4 Vanishing weights

Since $w(\vec\theta) \ge 0$ for every $\vec\theta \in \Omega$, y is non-negative, so that p_y(y) = 0 if y < 0. In principle, however, we cannot exclude that the case y = 0 has a finite probability. In particular, for finite support weight functions, p_y(y) could include the contribution from a Dirac delta distribution centered on zero.

Since y is the sum of weights at different positions, y = 0 can have a finite probability only if $w(\vec\theta)$ vanishes at some points. In turn, if P₀ = P(y = 0) is finite we could encounter situations where the numerator and the denominator of Eq. (5) vanish. In such cases we could not even define our estimator $\tilde f$. We also note that, in the continuous limit, P₀ is also the probability of having vanishing denominator in Eq. (5). As a result, if P₀ > 0, we have a finite probability of being unable to evaluate our estimator!

So far we have implicitly assumed that P₀ vanishes. Actually, we can explicitly evaluate this probability using the expression

$$P(y=0) = \lim_{y \rightarrow 0} \int_0^y p_y(y') \, \rm dy' .$$ (26)

From the properties of the Laplace transform this expression can be written as

$$P(y=0) = \lim_{s \rightarrow \infty} Y(s) = \lim_{s \rightarrow \infty} \rm e^{\rho Q(s)} .$$ (27)

We now observe that

$$\lim_{s \rightarrow \infty} \left[ \rm e^{-s w(x)} - 1 \rig... ...}\ w \neq 0 , \\ 0 & \textrm{otherwise.} \end{array}\right.$$ (28)

As a result, in the limit considered, -Q(s) approaches the area of the support of w. Calling this area $\pi_w$, we find finally

$$P(y=0) = \rm e^{-\rho \pi_w} .$$ (29)

In the case where $w(\vec\theta)$ has infinite support (for example if w is a Gaussian), P₀ vanishes as expected.

The result expressed by Eq. (29) can actually be derived using a more direct approach. Since $w(\vec\theta) \geq 0$, the condition y = 0 requires that all weights except $w( \vec\theta_1 )$are vanishing. In turn, this happens only if there is no object inside the neighborhood of the point considered. Since the area of this neighborhood is $\pi_w$, the number of objects inside this set follows a Poisson distribution of mean $\rho \pi_w$, and thus the probability of finding no object is given precisely by Eq. (29).

As mentioned before, P₀ > 0 is a warning that in some cases we cannot evaluate our estimator $\tilde f$. In order to proceed in our analysis allowing for finite-support weight functions, we decide to explicitly exclude in our calculations cases where w + y = 0: in such cases, in fact, we could not define $\tilde f$. In practice when smoothing the data we would mark as "bad points'' the locations $\vec\theta$ where $\tilde f(\vec\theta)$ is not defined. In taking the ensemble average, then, we would exclude, for each possible configuration $\bigl\{ \vec\theta_n \bigr\}$, the bad points. In order to apply this prescription we need to modify p_y, the probability distribution for y, and explicitly exclude cases with w + y = 0. In other words, we define a new probability distribution for y given by

$$\tilde p_y(y) = \left\{ \begin{array}{cc} p_y(y) & \text... ...igr] / (1 - P_0) & \textrm{if $w = 0 .$ } \end{array}\right.$$ (30)

Hence, if w = 0, we set to zero P(y = 0) and then renormalize the distribution. An important consequence of the new prescription is that the probability distribution for y no longer is independent of w. In fact, the probability P(y = 0) vanishes for w = 0, while it is finite (for a finite-support weight) if $w \neq 0$. Using Eq. (30) in the definition of Y(s) we then find (see Eqs. (17) and (19))

$$Y(s) = \rm e^{\rho Q(s)} - \Delta(w) \frac{P_0}{1 - P_0} \left[ 1 - \rm e^{\rho Q(s)} \right] ,$$ (31)

where $\Delta(w)$ is 1 for w = 0 and vanishes for w > 0. Equation (31) replaces Eq. (19) for the cases when P₀ > 0, i.e. for weight functions with finite support.

In order to implement the new requirement $w + y \neq 0$, we still need to modify Eq. (12) [or, equivalently, Eq. (14)]. In fact, in deriving that result, we have assumed that all objects can populate the area $\Omega$ with uniform probability and in fact we have used a factor 1/A^N to normalize Eq. (7). Now, however, we must take into account the fact that objects cannot make a "void'' around the point $\vec\theta$. As a result, we need a further factor 1/(1-P₀) in front of Eqs. (12) and (14). In summary, the new set of equations is

   

$$\int_\Omega \left[ \rm e^{-s w(\vec\theta)} - 1 \right] \, \rm d^2 \theta ,$$ (32)

$$\rm e^{\rho Q(s)} - \Delta(w) \frac{P_0}{1 - P_0} \left[ 1 - \rm e^{\rho Q(s)} \right] ,$$ (33)

$$\frac{\rho}{1 - P_0} \int_0^\infty \rm e^{-w s} Y(s) \, \rm ds ,$$ (34)

where, we recall, $P_0 = \rm e^{-\rho \pi_w}$ vanishes for weight functions with infinite support $\pi_w$.