Recently, papers have appeared that champion the Bayesian approach to the analysis of experimental data. From reading these papers, the physicist could be forgiven for believing that Bayesian methods reveal deep truths about physical systems and are the correct paradigm for the analysis of all experimental data. This paper makes a contrary argument and is deliberately provocative. It is argued that the Bayesian approach to reconstruction of chaotic time series is fundamentally flawed, and the apparent successes result not from any degree of correctness of the paradigm, but by an accidental and unintended property of an algorithm. We also argue that (non-Bayesian) shadowing techniques provide all the information the erroneous Bayesian methods obtain, but much more efficiently, and also provide a wealth of additional useful information.