Let's Practice What We Preach: Likelihood Methods for Monte Carlo Data

Let’s Practice What We Preach:
Likelihood Methods for Monte Carlo Data

Xiao-Li Meng

Department of Statistics, Harvard University

September 24, 2011

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Xiao-Li Meng (Harvard) MCMC+likelihood September 24, 2011 1 / 23

Let’s Practice What We Preach:
Likelihood Methods for Monte Carlo Data

Xiao-Li Meng

Department of Statistics, Harvard University

September 24, 2011

Based on
Kong, McCullagh, Meng, Nicolae, and Tan (2003, JRSS-B, with
discussions);
Kong, McCullagh, Meng, and Nicolae (2006, Doksum Festschrift);
Tan (2004, JASA); ..., Meng and Tan (201X)
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Importance sampling (IS)

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Estimand:
q1 (x)
c1 = q1 (x)µ(dx) = p2 (x)µ(dx).
Γ Γ p2 (x)

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Estimand:
q1 (x)
c1 = q1 (x)µ(dx) = p2 (x)µ(dx).
Γ Γ p2 (x)

Data: {Xi2 , i = 1, . . . n2 } ∼ p2 = q2 /c2

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Estimand:
q1 (x)
c1 = q1 (x)µ(dx) = p2 (x)µ(dx).
Γ Γ p2 (x)

Data: {Xi2 , i = 1, . . . n2 } ∼ p2 = q2 /c2
Estimating Equation (EE):

c1 q1 (X )
r≡ = E2 .
c2 q2 (X )

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Estimand:
q1 (x)
c1 = q1 (x)µ(dx) = p2 (x)µ(dx).
Γ Γ p2 (x)

Data: {Xi2 , i = 1, . . . n2 } ∼ p2 = q2 /c2

c1 q1 (X )
r≡ = E2 .
c2 q2 (X )

The EE estimator:
n2
1 q1 (Xi2 )
ˆ=
r
n2 q2 (Xi2 )
i=1

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Estimand:
q1 (x)
c1 = q1 (x)µ(dx) = p2 (x)µ(dx).
Γ Γ p2 (x)

Data: {Xi2 , i = 1, . . . n2 } ∼ p2 = q2 /c2

c1 q1 (X )
r≡ = E2 .
c2 q2 (X )

The EE estimator:
n2
1 q1 (Xi2 )
ˆ=
r
n2 q2 (Xi2 )
i=1

Standard IS estimator for c1 when c2 = 1. logo


What about MLE?

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What about MLE?

The “likelihood” is:
n2
f (X12 . . . Xn2 2 ) = p2 (Xi2 ) — free of the estimand c1 !
i=1

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What about MLE?

n2
i=1

So why are {Xi2 , i = 1, . . . n2 } even relevant?
Violation of likelihood principle?

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What about MLE?

n2
i=1

So why are {Xi2 , i = 1, . . . n2 } even relevant?
Violation of likelihood principle?
What are we “inferring”?
What is the “unknown” model parameter?

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Bridge sampling (BS)

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Data: {Xij , i = 1, . . . , nj } ∼ pj = qj /cj , j = 1, 2

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Estimating Equation (Meng and Wong, 1996):

c1 E2 [α(X )q1 (X )]
r≡ = , ∀α: 0<| αq1 q2 dµ| < ∞
c2 E1 [α(X )q2 (X )]

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c1 E2 [α(X )q1 (X )]
r≡ = , ∀α: 0<| αq1 q2 dµ| < ∞
c2 E1 [α(X )q2 (X )]

Optimal choice: αO (x) ∝ [n1 q1 (x) + n2 rq2 (x)]−1

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c1 E2 [α(X )q1 (X )]
r≡ = , ∀α: 0<| αq1 q2 dµ| < ∞
c2 E1 [α(X )q2 (X )]

Optimal choice: αO (x) ∝ [n1 q1 (x) + n2 rq2 (x)]−1
Optimal estimator Ô , the limit of
r
n2
1 q1 (Xi2 )
n2 (t)
s1 q1 (Xi2 )+s2 Ô q2 (Xi2 )
r
(t+1) i=1
Ô
r = n1
1 q2 (Xi1 )
n1 (t)
s1 q1 (Xi1 )+s2 Ô q2 (Xi1 )
r
i=1
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What about MLE?

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What about MLE?

2 nj
qj (Xij ) −n −n
∝ c1 1 c2 2 — free of data!
cj
j=1 i=1

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What about MLE?

2 nj
qj (Xij ) −n −n
cj
j=1 i=1

What went wrong: cj is not “free parameter” because
cj = Γ qj (x)µ(dx) and qj is known.

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What about MLE?

2 nj
qj (Xij ) −n −n
cj
j=1 i=1

So what is the “unknown” model parameter?

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What about MLE?

2 nj
qj (Xij ) −n −n
cj
j=1 i=1

Turns out ˆO is the same as Bennett’s (1976) optimal acceptance
r
ratio estimator, as well as Geyer’s (1994) reversed logistic regression
estimator.

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What about MLE?

2 nj
qj (Xij ) −n −n
cj
j=1 i=1

Turns out ˆO is the same as Bennett’s (1976) optimal acceptance
r
ratio estimator, as well as Geyer’s (1994) reversed logistic regression
estimator.
So why is that? Can it be improved upon without any “sleight of
hand”?
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Pretending the measure is unknown!

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Because
c= q(x)µ(dx),
Γ
and q is known in the sense that we can evaluate it at any sample
value, the only way to make c “unknown” is to assume the underlying
measure µ is “unknown”.

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Because
c= q(x)µ(dx),
Γ
This is natural because Monte Carlo simulation means we use samples
to represent, and thus estimate/infer, the underlying population
q(x)µ(dx), and hence estimate/infer µ since q is known.

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Because
c= q(x)µ(dx),
Γ
This is natural because Monte Carlo simulation means we use samples
to represent, and thus estimate/infer, the underlying population
q(x)µ(dx), and hence estimate/infer µ since q is known.
Monte Carlo integration is about ﬁnding a tractable discrete µ to
ˆ
approximate the intractable µ.

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Importance Sampling Likelihood

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Estimand: c1 = Γ q1 (x)µ(dx)

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−1
Data: {Xi2 , i = 1, . . . n2 } ∼ i.i.d. c2 q2 (x)µ(dx)

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−1
Likelihood for µ:
n2
−1
L(µ) = c2 q2 (Xi2 )µ(Xi2 )
i=1

Note that c2 is a functional of µ.

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−1
Likelihood for µ:
n2
−1
L(µ) = c2 q2 (Xi2 )µ(Xi2 )
i=1

Note that c2 is a functional of µ.
The nonparametric MLE of µ is
ˆ
P(dx)
µ(dx) =
ˆ , ˆ
P — empirical measure
q2 (x)
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Thus the MLE for r ≡ c1 /c2 is
n2
1 q1 (Xi2 )
ˆ=
r q1 (x)ˆ(dx) =
µ
n2 q2 (Xi2 )
i=1

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n2
1 q1 (Xi2 )
ˆ=
r q1 (x)ˆ(dx) =
µ
n2 q2 (Xi2 )
i=1

When c2 = 1, q2 = p2 , standard IS estimator for c1 is obtained.

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n2
1 q1 (Xi2 )
ˆ=
r q1 (x)ˆ(dx) =
µ
n2 q2 (Xi2 )
i=1

When c2 = 1, q2 = p2 , standard IS estimator for c1 is obtained.
{X(i2) , i = 1, . . . n2 } is (minimum) suﬃcient for µ on
x ∈ S2 = {x : q2 (x) > 0}, and hence c1 is guaranteed to be
ˆ
consistent only when S1 ⊂ S2 .

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Bridge Sampling Likelihood

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Estimand: ∝ cj = Γ qj (x)µ(x), j = 1, . . . , J.

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Data: {Xij , 1 ≤ i ≤ nj } ∼ cj−1 qj (x)µ(dx), 1 ≤j ≤J

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Data: {Xij , 1 ≤ i ≤ nj } ∼ cj−1 qj (x)µ(dx), 1 ≤ j ≤ J
nj
Likelihood for µ: L(µ) = J j=1
−1
i=1 cj qj (Xij )µ(Xij )

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Data: {Xij , 1 ≤ i ≤ nj } ∼ cj−1 qj (x)µ(dx), 1 ≤ j ≤ J
nj
Likelihood for µ: L(µ) = J j=1
−1
i=1 cj qj (Xij )µ(Xij )
Writing θ(x) = log µ(x), then
J
log L(µ) = n ˆ
θ(x)d P − nj log cj (θ),
Γ j=1

ˆ
P is the empirical measure on {Xij , 1 ≤ i ≤ nj , 1 ≤ j ≤ J}.

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ˆ
MLE for µ given by equating the canonical suﬃcient statistics P to
its expectation:
J
ˆ
nP(dx) = nj cj−1 qj (x)ˆ(dx),
ˆ µ
j=1

ˆ
nP(dx)
µ(dx) =
ˆ J
. (A)
ˆ−1
j=1 nj cj qj (x)

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ˆ
its expectation:
J
ˆ
ˆ µ
j=1

ˆ
nP(dx)
µ(dx) =
ˆ J
. (A)
ˆ−1
j=1 nj cj qj (x)

Consequently, the MLE for {c1 , . . . , cJ } must satisfy
J nj
qr (xij )
cr =
ˆ qr (x) d µ =
ˆ J
. (B)
Γ j=1 i=1 s=1 ˆ−1
ns cs qs (xij )

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ˆ
its expectation:
J
ˆ
ˆ µ
j=1

ˆ
nP(dx)
µ(dx) =
ˆ J
. (A)
ˆ−1
j=1 nj cj qj (x)

Consequently, the MLE for {c1 , . . . , cJ } must satisfy
J nj
qr (xij )
cr =
ˆ qr (x) d µ =
ˆ J
. (B)
Γ j=1 i=1 s=1 ˆ−1
ns cs qs (xij )

(B) is the “dual” equation of (A), and is also the same as the logo
equation for optimal multiple bridge sampling estimator (Tan 2004).

But We Can Ignore Less ...

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To restrict the parameter space for µ by using some knowledge of the
known µ, that it, to set up a sub-model.

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The new MLE has a smaller asymptotic variance under the submodel
than under the full model.

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Examples:

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Examples:
Group-invariance submodel

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Examples:
Linear submodel

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Examples:
Linear submodel
Log-linear submodel

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An Universally Improved IS

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Estimand: r = c1 /c2 ; cj = Rd qj (x)µ(dx)

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Estimand: r = c1 /c2 ; cj = R d qj (x)µ(dx)
−1
Data: {Xi2 , i = 1, . . . n2 } i.i.d ∼ c2 q2 µ(dx)

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−1
Taking G = {Id , −Id } leads to

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−1
n2
1 q1 (Xi2 ) + q1 (−Xi2 )
ˆG =
r .
n2 q2 (Xi2 ) + q2 (−Xi2 )
i=1

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−1
n2
1 q1 (Xi2 ) + q1 (−Xi2 )
ˆG =
r .
n2 q2 (Xi2 ) + q2 (−Xi2 )
i=1

Because of the Rao-Blackwellization, V(ˆG ) ≤ V(ˆ).
r r

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−1
n2
1 q1 (Xi2 ) + q1 (−Xi2 )
ˆG =
r .
n2 q2 (Xi2 ) + q2 (−Xi2 )
i=1

r r
Need twice as many evaluations, but typically this is a small insurance
premium.

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−1
n2
1 q1 (Xi2 ) + q1 (−Xi2 )
ˆG =
r .
n2 q2 (Xi2 ) + q2 (−Xi2 )
i=1

r r
premium.
Consider S1 = R & S2 = R + . Then ˆG is consistent for r :
r
n2 n2
1 q1 (Xi2 ) 1 q1 (−Xi2 )
ˆG =
r + .
n2 q2 (Xi2 ) n2 q2 (Xi2 )
i=1 i=1
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−1
n2
1 q1 (Xi2 ) + q1 (−Xi2 )
ˆG =
r .
n2 q2 (Xi2 ) + q2 (−Xi2 )
i=1

r r
premium.
Consider S1 = R & S2 = R + . Then ˆG is consistent for r :
r
n2 n2
1 q1 (Xi2 ) 1 q1 (−Xi2 )
ˆG =
r + .
n2 q2 (Xi2 ) n2 q2 (Xi2 )
i=1 i=1
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∞
But standard IS ˆ only estimates
r 0 q1 (x)µ(dx)/c2 .

There are many more improvements ...

Deﬁne a sub-model by requiring µ to be G-invariant, where G is a
ﬁnite group on Γ.

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The new MLE of µ is
ˆ
nP G (dx)
µG (dx) =
ˆ J
,
ˆ−1 G
j=1 nj cj q j (x)

ˆ ˆ
where P G (A) = aveg ∈G P(gA); q j G (x) = aveg ∈G qj (gx).

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ˆ
nP G (dx)
µG (dx) =
ˆ J
,
ˆ−1 G
j=1 nj cj q j (x)

ˆ ˆ
When the draws are i.i.d. within each ps dµ,

µG = E [ˆ| GX ],
ˆ µ

i.e., the Rao-Blackwellization of µ given the orbit.
ˆ

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ˆ
nP G (dx)
µG (dx) =
ˆ J
,
ˆ−1 G
j=1 nj cj q j (x)

ˆ ˆ
When the draws are i.i.d. within each ps dµ,

µG = E [ˆ| GX ],
ˆ µ

i.e., the Rao-Blackwellization of µ given the orbit.
ˆ
Consequently,

cj G =
ˆ qj (x)µG (dx) = E [ˆj |GX ].
c logo
Γ


Using Groups to model trade-oﬀ

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If G1 G2 , then
G1 G2
Var c ≤ Var c .

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If G1 G2 , then
G1 G2
Var c ≤ Var c .

The statistical eﬃciency increases with the size of Gi , but so does the
computational cost needed for function evaluation (but not for
sampling, because there are no additional samples involved).

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Linear submodel: stratiﬁed sampling (Tan 2004)

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i.i.d
Data: {Xij , 1 ≤ i ≤ nj } ∼ pj (x)µ(dx), 1 ≤ j ≤ J.

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i.i.d
The sub-model has parameter space

µ: pj (x) µ(dx), 1 ≤ j ≤ J, are equal (to 1).
Γ

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i.i.d

Γ

J nj
Likelihood for µ: L(µ) = j=1 i=1 pj (Xij )µ(Xij )

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i.i.d

Γ

J nj
Likelihood for µ: L(µ) = j=1 i=1 pj (Xij )µ(Xij )
The MLE is
ˆ
P(dx)
µlin (dx) =
ˆ J
,
j=1 πj pj (x)
ˆ
where πj s are MLEs from a mixture model:
ˆ
i.i.d J
the data ∼ j=1 πj pj (·) with πj s unknown
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So why MLE?

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So why MLE?
Goal: to estimate c = Γ q(x)µ(dx).

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So why MLE?
For an arbitrary vector b, consider the control-variate estimator
(Owen and Zhou 2000)
J nj
q(xji ) − b g (xji )
cb ≡
ˆ J
,
j=1 i=1 s=1 ns ps (xji )

where g = (p2 − p1 , . . . , pJ − p1 ) .

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So why MLE?
J nj
cb ≡
ˆ J
,

where g = (p2 − p1 , . . . , pJ − p1 ) .
A more general class: for J λj (x) ≡ 1 and J λj (x)bj (x) ≡ b,
j=1 j=1
consider (Veach and Guibas 1995 for bj ≡ 0; Tan, 2004)
J nj
1 q(xji ) − bj (xji )g (xji )
cλ,B =
ˆ λj (xji )
nj pj (xji )
j=1 i=1

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So why MLE?
J nj
cb ≡
ˆ J
,

where g = (p2 − p1 , . . . , pJ − p1 ) .
A more general class: for J λj (x) ≡ 1 and J λj (x)bj (x) ≡ b,
j=1 j=1
consider (Veach and Guibas 1995 for bj ≡ 0; Tan, 2004)
J nj
1 q(xji ) − bj (xji )g (xji )
cλ,B =
ˆ λj (xji )
nj pj (xji )
j=1 i=1

Should cλ,B be more eﬃcient than cb ? Could there be something
ˆ ˆ logo

even more eﬃcient?

Three estimators for c = Γ q(x) µ(dx):

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IS: 1
n
q(xi )
J
,
n j=1 πj pj (xi )
i=1

where πj = nj /n are the true proportions.

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IS: 1
n
q(xi )
J
,
n j=1 πj pj (xi )
i=1


Reg: n ˆ
1 q(xi ) − β g (xi )
J
,
n j=1 πj pj (xi )
i=1

ˆ
where β is the estimated regression coeﬃcient, ignoring stratiﬁcation.

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IS: 1
n
q(xi )
J
,
n j=1 πj pj (xi )
i=1


Reg: n ˆ
1 q(xi ) − β g (xi )
J
,
n j=1 πj pj (xi )
i=1

ˆ

Lik: 1
n
q(xi )
J
,
n j=1 πj pj (xi )
ˆ
i=1

where πj s are the estimated proportions, ignoring stratiﬁcation.
ˆ
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IS: 1
n
q(xi )
J
,
n j=1 πj pj (xi )
i=1


Reg: n ˆ
1 q(xi ) − β g (xi )
J
,
n j=1 πj pj (xi )
i=1

ˆ

Lik: 1
n
q(xi )
J
,
n j=1 πj pj (xi )
ˆ
i=1

where πj s are the estimated proportions, ignoring stratification.
ˆ
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Which one is most efficient? Least efficient?

Let’s ﬁnd it out ...

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Γ = R10 and µ is Lebesgue measure.

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The integrand is
10 10
q(x) = 0.8 φ(x j ) + 0.2 ψ(x j ; 4) ,
j=1 j=1

where φ(·) is standard normal density and ψ(·; 4) is t4 density.

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The integrand is
10 10
q(x) = 0.8 φ(x j ) + 0.2 ψ(x j ; 4) ,
j=1 j=1

Two sampling designs:

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The integrand is
10 10
q(x) = 0.8 φ(x j ) + 0.2 ψ(x j ; 4) ,
j=1 j=1

(i) q2 (x) with n draws, or

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The integrand is
10 10
q(x) = 0.8 φ(x j ) + 0.2 ψ(x j ; 4) ,
j=1 j=1

(i) q2 (x) with n draws, or
(ii) q1 (x) and q2 (x) each with n/2 draws,
where
10 10
q1 (x) = φ(x j ), q2 (x) = ψ(x j ; 1)
j=1 j=1 logo


A little surprise?

Table: Comparison of design and estimator

one sampler two samplers
IS Reg Lik IS Reg Lik
Sqrt MSE .162 .00942 .00931 .0175 .00881 .00881
Std Err .162
.00919 .00920 .0174 .00885 .00884
√
Note: Sqrt MSE is mean squared error of the point estimates and
√
Std Err is mean of the variance estimates from 10000 repeated
simulations of size n = 500.

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Comparison of eﬃciency:

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Statistical eﬃciency: IS < Reg ≈ Lik

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IS is a stratiﬁed estimator, which uses only the labels.

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Reg is conventional method of control variates.

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Reg is conventional method of control variates.
Lik is constrained MLE, which uses pj s but ignores the labels;
it is exact if q = pj for any particular j.

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Building intuition ...

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Suppose we make n = 2 draws, one from N(0, 1) and one from
Cauchy (0, 1), hence π1 = π2 = 50%.

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Cauchy (0, 1), hence π1 = π2 = 50%.
Suppose the draws are {1, 1}, what would be the MLE (ˆ1 , π2 )?
π ˆ

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Cauchy (0, 1), hence π1 = π2 = 50%.
π ˆ
π ˆ

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Cauchy (0, 1), hence π1 = π2 = 50%.
π ˆ
π ˆ
π ˆ

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What Did I Learn?

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What Did I Learn?

Model what we ignore, not what we know!

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What Did I Learn?

Model comparison/selection is not about which model is true (as all
of them are “true”), but which model represents a better compromise
among human, computational, and statistical eﬃciency.

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What Did I Learn?

Model comparison/selection is not about which model is true (as all
of them are “true”), but which model represents a better compromise
among human, computational, and statistical eﬃciency.
There is a cure for our “schizophrenia” — we now can analyze Monte
Carlo data using the same sound statistical principles and methods for
analyzing real data.

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If you are looking for theoretical research topics ...

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RE-EXAM OLD ONES AND DERIVE NEW ONES!

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Prove it is MLE, or a good approximation to MLE.

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Or derive MLE or a cost-eﬀective approximation to it.

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Markov chain Monte Carlo (Tan 2006, 2008)

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Markov chain Monte Carlo (Tan 2006, 2008)
More ......

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Let's Practice What We Preach: Likelihood Methods for Monte Carlo Data

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