Procesamiento y análisis de la información:

More complicated

mcmc

kernels are constructed by combining element- ary

mcmc

kernels. Indeed, let .Pzm/m2M be a countable collection of

-invariant kernels withPzmD zP ..m; /; /2K₁.X;X/, form2 MWD

NM. Additionally, letˇ 2K₁.M;M/be another stochastic kernel which

will be used to determine the choice of kernelPm.

In this case, we can include the indicesM into the state space to justify

kernels of the form

P ..m; x/;dm0dx0/_WDˇ.m;dm0/P ..mz 0; x/;dx0/:

This implies amixtureof (elementary) kernels on the marginal spaceX.

As a special case, we can justifycompositionsof kernels (on the marginal

space),P DP₁ PM;by setting

3.4 Example (Gibbs sampler). Assume that we can decompose the state space as X DX

1WJ for some J 2 N and write X D X1WJ. Define Sets J₁; : : : ; JM NJ such thatS

m2MJmDNJ. Furthermore, define themth Gibbs kernel via

z

P ..m; x/;dy/_WD•xJm.dyJm/˘m.y Jm;dyJm/;

where X_J

m WD .Xj/j2Jm,X Jm WD.Xj/j2JnJm and where˘m.x Jm; /

denotes the full conditional distribution ofX_J

m under. In this case, the

Markov chain induced byP Dˇ˝ zP is called aGibbs sampler(S. Geman & Geman, 1984). In particular, it is called a

random-scanGibbs sampler ifˇ.m;/is non-degenerate,

deterministic-scanGibbs sampler ifˇis as given in Equation3.4.

3.5 Example (partially-collapsed Gibbs sampler).Deterministic-scan Gibbs samplers are often presented as updating every component ofX ex- actly once per iteration, i.e. taking.J_m/_m2Mto be a partition ofN_J. However, large reductions in the asymptotic variance are achievable ifJ_m\J_mC1 ¤ ;, and in this case, the Gibbs sampler is often referred to as apartially-collapsed Gibbs sampler (Van Dyk & Park, 2008).

3.2.3

Generic

mcmc

Kernel

Clearly, sampling from a tractable distribution on a sufficiently small, finite state space is feasible and we would not need

mcmc

algorithms for this task. However, the only

mcmc

kernel capable of sampling (approximately) from a higher-dimensional distribution on general state space is the

Gibbs-sampling kernel presented in Example 3.4. Unfortunately, this kernel requires sampling from full conditional distributions under

which are often intractable.

In this subsection, we construct a generic-invariant

mcmc

kernel

which does not require sampling from full conditional distributions under

. Unsurprisingly, this generic kernel is based around a state-space

extension pioneered by Tjelmeland (2004). That is, it can be viewed as being invariant with respect to a distribution on an extended space which (1) admits as a marginal, (2) has full conditional distributions from

3.2 Generic

mcmc

Kernel as a Gibbs sampler targeting this extended distribution. As we shall see, this generic kernel admits most known

mcmc

kernels as a special case.

The generic

mcmc

kernel is again based around (a normalised version of) the extended measureN D N = .N 1/2 M₁.Xx/from Chapter 1. Recall

thatXxDXKZ, whereK DNN andZDXN Y, whereYis the set

of values taken by some auxiliary variables,Y.

Note that in the previous section, we used the framework from Chapter 1 to interpret the entire

mcmc

algorithm as a special case of

is

. Here, we employ the same framework again but at a lower level to construct the

-invariant kernel,P.

Further Extended Target Distribution. Recall that the extended dis-

tribution can be seen as a distribution over some auxiliary variablesY, a

pool ofN candidates forX,X DX1WN, and an indexK. It is construc-

ted in such a way that ifXx D .X; K;Z/ D .X; K; X1WN; Y / N then XK _{DX (Assumption 1.12). Approximations of were previously}

constructed by integrating out.X; K/and thus averaging over all possible

candidates in an

is

scheme.

Here, instead of integrating.X; K/out, we generate a new value for

the indexK, denotedKz, and a ‘new’ candidate,Xz DXKz, in such a way

that again,Xz under the extended distribution defined below. More

precisely, writingXz _WD.Xx;K;z X /z and recalling thatZx D .K;Z/, the

generic

mcmc

kernel targets the extended distribution

Q

.dxQ/_{WD N}.dxN/ .xN;dkQdx/Q

D.dx/˘ .x;x dzN/ .xN;dkQdx/Q

on_Xz _{WD xX}KX. Here, 2 K₁.Xx;KX/is chosen such that z

X Q ) XKz D zX : (3.5)

Note that such a kernel exists because we can always (and will often) take

.xN;dkQdx/Q D Nc.z;dkQdx/Q , whereNc.z; /is the full conditional

distribution of.K; X /underN.

Dominating Measure. Throughout the remainder of this chapter, we

assume that N has a density wx with respect to a suitable dominating

measure N _{which admits the factorisation} N

where 2 M¢.Z/and 2 K¢.Z;K/. This factorisation is required to

obtain the following explicit representation ofNc. ForxN D.x; k;z/, write wk.z/_{WD x}w.x/ .N z;fkg/. We can then represent the full conditional dis-

tribution of.X; K/underN asNc.z;dkdx/ D Nz.dk/•xk.dx/, where

N z.fkg/WD wk.z/ PN nD1wn.z/ :

Finally, note the measure does not depend onk. It therefore gen-

eralises the symmetric dominating measure required in Green (1995) as discussed in the Subsection 3.3.2,

Summary. By Equation 3.5, a-invariant kernel is given by

P .x; A/ WD Z x ZKA x ˘ .x;dzN/ .xN;dkQdx/;Q

for all.x; A/2 XB.X/. It is summarised in Algorithm 3.6.

3.6 Algorithm (generic

mcmc

kernel). GivenX , (1) sampleZx x˘ .x; /and setXx WD.X;Zx/,

(2) sample.K;z X /z .xN; /, (3) outputX WD zX(DXKz).

Note that the generic

mcmc

kernel is just a Gibbs kernel targeting the extended distributionQ. The advantage of this state-space extension is

that the kernels_˘x _{andcan often be chosen in such a way that sampling}

from them – and thus sampling from this Gibbs kernel – is feasible even if sampling from full conditional distributions under is not.

Finally, we note that the generic

mcmc

kernel is-invariant by con-

struction. In particular, this insight immediately shows that all the

mcmc

kernels mentioned in this chapter are-invariant (since they are all spe-

cial cases of this approach) without appealing (explicitly) to sufficient conditions such asdetailed balance. However, we reiterate the fact that

-invariance is not sufficient for obtaining useful

mcmc

kernels. Indeed,

takingq WD in Example 1.14 can be viewed as initialising the Markov

chain from and then applying the trivial kernelP .x;/_WD •x. This

3.2 Generic

mcmc

Kernel For later use, we define the following terminology, borrowed from Nicholls, Fox and Watt (2012), which will be justified in the next section.

3.7 Definition (randomised

mcmc

kernel). Any instance of the gen- eric

mcmc

kernel is called ‘randomised’ if it makes use of further auxiliary variables, i.e. ifY ¤ ;.

In document Aplicación de estrategia “Resolución de Problemas” para mejora de aprendizajes significativos en los estudiantes del cuarto grado “A” de la I E “César Abraham Vallejo” (página 85-107)