Uhm, it's similar, but not the same.
If I understand it correctly, selection bias is when 20 researchers make an experiment with green jelly beans, 19 of them don't find significant correlation, 1 of them finds it... and only the 1 publishes, and the 19 don't. The essence is that we had 19 pieces of evidence against the green jelly beans, only 1 piece of evidence for the green jelly beans, but we don't see those 19 pieces, because they are not published. Selection = "there is X and Y, but we don't see Y, because it was filtered out by the process that gives us information".
But imagine that you are the first researcher ever who has researched the jelly beans. And you only did one experiment. And it happened to succeed. Where is the selection here? (Perhaps selection across Everett branches or Tegmark universes. But we can't blame the scientific publishing process for not giving us information from the parallel universes, can we?)
In this case, base rate neglect means ignoring the fact that "if you take a random thing, the probability that this specific thing causes acne is very low". Therefore, even if the experiment shows a connection with p = 0.05, it's still more likely that the result just happened randomly.
The proper reasoning could be something like this (all number pulled out of the hat) -- we already have pretty strong evidence that acne is caused by food; let's say there is a 50% probability for this. With enough specificity (giving each fruit a different category, etc.), there are maybe 2000 categories of food. It is possible that more then one of them cause acne, and our probability distribution for that is... something. Considering all this information, we estimate a prior probability let's say 0.0004 that a random food causes acne. -- Which means that if the correlation is significant on level p = 0.05, that per se means almost nothing. (Here one could use the Bayes theorem to calculate that the p = 0.05 successful experiment shows the true cause of acne with probablity cca 1%.) We need to increase it to p = 0.0004 just to get a 50% chance of being right. How can we do that? We should use a much larger sample, or we should repeat the experiment many times, record all the successed and failures, and do a meta-analysis.
But imagine that you are the first researcher ever who has researched the jelly beans. And you only did one experiment. And it happened to succeed. Where is the selection here?
That's a different case -- you have no selection bias here, but your conclusions are still uncertain -- if you pick p=0.05 as your threshold, you're clearly accepting that there is a 5% chance of a Type I error: the green jelly beans did nothing, but the noise happened to be such that you interpreted it as conclusive evidence in favor of your hypothesis.
But that all is fine -- the...
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