Say there is a hypothesis that A causes B (A -> B), and some likelihood that the hypothesis is correct (AB1%).
Now, an experiment is run that claims to find a correlation between A and B.
What I would like to know is whether my new probability (AB2%) is greater than, or less than, my previous probability (AB1%).
Common sense tells me that AB2% must be greater than AB1% -- that there's no way that finding a correlation would make it less likely that A -> B.
Is my common sense correct? I would like to account for the possibilities of sampling bias, huge experimental error, reporting bias, etc. -- everything except for falsified data.
For example, knowing that things like these can happen:
- extremely biased experiment in which any correlation is much more likely to be due to biased sampling than to A -> B.
- very noisy experimental design, where random correlations between noise is much more likely than observing any true correlation
- totally random data, cherry-picked to find correlations
how would I incorporate my knowledge of those (an any other) factors into an estimate for AB2%?
Is this statement true: a correlation is found, therefore it's more likely that there's a real causal link between two variables than it was before the correlation was found?
I'm not a statistician -- please be gentle! :)