; φ(x) ≡ (φ1(x), . . . , φl̄(x))′ are the features; and ψ(θ) is the
log-partition.
The conjugate distribution for the natural parameters is
Θ|η, ν ∼ Conj (ψ (·) ,η, ν) ⇔ fψ(θ|η, ν) = g(η, ν)eν(θ′η−ψ(θ)) (23.17)
where g(η, ν) normalizes the pdf to integrate to one
g(η, ν) =1∫
Rl̄ eν(θ′η−ψ(θ))dθ(23.18)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Feb-21-2017 - Last update
Quant Toolbox > 23. Bayesian statistics > Analytical resultsConjugate distribution
Properties of the conjugate distribution
• The conjugate (23.17) in general does not belong to the exponentialfamily.
• The hyperparameter η ∈ Rl̄ satisfies
η = ∇θψ(Mod{Θ}) (23.19)
• The parameter η is the most likely value of the expected featuresimplied by the conjugate
η ≈ E{φ(X)} (23.20)
Example: normal distribution with unit variance
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Feb-21-2017 - Last update
Quant Toolbox > 23. Bayesian statistics > Analytical resultsConjugate distribution
Properties of the conjugate distribution
• If the model (24.3) is exponential (24.17) and the prior (24.4) isconjugate (24.18), then the posterior (24.16) is also conjugate
X ∼ Exp(θ, φ(·), h(·))Θ|ηpri , νpri ∼ Conj (ψ (·) ,ηpri , νpri)
}⇒ Θ|ηpos , νpos ∼ Conj (ψ (·) ,ηpos , νpos) (24.25)
with new parameters
ηpos ≡νpri
νpri + 1ηpri +
1
νpri + 1φ(x), νpos ≡ νpri + 1 (24.26)
• The predictive distribution reads
fpred(x) = h(x)g(ηpos , νpos)
g( 11+νpos
φ(x) +νpos
1+νposηpos , 1 + νpos)
(24.27)
Example: normal distribution with unit variance
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Feb-21-2017 - Last update