High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation

The ratio between two probability density functions is an important component of various tasks, including selection bias correction, novelty detection and classification. Recently, several estimators of this ratio have been proposed. Most of these methods fail if the sample space is high-dimensional, and hence require a dimension reduction step, the result of which can be a significant loss of information. Here we propose a simple-to-implement, fully nonparametric density ratio estimator that expands the ratio in terms of the eigenfunctions of a kernel-based operator; these functions reflect the underlying geometry of the data (e.g., submanifold structure), often leading to better estimates without an explicit dimension reduction step. We show how our general framework can be extended to address another important problem, the estimation of a likelihood function in situations where that function cannot be well-approximated by an analytical form. One is often faced with this situation when performing statistical inference with data from the sciences, due the complexity of the data and of the processes that generated those data. We emphasize applications where using existing likelihood-free methods of inference would be challenging due to the high dimensionality of the sample space, but where our spectral series method yields a reasonable estimate of the likelihood function. We provide theoretical guarantees and illustrate the effectiveness of our proposed method with numerical experiments.Comment: With supplementary materia

New Image Statistics for Detecting Disturbed Galaxy Morphologies at High Redshift

Testing theories of hierarchical structure formation requires estimating the distribution of galaxy morphologies and its change with redshift. One aspect of this investigation involves identifying galaxies with disturbed morphologies (e.g., merging galaxies). This is often done by summarizing galaxy images using, e.g., the CAS and Gini-M20 statistics of Conselice (2003) and Lotz et al. (2004), respectively, and associating particular statistic values with disturbance. We introduce three statistics that enhance detection of disturbed morphologies at high-redshift (z ~ 2): the multi-mode (M), intensity (I), and deviation (D) statistics. We show their effectiveness by training a machine-learning classifier, random forest, using 1,639 galaxies observed in the H band by the Hubble Space Telescope WFC3, galaxies that had been previously classified by eye by the CANDELS collaboration (Grogin et al. 2011, Koekemoer et al. 2011). We find that the MID statistics (and the A statistic of Conselice 2003) are the most useful for identifying disturbed morphologies. We also explore whether human annotators are useful for identifying disturbed morphologies. We demonstrate that they show limited ability to detect disturbance at high redshift, and that increasing their number beyond approximately 10 does not provably yield better classification performance. We propose a simulation-based model-fitting algorithm that mitigates these issues by bypassing annotation.Comment: 15 pages, 14 figures, accepted for publication in MNRA

Rethinking Hypothesis Tests

Null Hypothesis Significance Testing (NHST) have been a popular statistical tool across various scientific disciplines since the 1920s. However, the exclusive reliance on a p-value threshold of 0.05 has recently come under criticism; in particular, it is argued to have contributed significantly to the reproducibility crisis. We revisit some of the main issues associated with NHST and propose an alternative approach that is easy to implement and can address these concerns. Our proposed approach builds on equivalence tests and three-way decision procedures, which offer several advantages over the traditional NHST. We demonstrate the efficacy of our approach on real-world examples and show that it has many desirable properties

Theoretical and experimental evidence of level repulsion states and evanescent modes in sonic crystal stubbed waveguides

The complex band structures calculated using the Extended Plane Wave Expansion (EPWE) reveal the presence of evanescent modes in periodic systems, never predicted by the classical \omega(\vec{k}) methods, providing novel interpretations of several phenomena as well as a complete picture of the system. In this work we theoretically and experimentally observe that in the ranges of frequencies where a deaf band is traditionally predicted, an evanescent mode with the excitable symmetry appears changing drastically the interpretation of the transmission properties. On the other hand, the simplicity of the sonic crystals in which only the longitudinal polarization can be excited, is used to interpret, without loss of generality, the level repulsion between symmetric and antisymmetric bands in sonic crystals as the presence of an evanescent mode connecting both repelled bands. These evanescent modes, obtained using EPWE, explain both the attenuation produced in this range of frequencies and the transfer of symmetry from one band to the other in good agreement with both experimental results and multiple scattering predictions. Thus, the evanescent properties of the periodic system have been revealed necessary for the design of new acoustic and electromagnetic applications based on periodicity