This document discusses Bayesian estimation of conditional moment models. It presents several approaches for completing conditional moment models for Bayesian processing, including using non-parametric parts, empirical likelihood Bayesian tools, or maximum entropy alternatives. It also discusses simplistic ABC alternatives and innovative aspects of introducing tolerance parameters for misspecification and cancelling conditional aspects. Unconditional and conditional model comparison using empirical likelihoods and Bayes factors is proposed.

1. Bayesian estimation of conditional
moment models: Discussion
Christian P. Robert
(Paris Dauphine PSL & Warwick U.)
10th French Econometrics Conference

2. [conditional] moment models
Model partly defined by moment conditions
Requires completion by non-parametric part for Bayesian
processing
Or use of empirical likelihood Bayesian tools
[Schennach, 2005; Chib, Shin, Simoni, 2017, 2018]
Warnin: conditional aspects secondary to the discussion
why is Schennach (2005) unknown to the BNP community?

3. [conditional] moment models
Model partly defined by moment conditions
Requires completion by non-parametric part for Bayesian
processing
Or use of empirical likelihood Bayesian tools
[Schennach, 2005; Chib, Shin, Simoni, 2017, 2018]
Warnin: conditional aspects secondary to the discussion
why is Schennach (2005) unknown to the BNP community?

4. Empirical likelihood
Given dataset x1, . . . , xn and moment constraints
E[g(X, θ)] = 0
empirical likelihood defined as
emp
(θ|y) =
n
i=1
ˆpi
with (ˆp1, . . . , ˆpn) minimising
n
i=1
pi log(pi)
under the constraint
n
i=1
pig(xi, θ) = 0
[Owen, 1988]

5. Bayesian empirical likelihood
Under prior π(θ), substitute empirical likelihood to true
likelihood
πemp
(θ|y) ∝ π(θ) emp
(θ|y)
[Lazar, 2005; Mengersen, Pudlo, Robert, 2013]

6. Maximum entropy alternative?
Given a reference measure dµ, use the least informative
sampling distribution [if it exists]
f0(x|θ) =
dPθ(x)
dµ
= exp{λ(θ)T
g(x, θ)}
with λ(θ) Lagrangian for constraint
[Jaynes, 2008]
and associate with prior π(θ)
Warnin: Lagrangian differs from BETEL representation of
empirical likelihood
Unlikely to be a generative model

7. Maximum entropy alternative?
Given a reference measure dµ, use the least informative
sampling distribution [if it exists]
f0(x|θ) =
dPθ(x)
dµ
= exp{λ(θ)T
g(x, θ)}
with λ(θ) Lagrangian for constraint
[Jaynes, 2008]
and associate with prior π(θ)
Warnin: Lagrangian differs from BETEL representation of
empirical likelihood
Unlikely to be a generative model

8. Simplistic ABC alternative?
In case generative model available [standard ABC]
Generate θ from π(θ)
Generate pseudo-sample y∗(θ)
Compute
ρ(θ) =
1
n
n
i=1
{g(yi, θ) − g(y∗
i (θ), θ)}
Select smallest 1% in terms of distance ||ρ(θ)||
[Rubin, 1984; Tavar´e et al., 1998]

9. Simplistic ABC alternative?
In case generative model unavailable [standard ABC
inapplicable]
Generate θ from π(θ)
Compute
ρ(θ) =
1
n
n
i=1
g(yi, θ)
Select smallest 1% in terms of distance ||ρ(θ)||
[Rubin, 1984; Tavar´e et al., 1998]

10. Innovative aspects
introduction of tolerance (nuisance) extra-parameters when
some constraints do not hold (misspecification)
Chib-Jeliazkov representation of the marginal empirical
likelihood
Bernstein-von Mises theorem (aka CLT) under correct
specification and under misspecification
cancelling conditional aspects by choice of (sub) functional
basis [with extra parameter K]
choice of K removing conditional structure based solely on
asymptotic and greatly increasing the number of constraints
hence presumably lowering efficiency [and increasing cost?]
Role of prior on indicators?

11. Innovative aspects
introduction of tolerance (nuisance) extra-parameters when
some constraints do not hold (misspecification)
Chib-Jeliazkov representation of the marginal empirical
likelihood
Bernstein-von Mises theorem (aka CLT) under correct
specification and under misspecification
cancelling conditional aspects by choice of (sub) functional
basis [with extra parameter K]
choice of K removing conditional structure based solely on
asymptotic and greatly increasing the number of constraints
hence presumably lowering efficiency [and increasing cost?]
Role of prior on indicators?

12. Unconditional model comparison
Novel and exciting aspect: to compare models (or rather
moment restrictions) by genuine Bayes factors derived from
empirical likelihoods.
Grand (encompassing) model obtained by considering all
moment restrictions at once and use of spike & slab prior on
indicators of active constraints
[George and Foster, 1997]
First sounds like too restrictive, except extra-parameters there
to monitor constraints that actually hold
unclear whether or not priors on extra-parameters can be
automatically derived from single prior
how much impact value of Bayes factor

13. Conditional model comparison
Extension without grand model is major improvement, thanks
to the candidate’s formula
log m (y) = log π (θ∗
) + log p (y|θ∗
) − log π (θ∗
|y)
[Besag, 1989; Chib, 1995]
Plus some consistency results
Requires simulation from eMCMC for each subset of
constraints, with a potential combinatoric explosion