A fundamental endeavor in exoplanetary research is to characterize the bulk compositions of planets via measurements of their masses and radii. With future sample sizes of hundreds of planets to come from TESS and PLATO, we develop a statistical method that can flexibly yet robustly characterize these compositions empirically, via the exoplanet M–R relation. Although the M–R relation has been explored in many prior works, they mostly use a power-law model, with assumptions that are not flexible enough to capture important features in current and future M–R diagrams. To address these shortcomings, a nonparametric approach is developed using a sequence of Bernstein polynomials. We demonstrate the benefit of taking the nonparametric approach by benchmarking our findings with previous work and showing that a power law can only reasonably describe the M–R relation of the smallest planets and that the intrinsic scatter can change non-monotonically with different values of a radius. We then apply this method to a larger data set, consisting of all the Kepler observations in the NASA Exoplanet Archive. Our nonparametric approach provides a tool to estimate the M–R relation by incorporating heteroskedastic measurement errors into the model. As more observations will be obtained in the near future, this approach can be used with the provided R code to analyze a larger data set for a better understanding of the M–R relation.

Though they are the most abundant stars in the Galaxy, M dwarfs form only a small subset of known stars hosting exoplanets with measured radii and masses. In this paper, we analyze the mass-radius (M-R) relationship of planets around M dwarfs using M-R measurements for 24 exoplanets. In particular, we apply both parametric and nonparametric models and compare the two different fitting methods. We also use these methods to compare the results of the M dwarf M-R relationship with that from the Kepler sample. Using the nonparametric method, we find that the predicted masses for the smallest and largest planets around M dwarfs are smaller than similar fits to the Kepler data, but that the distribution of masses for 3 R ⊕ planets does not substantially differ between the two datasets. With future additions to the M dwarf M-R relation from the Transiting Exoplanet Survey Satellite and instruments like the Habitable zone Planet Finder, we will be able to characterize these differences in more detail. We release a publicly available Python code called MRExo a) which uses the nonparametric algorithm introduced by Ning et al. (2018) to fit the M-R relationship. Such a nonparametric fit does not assume an underlying power law fit to the measurements and hence can be used to fit an M-R relationship that is less biased than a power-law. In addition MRExo also offers a tool to predict mass from radius posteriors, and vice versa.

A general non-threshold BPS (F, D p ) (or (D p−2 , D p )) bound state can be described by a boundary state with a quantized world-volume electric (or magnetic) flux and is characterized by a pair of integers (m, n). With this, we calculate explicitly the interaction amplitude between two such non-threshold bound states with a separation Y when each of the states is characterized by a pair of integers (m i , n i ) with i = 1, 2. With this result, one can show that the non-degenerate (i.e., m i n i = 0) interaction is in general attractive for the case of (D p−2 , D p ) but this is true and for certain only at large separation for the case of (F, D p ). In either case, this interaction vanishes only if m 1 /n 1 = m 2 /n 2 and n 1 n 2 > 0. We also study the analytic structure of the corresponding amplitude and calculate in particular the rate of pair production of open strings in the case of (F, D p ).

We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors. (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving a ℓ2,1-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein-von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Rényi divergence of order 1/2. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.

Bayesian methods provide a natural means for uncertainty quantification, that is, credible sets can be easily obtained from the posterior distribution. But is this uncertainty quantification valid in the sense that the posterior credible sets attain the nominal frequentist coverage probability? This paper investigates the frequentist validity of posterior uncertainty quantification based on a class of empirical priors in the sparse normal mean model. In particular, we show that our marginal posterior credible intervals achieve the nominal frequentist coverage probability under conditions slightly weaker than needed for selection consistency and a Bernstein-von Mises theorem for the full posterior, and numerical investigations suggest that our empirical Bayes method has superior frequentist coverage probability properties compared to other fully Bayes methods.