Consistent Kernel Mean Estimation for Functions of Random Variables

We provide a theoretical foundation for non-parametrically estimating functions of random variables using kernel mean embeddings. We show that for any continuous function f, consistent estimators of the mean embedding of a random variable X lead to consistent estimators of the mean embedding of f(X). For Gaussian kernels and sufficiently smooth functions we also provide rates of convergence. Our results also apply for functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate mean embeddings of their marginal distributions. In either case, our results cover both mean embeddings expressed based on i.i.d. samples as well as reduced set expansions in terms of dependent expansion points. The latter serves as a justification for using such expansions to limit memory resources when we use the approach as a basis for probabilistic programming.Carl-Johann Simon-Gabriel is supported by a Google European Fellowship in Causal Inference

Reconstructed historical distribution and phylogeography unravels non-steppic origin of Caucasotachea vindobonensis (Gastropoda: Helicidae)

Existing data on the phylogeography of European taxa of steppic provenance suggests that species were widely distributed during glacial periods but underwent range contraction and fragmentation during interglacials into “warm-stage refugia.” Among the steppe-related invertebrates that have been examined, the majority has been insects, but data on the phylogeography of snails is wholly missing. To begin to fill this gap, phylogeographic and niche modeling studies on the presumed steppic snail Caucasotachea vindobonensis were conducted. Surprisingly, reconstruction of ancestral areas suggests that extant C. vindobonensis probably originated in the Balkans and survived there during the Late Pleistocene glaciations, with a more recent colonization of the Carpatho-Pannonian and the Ponto-Caspian regions. In the Holocene, C. vindobonensis colonized between the Sudetes and the Carpathians to the north, where its recent and current distribution may have been facilitated by anthropogenic translocations. Together, these data suggest a possible non-steppic origin of C. vindobonensis. Further investigation may reveal the extent to which the steppic snail assemblages consist partly of Holocene newcomers

Feasibility of Preparing Nesting Box and Luring Large Solitary Carpenter Bee, Xylocopa Valga

Xylocopa valga, commonly called the carpenter bee and the largest bee with metallicviolet hair cover, is extremely rarely observed in Poland. We hypothesize that a stable and possibly long-term population of X. valga can be maintained in Poland through the creation of suitable nesting conditions. X. valga has been observed since the spring of 2014 in Wisznice (south-eastern Poland). A nesting box made out of 25 wooden blocks with drilled holes was hung about 2.5 meters above the ground. X. valga were interested in the blocks made of willow wood but did not nest in the beech, alder and pine. The carpenter bees chose holes made with drill bits of 10, 15, 20 mm in diameter and a length of 10, 15 and 20 cm. X. valga flying in the same direction most often visited the flora taxa: Aquilegia vulgaris, Ballota nigra, Consolida ajacis, Delphinium consolida, Deutzia scabra, Catalpa spp., Wisteria spp., Robinia ambigua, Stachys spp. and Trifolium pretense. X. valga is a solitary bee, but unlike most other solitary bees it demonstrates aspects of social behavior. It was observed to display cohabitative behavior involving the use of a single hole by more than one female. The females showed aggressive defensive behavior and if approached too closely started buzzing loudly. The information obtained during the long-term observation shows that X. valga can be maintained in partly artificial conditions to increase and stabilize the bee population