even if you took some drug that wasn't just water, your case is anecdotal evidence. It should hold the same weight as any single case in any of the clinical trials.

Ah, but that's not quite true. If a remedy helps you personally, it is less than a "single case" for clinical trial purposes, because it was not set up properly for that. You would do well to ignore it completely if you are calculating your priors for the remedy to work for you.

However, it is much more than nothing for you personally, once you have seen it working for you once. Now you have to update on the evidence. It's a different step in the Bayesian reasoning: calculating new probabilities given that a certain event (in this case one very unlikely apriori -- that the remedy works) actually happened.