final report - probability.ca

2.4 Metropolis-within-Gibbs algorithms
In this section we move our consideration from Metropolis algorithms that make updates for all components simultaneously
to partially updating algorithms. This means that the updates are chosen to be lower dimensional than
the target density itself. Optimal scaling in this context requires two choices, one for the scaling of the proposal
density, and the second for the dimensionality of the proposal. The paper by Neal and Roberts, [NR05], proves
optimality results for the Metropolis-within-Gibbs algorithm with random walk and Langevin updates separately.
In the first instance, they find that full-dimensional random walk Metropolis updates are not better asymptotically
than smaller dimensional updating schemes, and thus, with the added factor of computational cost for higher dimensional
updates (computational overhead at every iteration is a non-decreasing function of the proportion of the
density being updated), it is always optimal to use lower dimensional updates when possible. On the other hand,
they find that full dimensional Langevin algorithms are worthwhile in general, although one needs to consider the
computational overhead associated with their implementation when deciding on optimality.
For RWM/MALA-within-Gibbs algorithms, the implementation of the MCMC algorithm requires choosing d·c d
components, where c d denotes the fixed value of updating proportion, chosen at random at each iteration, and then
attempt to update them jointly according to the RWM/MALA mechanism, respectively. Hence, the two algorithms
(RWM and MALA, respectively) propose new values for coordinate i according to
Y (d)
i = x (d)
i
Y (d)
i = x (d)
i + χ (d)
i
+ χ (d)
i σ d,cd Z i ,
{
σ d,cd Z i + σ2 d,c d
2
∂
(
log π d x (d))} ,
∂x i
where 1 ≤ i ≤ d. The random variables {Z i } i∈[1,d]
are i.i.d. with distributions Z ∼ N (0, 1), and each of the
characteristic functions χ (d)
i are chosen independently of the Z i ’s. At each iteration, a new random subset A is
chosen from {1, 2, . . . , d}, and χ d i = 1 if i ∈ A, and is zero otherwise. The Metropolis-Hastings acceptance rule for
the two above setup when applied to the i.i.d. components target (1.8) yields the acceptance probability
⎧ (
α c d
d
(x (d) , Y (d)) ⎨ π Y (d)) ) ⎫
q
(Y (d) , x (d) ⎬
= min
⎩ 1, π ( x (d)) q
(x (d) , Y (d)) ⎭ ,
where q (·, ·) is the chosen proposal density. In case of rejection, we choose X (d)
m
= X (d)
m−1 . As we had done for the
cases considered previously, we can use the notion of average acceptance rate as means for measuring efficiency and
optimality of the algorithm. We denote
[ (
a c d
d
(l) = E π α c d
d
X d , Y d)]
⎡ ⎧ (
⎨ π d Y d) ) ⎫⎤
q
(Y d , x d ⎬
= E π
⎣min
⎩ 1, (
π d (x d ) q x d , Y d) ⎦ ,
⎭
where σd,c 2 d
= l 2 d −s . Furthermore, as always, it is assumed throughout that the algorithm starts at stationarity.
The scaling exponent s is given by s = 1 for RWM and s = 1 3
for MALA.
2.4.1 RWM-within-Gibbs
Consider first the RWM-within-Gibbs algorithm applied to the i.i.d. target density. As was done in [RR98], it is
helpful to express the density in the exponential target form
(
π x (d)) =
d∏ (
f
i=1
x (d)
i
)
=
d∏
i=1
{ (
exp g
The proposal standard deviation is denoted by σ (d) = √ l
d−1
, for some l > 0. Since the target is again formed
of i.i.d. components, all individual components of the target density are equivalent. For convenience, it will be
again useful to consider the first component of the resulting algorithm, with a time-increasing factor of d, which is
denoted by Ut,1, d obtained from
U (d)
t = ( )
X [dt],1 , X [dt],2 , . . . X [dt],d .
x (d)
i
)}
,
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