Identifiability of causal graphs under nonadditive conditionally parametric causal models

Causal discovery from observational data is a very challenging, often impossible, task. However, estimating the causal structure is possible under certain assumptions on the data-generating process. Many commonly used methods rely on the additivity of the noise in the structural equation models. Additivity implies that the variance or the tail of the effect, given the causes, is invariant; the cause only affects the mean. In many applications, it is desirable to model the tail or other characteristics of the random variable since they can provide different information about the causal structure. However, models for causal inference in such cases have received only very little attention. It has been shown that the causal graph is identifiable under different models, such as linear non-Gaussian, post-nonlinear, or quadratic variance functional models. We introduce a new class of models called the Conditional Parametric Causal Models (CPCM), where the cause affects the effect in some of the characteristics of interest.We use the concept of sufficient statistics to show the identifiability of the CPCM models, focusing mostly on the exponential family of conditional distributions.We also propose an algorithm for estimating the causal structure from a random sample under CPCM. Its empirical properties are studied for various data sets, including an application on the expenditure behavior of residents of the Philippines

Linear prediction of point process times and marks

In this paper, we are interested in linear prediction of a particular kind of stochastic process, namely a marked temporal point process. The observations are event times recorded on the real line, with marks attached to each event. We show that in this case, linear prediction extends straightforwardly from the theory of prediction for stationary stochastic processes. Following classical lines, we derive a Wiener-Hopf-type integral equation to characterise the linear predictor, extending the "model independent origin" of the Hawkes process (Jaisson, 2015) as a corollary. We propose two recursive methods to solve the linear prediction problem and show that these are computationally efficient in known cases. The first solves the Wiener-Hopf equation via a set of differential equations. It is particularly well-adapted to autoregressive processes. In the second method, we develop an innovations algorithm tailored for moving-average processes. A small simulation study on two typical examples shows the application of numerical schemes for estimation of a Hawkes process intensity

Tail asymptotics and precise large deviations for some Poisson cluster processes

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [18] and [5] notably, relying on Karamata's Tauberian Theorem to do so. We use these asymptotics to derive precise large deviation results in the fashion of [30] for the above-mentioned processes