Bayesian Inference on the Radio-quietness of Gamma-ray Pulsars

We demonstrate for the first time using a robust Bayesian approach to analyse the populations of radio-quiet (RQ) and radio-loud (RL) gamma-ray pulsars. We quantify their differences and obtain their distributions of the radio-cone opening half-angle $\delta$ and the magnetic inclination angle $\alpha$ by Bayesian inference. In contrast to the conventional frequentist point estimations that might be non-representative when the distribution is highly skewed or multi-modal, which is often the case when data points are scarce, Bayesian statistics displays the complete posterior distribution that the uncertainties can be readily obtained regardless of the skewness and modality. We found that the spin period, the magnetic field strength at the light cylinder, the spin-down power, the gamma-ray-to-X-ray flux ratio, and the spectral curvature significance of the two groups of pulsars exhibit significant differences at the 99\% level. Using Bayesian inference, we are able to infer the values and uncertainties of $\delta$ and $\alpha$ from the distribution of RQ and RL pulsars. We found that $\delta$ is between $10^\circ$ and $35^\circ$ and the distribution of $\alpha$ is skewed towards large values.