Mendelian randomization uses genetic variants as instrumental variables to make causal inferences on the effect of an exposure on an outcome. Due to the recent abundance of high-powered genome-wide association studies, many putative causal exposures of interest have large numbers of independent genetic variants with which they associate, each representing a potential instrument for use in a Mendelian randomization analysis. Such polygenic analyses increase the power of the study design to detect causal effects; however, they also increase the potential for bias due to instrument invalidity. Recent attention has been given to dealing with bias caused by correlated pleiotropy, which results from violation of the "instrument strength independent of direct effect" assumption. Although methods have been proposed that can account for this bias, a number of restrictive conditions remain in many commonly used techniques. In this paper, we propose a Bayesian framework for Mendelian randomization that provides valid causal inference under very general settings. We propose the methods MR-Horse and MVMR-Horse, which can be performed without access to individual-level data, using only summary statistics of the type commonly published by genome-wide association studies, and can account for both correlated and uncorrelated pleiotropy. In simulation studies, we show that the approach retains type I error rates below nominal levels even in high-pleiotropy scenarios. We demonstrate the proposed approaches in applied examples in both univariable and multivariable settings, some with very weak instruments.