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We present a novel fully Bayesian analysis to constrain short gamma-ray burstjet structures associated with cocoon, wide-angle and simple top-hat jetmodels, as well as the binary neutron star merger rate. These constraints aremade given the distance and inclination information from GW170817, observedflux of GRB170817A, observed rate of short gamma-ray bursts detected by Swift,and the neutron star merger rate inferred from LIGO's first and secondobserving runs. A separate analysis is conducted where a fitted short gamma-rayburst luminosity function is included to provide further constraints. The jetstructure models are further constrained using the observation of GW190425 andwe find that the assumption that it produced a GRB170817-like short gamma-rayburst that went undetected due to the jet geometry is consistent with previousobservations. We find and quantify evidence for low luminosity and wide-angledjet structuring in the short gamma-ray burst population, independently fromafterglow observations, with log Bayes factors of $0.45{-}0.55$ for such modelswhen compared to a classical top-hat jet. Slight evidence is found for aGaussian jet structure model over all others when the fitted luminosityfunction is provided, producing log Bayes factors of $0.25{-}0.9\pm0.05$ whencompared to the other models. However without considering GW190425 or thefitted luminosity function, the evidence favours a cocoon-like model with logBayes factors of $0.14\pm0.05$ over the Gaussian jet structure. We provide newconstraints to the binary neutron star merger rates of$1{-}1300$Gpc$^{-3}$yr$^{-1}$ or $2{-}680$Gpc$^{-3}$yr$^{-1}$ when a fittedluminosity function is assumed.

Unpacking merger jets: a Bayesian analysis of GW170817, GW190425 and electromagnetic observations of short gamma-ray bursts