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Update Bayes and Certainty authored by Peter Menegay's avatar Peter Menegay
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Bayes-and-Certainty.md
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...@@ -9,3 +9,9 @@ We've already discussed the fact that Bayes pulls in favor of certainty. If we c ...@@ -9,3 +9,9 @@ We've already discussed the fact that Bayes pulls in favor of certainty. If we c
See what's happening here? The combined opinion gets pulled very strongly toward the 1st opinion the more certain the first opinion becomes. It seems a little strange that what seems like small differences in the first opinion should have such a pronounced effect. See what's happening here? The combined opinion gets pulled very strongly toward the 1st opinion the more certain the first opinion becomes. It seems a little strange that what seems like small differences in the first opinion should have such a pronounced effect.
The Bayes eqn will confirm the truth of this but this situation begs for a more intuitive explanation. Although 99% and 99.9% don't seem like much of a difference, they represent a huge difference in sample sizes. To be able to say 99%, one must perform at least 100 experiments, 99 of which succeeded and 1 which failed. To be able to say 99.9%, one must perform at least 1000 experiments where 999 succeeded and 1 failed. And so on. The Bayes eqn will confirm the truth of this but this situation begs for a more intuitive explanation. Although 99% and 99.9% don't seem like much of a difference, they represent a huge difference in sample sizes. To be able to say 99%, one must perform at least 100 experiments, 99 of which succeeded and 1 which failed. To be able to say 99.9%, one must perform at least 1000 experiments where 999 succeeded and 1 failed. And so on.
The evidential difference between 99 and 99.9% is now clearly huge. By performing 1000 experiments a scientist has built an almost unassailable advantage vs someone who would challenge his results with just 100 experiments. The challenger would also have to perform 1000 experiments just to compete and many more to show that the first scientist's results were conclusively wrong.
This also explains why any opinion, except 0%, combined with 100% yields 100% in Bayes. The one exception, the 0% opinion, combined with 100% is not computable and yields no result. In the sense outlined here, a 100% opinion doesn't really exist because it implies that the scientist must perform an infinite number of experiments to attain it. The same is true for 0%. Anyone who performs, say 1000 experiments, and claims that all of them succeeded (thus yielding a "100%" success rate) can be challenged as simply not having performed enough experiments.
Keep in mind that a probability is something that _can_ happen through random chance. We are not talking about logical or mathematical statements, for example. You cannot say that 2+2=4 100% of the time and call this a probabilistic statement. Such a statement is always true and there is no random event that can alter it. And, needless to say, statements of judgement are not probabilities either. If someone says they are 100% sure it will rain tomorrow, they are either rounding, estimating and exaggerating, or using 100% as an English synonym for "very certain". Whenever a probability is used, it must include the random chance that the opposite event will take place, however small it might be.
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