Information geometry and sufficient statistics

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the Amari-Chentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows.Comment: 37 p, final version, minor corrections, improved presentatio

A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on Rd. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W²'¹ i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined

Does ORTO-15 produce valid data for 'Orthorexia Nervosa'? A mixed-method examination of participants' interpretations of the fifteen test items

Purpose: Orthorexia Nervosa (ON) is defined as a pathological eating behaviour stemming from being “healthy” or “pure”. Survey-based studies typically rely on the ORTO-15 questionnaire or its variations to detect orthorexia. However, frequent post-hoc adjustments to the ORTO-15 suggest psychometric problems. In this study, we explored people’s cognitions about the ORTO-15 items to (1) identify problems specific to ORTO-15 items and (2) explore participants’ understanding of ON symptoms. Methods: Fifty adult participants (40% male, mean age = 34.0 ± 14.4 years) completed the ORTO-15, the Eating Attitudes Test (EAT-26) and the Obsessive–Compulsive Inventory–Revised edition (OCI-R). Qualitative data were collected using the modified “think aloud” protocol, which asked participants to ‘verbalise’ their responses to the ORTO-15 items. These qualitative responses were first analysed conjunctively with the quantitative responses; then subjected to thematic analysis. Results: ORTO-15 identified 64% of the participants for orthorexic tendencies. In most cases (76%), participants reported no issues completing the ORTO-15. However, in some cases, qualitative responses differed from quantitative ones. When people encountered problems, it was because of poor psychometric construction: lack of clarity, ambiguous wording and multiple statements in a single item. Elaborations around the ORTO-15 items formed four major themes: “preoccupation with physical appearance”, “control”, “food is fuel” and “alone, not isolated”. Conclusion: Even though in the majority of cases there were no issues with completing ORTO-15, thematic analysis revealed several discrepancies between our participants’ perceptions of the ORTO-15 items and the previously proposed diagnostic criteria for ON. The results suggest that ORTO-15 is, at best, a mediocre screening tool for ON, which is sensitive to diet but fails to have sufficient level of specificity to detect the pathological stage. More accurate instruments are needed to further research on ON. Level of evidence: V (cross-sectional descriptive study with qualitative analysis)