Compared to points being stuck in attractor regions permanently, we now describe `ghost points', which could be seen as the complete counter part to attractor regions. De nition 5.4.1. A point xg is called a ghost point if, after being sampled via some
wavelet ψj,i, it has an associated survival time tg ≡ 0.
From an inverse sampling algorithm for d(xs) = 0, where s is a current point in time
and t ≥ s is a survival point in time of xs,
t = s − rf (xs) c(xs)
log us, (5.4.22)
we can clearly see that for c(xs) > 0 and f(xs) = 0 the associated survival time is
equal to the previous one t = s, indicating that a point xs has not advanced the
process in time at all. So ghost points could be seen as intermediate sample points xg that were sampled from the zero probability region, xg ∈/ supp(f) ∪ supp(g).
A sampled ghost point xgexists for a zero amount of time and although it is sampled,
the WMC process leaves that point immediately. Clearly, these points are not desired, as computational power is wasted on sampling them in the rst place only to nd out that this survival time is zero and hence WMC has not advanced in time towards the target sample. The most important question here is, why are these points being sampled in a rst place?
This phenomena could be demonstrated using two uniform distributions. In Figure 5.5 intermediate points were sampled in regions of zero probability. This is a very serious issue as it contradicts the claim that all intermediate points xt come from
intermediate distributions ft(·), t ∈ [0, 1]. The only way that a point sampled
from f(·) is moved to the region of g(·) is if chosen wavelet ψj,i envelopes at least
−10 −5 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Density / Time ● ● ● ● ● ● ● ● ● ● ● Starting Target Optimal path Ghost points Survival time Inter. sample
Figure 5.5: Demonstration of the existence of ghost points using two uniform distribution with K = 2 and Nd = 200. Given that PDF and survival time takes
values between 0 and 1, the vertical axis corresponds to both. Given that f(xg) = 0
and g(xg) = 0, the survival time of ghost points is 0.
example where a chosen wavelet includes a zero density region in its support. This means that there exists a non-zero probability that a sampled intermediate point will fall in the zero density region of ft(·). This is exactly what happens in practice,
leading to many points being sampled from regions of zero density. As intermediate points xtgenerated by the WMC process do not necessarily come from a distribution
with density ft(·), Theorem 3.3.2 is put into question, requiring one to update and
reformulate assumptions of Proof 3.3.
A simulation was performed using two uniform distributions identical to as in Figure 5.5. The idea was to produce 100 samples from the target distribution U[5, 6] using samples from U[−5, −3] and to monitor how many intermediate points were produced from zero density regions, i.e. how many points were generated that did
−5 0 5 10 −0.5 0.0 0.5 1.0 ● ●
Figure 5.6: Daubechies K = 2 transition wavelet ψ−2,−1 partially envelopes f(·) and
fully envelope g(·), however it also covers the zero density region in between. The selection of such a wavelet would potentially lead to points being sampled from the zero-density region.
not follow ft(·) at a given particular time. Daubechies wavelets with K = 2 and
Nd = 200 were used. Out of 910 sample points generated, 772 were sampled from
the zero-density region. That means that around 84% of the computing power was wasted on points that should not have been generated in the rst place. On average, there were 7.72 ghost points generated to produce one sample from the target g(·). In a situation when the choice of a starting and a target distribution creates zero density regions, points that fall into them are immediately classi ed as ghost points. However, the situation is less clear when the starting and target densities have in nite support but there are regions of near zero-density. For instance, if we choose the starting distribution to be N (−5, 1) and the target to be N (5, 1), then all intermediate densities ft(·) formed will have a near zero-density region in between
−15 −10 −5 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Density / Time ● ● ● ● ● ● ● ● Starting Target Optimal path Ghost points Survival time
Figure 5.7: Example of semi-ghost points being generated in a situation when the support is in nite, supp(ft) = R, and there are regions of very low density.
x = −5 and x = 5 and as |x| → ∞ regions. Although intermediate points sampled via WMC do belong to the support of the density ft(·), it should be quite unlikely
that points are sampled from regions of low density. Unfortunately, for the same exact reason as before, points are being generated from low density regions and Figure 5.7 illustrates this problem quite clearly. Instead of taking an optimal path, semi-ghost points were generated leading to an ine cient algorithm.
De nition 5.4.2. A point xsg is called a semi-ghost point if after being sampled
via some wavelet ψj,i it has an associated survival time 0 < tsg 1.
The examples demonstrated in Figure 5.5 and 5.7 use Daubechies wavelets with K = 2, jmin = −8 and jmax= 11, which is more than enough to cover all the details
of the dierence function. One might speculate that the ghost point phenomena could be tied to the nite computing power nature and imprecise implementation,
however it is not the case. For example, what if wavelet coe cients dj,i could be
computed exactly and one had an access to levels jmin = −∞and jmax = +∞? Even
in this perfect scenario, there would always exist a positive probability q−2,−1 > 0
for the wavelet ψ−2,−1 to be selected as in Figure 5.6, therefore there would always
exist a possibility that an intermediate point would be sampled from a region that does not belong to the support of the intermediate density ft(·).