The variational framework for learning inducing variables (Titsias, 2009a) has had a large impact on the Gaussian process literature. The framework may be interpreted as minimizing a rigorously defined Kullback-Leibler divergence between the approxim
ating and posterior processes. To our knowledge this connection has thus far gone unremarked in the literature. In this paper we give a substantial generalization of the literature on this topic. We give a new proof of the result for infinite index sets which allows inducing points that are not data points and likelihoods that depend on all function values. We then discuss augmented index sets and show that, contrary to previous works, marginal consistency of augmentation is not enough to guarantee consistency of variational inference with the original model. We then characterize an extra condition where such a guarantee is obtainable. Finally we show how our framework sheds light on interdomain sparse approximations and sparse approximations for Cox processes.
We report on a search for Interstellar CH2D+. Four transitions occur in easily accessible portions of the spectrum; we report on emission at the frequencies of these transitions toward high column density star-forming regions. While the observations
can be interpreted as being consistent with a detection of the molecule, further observations will be needed to secure that identification. The CH2D+ rotational spectrum has not been measured to high accuracy. Lines are weak, as the dipole moment induced by the inclusion of deuterium in the molecule is small. Astronomical detection is favored by observations toward strongly deuterium-fractionated sources. However, enhanced deuteration is expected to be most significant at low temperatures. The sparseness of the available spectrum and the low excitation in regions of high fractionation make secure identification of CH2D+ difficult. Nonetheless, owing to the importance of CH3+ to interstellar chemistry, and the lack of rotational transitions of that molecule owing to its planar symmetric structure, a measure of its abundance would provide key data to astrochemical models.
