On Data Augmentation in Point Process Models Based on Thinning
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Many models for point process data are defined through a thinning procedure where locations of a base process (often Poisson) are either kept (observed) or discarded (thinned) according to some probabilistic or deterministic rule. The objective of this article is to present a universal theoretical framework that allows the derivation of the joint distribution of thinned and observed locations from the density of the base point process along with the formula that describes how points are discriminated. This theory is essential in practice when designing inference schemes based on data augmentation where observed locations are augmented with thinned locations in order to circumvent some intractability in the likelihood function of the marginal model. Such schemes have been employed in the recent literature, but the absence of a proper theoretical framework has led to conceptual flaws being introduced and carried on in subsequent publications. This has motivated us to propose a theoretical approach to this problem in order to avoid those pitfalls in the future. The results described in this paper are general enough to enable future authors in creating even more flexible models based on thinning and use the tools described here to obtain a principled way of carrying inference.
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