ABC Shadow algorithm: a tool for statistical analysis of spatial patterns

Abstract

This paper presents an original ABC algorithm, ABC Shadow, that can be applied to sample posterior densities that are continuously differentiable. The proposed algorithm solves the main condition to be fulfilled by any ABC algorithm, in order to be useful in practice. This condition requires enough samples in the parameter space region, induced by the observed statistics. The algorithm is tuned on the posterior of a Gaussian model which is entirely known, and then, it is applied for the statistical analysis of several spatial patterns. These patterns are issued or assumed to be outcomes of point processes. The considered models are: Strauss, Candy and area-interaction.

Appendix: Proof of the results

1.1 Proof of Theorem 1

• (i) Both density functions vanish outside $b(\theta , \varDelta /2)$. For $\psi \in b(\theta , \varDelta /2)$, the integral mean value theorem applied to the denominator of $q_\varDelta$ leads to

  $$\begin{aligned} \quad q_\varDelta (\theta \rightarrow \psi ) = \frac{ \frac{f(\mathbf{x}|\psi )}{c(\psi )} }{ V_\varDelta \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} } \end{aligned}$$

  for some $\theta ^* \in b(\theta , \varDelta /2)$. The positivity and the continuity of the density p (uniform continuity indeed, since $\varTheta$ is compact) allows us to do the following. Let $m(\mathbf{x}) := \inf _{\phi \in \varTheta } p(\mathbf{x}| \phi ) > 0$ since it is actually a minimum. For $A \in \mathcal{T}_{\varTheta }$ we have

  $$\begin{aligned}&\int _A \left| q_{\varDelta }( \theta \rightarrow \psi ) - U_{\varDelta }( \theta \rightarrow \psi ) \right| \hbox {d}\psi \\&\quad = \int _{A \cap b(\theta , \varDelta /2))} \left| \frac{\frac{f(\mathbf{x}|\psi )}{c(\psi )} }{ V_{\varDelta } \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} } - \frac{1}{V_\varDelta } \right| \hbox {d}\psi \\&\quad \le \frac{1}{V_\varDelta }\sup _{\phi \in \varTheta } \frac{c(\phi )}{f(\mathbf{x}|\phi )} \int _{A \cap b(\theta , \varDelta /2)} \\&\qquad \times \left| \frac{f(\mathbf{x}|\psi )}{c(\psi )} - \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} \right| \hbox {d} \psi \\&\quad \le \frac{\mu (A \cap b(\theta , \varDelta /2))}{V_\varDelta } m(\mathbf{x})^{-1} \\&\quad \quad \times \sup _{\hbox {d}(\psi , \theta ^*)< \varDelta } \left| \frac{f(\mathbf{x}|\psi )}{c(\psi )} - \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} \right| \\&\quad \le m(\mathbf{x})^{-1} \sup _{\hbox {d}(\psi , \theta ^*) < \varDelta } \left| \frac{f(\mathbf{x}|\psi )}{c(\psi )} - \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} \right| \end{aligned}$$

  where the last supremum is independent of $\psi$ and $\theta ^*$, and approaches to 0 as far as $\varDelta$ does. With the regularity condition on $p(\mathbf{x}|\cdot )$, we can tune up the inequality, using the (differential) mean value theorem, to

  $$\begin{aligned} \int _A&\left| q_{\varDelta }( \theta \rightarrow \psi ) - U_{\varDelta }( \theta \rightarrow \psi ) \right| \hbox {d} \psi \\&\quad \le m(\mathbf{x})^{-1} \varDelta \sup _{\psi ^* \in \varTheta } \left\| D_{\varTheta } p(\mathbf{x}|\psi ^*) \right\| \\&\quad := C_1(\mathbf{x}, p, \varTheta ) \varDelta \end{aligned}$$

  where $C_1(\mathbf{x}, p, \varTheta )$ is a constant depending on $\mathbf{x},p$ and $\varTheta$.

• (ii) As previously, the use of the integral mean value theorem gives the result, since

  $$\begin{aligned}&\sup _{\psi \in \varTheta } \left| \frac{q_\varDelta (\theta \rightarrow \psi | \mathbf{x})}{q_\varDelta (\psi \rightarrow \theta | \mathbf{x})} - \frac{ \frac{f(\mathbf{x}|\psi )}{c(\psi )}\mathbf{1}_{b(\theta ,\varDelta /2)}(\psi ) }{ \frac{f(\mathbf{x}|\theta )}{c(\theta )}\mathbf{1}_{b(\psi ,\varDelta /2)}(\theta ) } \right| \\&\quad \le \sup _{\psi \in b(\theta ,\varDelta /2)} \left| \frac{ \frac{f(\mathbf{x}|\psi )}{c(\psi )} / (V_\varDelta \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)}) }{ \frac{f(\mathbf{x}|\theta )}{c(\theta )} / (V_\varDelta \frac{f(\mathbf{x}|\psi ^*)}{c(\psi ^*)}) } - \frac{ \frac{f(\mathbf{x}|\psi )}{c(\psi )} }{ \frac{f(\mathbf{x}|\theta )}{c(\theta )} } \right| \\&\quad \le \sup _{\psi \in b(\theta ,\varDelta /2)} \left| \frac{ \frac{f(\mathbf{x}|\psi )}{c(\psi )} }{ \frac{f(\mathbf{x}|\theta )}{c(\theta )} } \right| \left| \frac{ \frac{f(\mathbf{x}|\psi ^*)}{c(\psi ^*)} }{ \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} } - 1 \right| \\ \le&M(\mathbf{x}) m(\mathbf{x})^{-2} \sup _{d(\theta ^*,\psi ^*) \le \varDelta } \left| \frac{f(\mathbf{x}|\psi ^*)}{c(\psi ^*)} - \frac{f(\mathbf{x}|\theta ^*)}{c(\theta ^*)} \right| \end{aligned}$$

  where $M(\mathbf{x}) := \sup _{\phi \in \varTheta } p(\mathbf{x}| \theta ) < \infty$ is a maximum and $\theta ^* \in b(\theta , \varDelta /2)$, $\psi ^* \in b(\psi , \varDelta /2)$ are obtained from the integral mean value theorem. As in (i), under the regularity condition on $p(\mathbf{x}|\cdot )$, this inequality can evolve to

  $$\begin{aligned}&\sup _{\psi \in \varTheta } \left| \frac{q_\varDelta (\theta \rightarrow \psi | \mathbf{x})}{q_\varDelta (\psi \rightarrow \theta | \mathbf{x})} - \frac{ \frac{f(\mathbf{x}|\psi )}{c(\psi )}\mathbf{1}_{b(\theta ,\varDelta /2)}(\psi ) }{ \frac{f(\mathbf{x}|\theta )}{c(\theta )}\mathbf{1}_{b(\psi ,\varDelta /2)}(\theta ) } \right| \\&\quad \le M(\mathbf{x}) m(\mathbf{x})^{-2} \varDelta \sup _{\psi ^* \in \varTheta } \left\| D_{\varTheta } p(\mathbf{x}|\psi ^*) \right\| \\&\quad := C_2(\mathbf{x}, p, \varTheta ) \varDelta \end{aligned}$$

  where $C_2(\mathbf{x}, p, \varTheta )$ is a constant depending on $\mathbf{x},p$ and $\varTheta$.

1.2 Proof of Proposition 1

If $n = 1$, the definition of the transition kernels, the introduction of the term $U_\varDelta (\theta \rightarrow \psi ) \alpha _i(\theta \rightarrow \psi ) - U_\varDelta (\theta \rightarrow \psi ) \alpha _i(\theta \rightarrow \psi )$, then the use of the triangle’s inequality and also the boundedness of functions $1_{A}(\cdot )$, $\alpha _i(\cdot )$ and $\alpha _s(\cdot )$ allow us to write

$$\begin{aligned}&| P_i(\theta ,A) - P_s(\theta ,A) | \\&\quad \le \int _{\psi \in b(\theta , \varDelta /2)} | q_\varDelta (\theta \rightarrow \psi ) \alpha _i(\theta \rightarrow \psi ) \\&\quad \quad - U_\varDelta (\theta \rightarrow \psi ) \alpha _s(\theta \rightarrow \psi ) | \hbox {d} \psi \\&\quad \quad + 1_{A}(\theta ) \int _{\psi \in b(\theta , \varDelta /2)} | q_\varDelta (\theta \rightarrow \psi ) [ 1 - \alpha _i(\theta \rightarrow \psi ) ] \\&\quad \quad - U_\varDelta (\theta \rightarrow \psi ) [ 1 - \alpha _s(\theta \rightarrow \psi ) ] | \hbox {d} \psi \\&\quad \le 3 \int _{\psi \in b(\theta , \varDelta /2)} | q_\varDelta (\theta \rightarrow \psi ) - U_\varDelta (\theta \rightarrow \psi ) | \hbox {d} \psi \\&\quad \quad + 2 \int _{\psi \in b(\theta , \varDelta /2)} U_\varDelta (\theta \rightarrow \psi ) | \alpha _i(\theta \rightarrow \psi ) \\&\qquad - \alpha _s(\theta \rightarrow \psi ) | \hbox {d} \psi \end{aligned}$$

Under the hypothesis of Theorem 1, and then applying Theorem 1(i) and Corollary 1, the transition kernels of the ideal and the shadow Markov chains, respectively, are uniformly close as well (say for any $\epsilon > 0$, there exists $\varDelta (\epsilon ,1) > 0$ such that we have $| P_s(\theta ,A) - P_i(\theta ,A) | < \epsilon$ if $\varDelta \le \varDelta (\epsilon ,1)$, independently of $\theta$ and A).

If $p(\mathbf{x}|\cdot ) \in \mathcal{C}^1(\varTheta )$, then the previous inequalities can be completed to

$$\begin{aligned}&| P_i(\theta ,A) - P_s(\theta ,A) | \\&\quad \le 3 C_1(\mathbf{x}, p, \varTheta ) \varDelta + 2 C_2(\mathbf{x}, p, \varTheta ) \varDelta := C_3(\mathbf{x}, p, \varTheta ) \varDelta \end{aligned}$$

giving a candidate expression for $\varDelta _0(\epsilon , 1) := C_4(\mathbf{x}, p, \varTheta ) \epsilon$. The constants $C_3$ and $C_4$ depend on $\mathbf{x},p$ and $\varTheta$.

For $n > 1$, we get by induction that

$$\begin{aligned}&|P_i^{(n)} - P_s^{(n)} | \\&\quad \le |P_i^{(n)} - P_i^{(n-1)} P_s^{(1)} | + |P_i^{(n-1)} P_s^{(1)} - P_s^{(n)} | \\&\quad \le |P_i^{(n-1)} | |P_i^{(1)} - P_s^{(1)} | + |P_i^{(n-1)} - P_s^{(n-1)} | | P_s^{(1)} | \\&\quad \le |P_i^{(1)} - P_s^{(1)} | + |P_i^{(n-1)} - P_s^{(n-1)} | \\&\quad \le \cdots \le n |P_i^{(1)} - P_s^{(1)} | < \epsilon \end{aligned}$$

uniformly in $\theta$ and A for all $\varDelta$ such that

$$\begin{aligned} \varDelta \le \varDelta _0(\epsilon , n) := \varDelta _0(\epsilon /n, 1) = C_4(\mathbf{x}, p, \varTheta ) \frac{\epsilon }{n}. \end{aligned}$$