On the causal interpretation of acyclic mixed graphs under multivariate normality

In multivariate statistics, acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden (i.e., latent or unobserved) variables. Indeed, under multivariate normality, every mixed graph corresponds to a set of covariance matrices that contains as a full-dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph. This digraph generally has some of its nodes corresponding to hidden variables. We seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model. Restricting to the tractable setting of chain graphs and multivariate normality, we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

The Maximum Likelihood Threshold of a Path Diagram

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops

CRIRES-VLT high-resolution spectro-astrometry as a tool in the search of small structures at the cores of Planetary Nebulae

The onset of the asymmetry in planetary nebulae (PNe) occurs during the short transition between the end of the asymptotic giant branch (AGB) phase and the beginning of the PN phase. Sources in this transition phase are compact and emit intensely in infrared wavelengths, making high spatial resolution observations in the infrared mandatory to investigate the shaping process of PNe. Interferometric VLTI IR observations have revealed compelling evidence of disks at the cores of PNe, but the limited sensitivity, strong observational constraints, and limited spatial coverage place severe limits on the universal use of this technique. Inspired by the successful detection of proto-planetary disks using spectro-astrometric observations, we apply here for the first time this technique to search for sub-arcsecond structures in PNe. Our exploratory study using CRIRES (CRyogenic high-resolution Infra-Red Echelle Spectrograph) commissioning data of the proto-PN IRAS 17516-2525 and the young PN SwSt 1 has revealed small-sized structures after the spectro-astrometric analysis of the two sources. In IRAS 17516-2525, the spectro-astrometric signal has a size of only 12 mas, as detected in the Brackett-gamma line, whereas the structures found in SwSt 1 have sizes of 230 mas in the [Fe III] line and 130 mas in the Brackett-gamma line. The spectroscopic observations required to perform spectro-astrometry of sources in the transition towards the PN phase are less time consuming and much more sensitive than VLTI IR observations. The results presented here open a new window in the search of the small-sized collimating agents that shape the complex morphologies of extremely axisymmetric PNe.Comment: 6 pages, 4 figure

The Distribution of the Maximum Likelihood Estimator in Invariant Gaussian Graphical Models and its Application to Likelihood Ratio Tests

The distribution of the maximum likelihood estimator of the covariance matrix in a class of invariant Gaussian graphical models is determined and seen to belong to the class of generalized Riesz distributions. For testing two nested models, the distribution of the likelihood ratio statistic under the null hypothesis is shown to be of Box type, so that accurate approximation techniques are applicable

Statistische Inferenz in invarianten graphischen Modellen mit Normalverteilungsannahme

Die vorliegende Arbeit beschäftigt sich mit statistischer Inferenz in invarianten graphischen Modellen mit Normalverteilungsannahme. Hierbei handelt es sich um Verteilungsfamilien, die zwei Arten von Restriktionen berücksichtigen. Zum einen spiegeln alle Modellverteilungen eine gewisse Abhängigkeitsstruktur wider, die von einem Graphen vorgegeben wird. Zum anderen werden die von einer endlichen Gruppe vorgegebenen Symmetrien zwischen den beteiligten Variablen berücksichtigt. Zunächst werden bekannte Ergebnisse für diese Modelle vorgestellt, unter anderem hinreichende Bedingungen für die Existenz des Maximum-Likelihood-Schätzers der Kovarianzmatrix sowie dessen explizite Form. Anschließend wird mit Hilfe einer Verallgemeinerung der klassischen Wishart-Verteilung die Verteilung des Maximum-Likelihood-Schätzers ermittelt und anhand eines Beispiels veranschaulicht. Außerdem wird die berechnete Verteilung dazu verwendet, Likelihood-Ratio-Tests für geschachtelte invariante graphische Modelle zu entwickeln

Abstentions in the German Bundesrat and ternary decision rules

Ternary decision rules are analyzed that allow for abstentions, in addition to Yea- and Nay-votes. The German Bundesrat serves as a prime example. We show that the decision-making efficiency of the Bundesrat would increase if abstentions were allowed. Generally, a formula for the mean success margin of a ternary decision rule is derived assuming selfdual and permutationally invariant distributions. Specifically, the ternary Penrose/Banzhaf model is discussed in detail. The influence probabilities of voters, and the influence sensitivity of the rule are evaluated