Is the paper online to read? I think without needing to go fully(?) into a Bayesian setting, we can see

why conclusions from studies with many tests are so often wrong. Does that paper make his case around something like microarray studies?

"In [Ioannidis'] example, he supposes that 100,000 gene polymorphisms are being tested for association with schizophrenia. If 10 polymorphisms truly are associated with schizophrenia, the pre-study probability that a given gene is associated is 0.0001. If a study has 60% power (β = 0.4) and significance level α = 0.05, the post-study probability that a polymorphism determined to be associated really is associated is 0.0012. There’s a 99.8% chance that the result is false."