The pattern maximum likelihood (PML) estimate, introduced by Orlitsky et al. (2004), is an estimate of an unknown underlying probability distribution from n i.i.d. samples taken from the distribution. The PML estimate involves solving a non-trivial optimization problem over the set of all probability mass functions (pmfs) of finite support. In recent work, we showed an interesting phase transition phenomenon in the PML estimate: at a certain threshold pmf support size Theta, the uniform distribution goes from being a local maximum to being a local minimum for the optimization problem in the estimate. In this work, we study the Bethe approximation to the PML estimate, introduced by Vontobel (2012), and exhibit a more complex phase transition phenomenon for this approximation.