Signal Processing - July 2017 - 67

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weight degeneracy problem. While the standard weights
can be used for estimation, the nonlinearly transformed
weights are crucially used for the adaptation step. The
latter can be carried out in different ways, with [32] advocating for a simple Gaussian proposal where both the
mean vector and the covariance matrix are adapted through
the iterations.
Layered AIS (LAIS) [23]: The adaptive process of the
LAIS algorithm is independent of the samples drawn at
each iteration. In particular, the algorithm can be seen as a
two-layer procedure in which the location parameters of
the proposals are adapted through one or several MCMC
steps with the target as the stationary distribution. In its
basic version, a single MCMC step is independently performed at each location parameter.
DM-PMC [24]: This algorithm meets the simplicity of
the standard PMC of [19] with a very high performance.
DM-PMC calculates the weights using (16) instead of
(15), which provides two important advantages, specifically, the variance of the estimators is decreased (see
[25]) and the resampling step with the DM weights promotes the replication of proposals in relevant parts of the
target that are underrepresented by the set of proposals

(i.e., the exploration is coordinated). DM-PMC generates
K samples per each of the N proposals (instead of one,
as in [19]). At each iteration, the population of KN
samples must be reduced to N via either global or local
resampling (LR).
■ AMIS [21]: In this algorithm, just one proposal is used
and adapted over the iterations. The adaptive procedure
consists of estimating the moments of the target with the
available set of K weighted samples and fitting the
moments of the proposal. Its key feature is the reweighting of all of the past samples with a temporal mixture
weight where the whole sequence of proposals is used in
the denominator.
■ Gradient APIS (GAPIS) [34]: Similar to the LAIS algorithm, GAPIS adapts N proposals by a process that is
independent of the samples. In its basic version, the location parameters of the proposals are adapted via a gradient
ascent of the target and the scale parameter by using the
Hessian of the target. An advanced implementation is proposed that adds a repulsive interaction among proposals to
promote a cooperative exploration of the target.
In Tables 3 and 4, six out of the seven previous algorithms
are outlined by means of pseudocodes. Note that we follow

Table 3. The pseudocodes of PMC, DM-PMC, and LAIS.
PMC

DM-PMC

LAIS
Initialization

J, N, K = 1
N
{i n, 1} N
n = 1 / {n n, 1, C n} n = 1

J, N, K,
N
{i n, 1} N
n = 1 / {n n, 1, C n} n = 1

J, N, K,
N
{i n, 1} N
n = 1 / {n n, 1, C n} n = 1

For j = 1, f, J:
1) Sampling
x n, j ~q n, j ( x n n, j, C n)

(k)
x n, j ~ q n, j ( x

n = 1, f, N

n = 1, f, N
k = 1 , f, K

(k)

n n , j , C n)

x n, j ~ q n , j ( x n n , j , C n)
n = 1, f, N
k = 1 , f, K

2) Weighting
w n, j =

r (x n, j)

q n, j (x n, j)

n = 1, f, N

(k)
r (x n, j)
N

(k)

w n, j =

(k)

r (x n, j)

(k)

w n, j =

1 / q (x (k) )
i, j
n, j
N i=1

n = 1 , f, N
k = 1, f, K

N

1 / q (x (k) )
i, j
n, j
N i=1

n = 1 , f, N
k = 1, f, K
3) Adaptation

Multinomial resampling with replacement over
w n, j N
{x n, j, wr n, j = N
}n=1
/ w i,j

Multinomial resampling with replacement over
(k)

(k)

{x n, j, wr n, j =

i=1

N
to update {n n, j + 1} n = 1 .

N

(k)
w n, j
K

/ /

j=1 m=1

(m)
w j, j

N, K

} n = 1, k = 1

One (or more) MCMC steps from n n, j to n n, j + 1 ,
with ru as a stationary distribution, for
n = 1, ..., N.

N
to update {n n, j + 1} n = 1 .

Outputs
{ x n, j , w n, j }
n = 1 , f, N
j = 1 , f, J

(k)
{ x n, j ,

(k)
w n, j }

(k)

n = 1 , f, N
k = 1 , f, K
j = 1, f, J

IEEE SIGNAL PROCESSING MAGAZINE

(k)

{ x n, j , w n, j }
n = 1 , f, N
k = 1, f, K
j = 1, f, J

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July 2017

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67
Table of Contents for the Digital Edition of Signal Processing - July 2017
