That is a good question for a statistician, and I am not a statistician.

One thing that leaps to mind, however, is two-boxing on Newcomb's Problem using assumptions about the prior probability of box B containing $1,000,000. Some new work using math that I don't begin to understand suggests that either response to Newcomb's problem is defensible using Bayesian nets.

There could be more trivial cases, too, where a person inputs unreasonable prior probabilities and uses cargo-cult statistics to support some assertion.

Also, it's struck me that a frequentist statistician might call most Bayesian uses of the theorem "abuses."

I'm not sure those are really good examples, but I hope they're satisfying.