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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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......@@ -83,8 +83,8 @@ Nevertheless, this idea asks us to consider what happens when two opinions diffe
So one idea to handle this is to weigh the opinions as follows:
1) Use the regular Bayes equation to combine the two probabilities. This is the one limit.
2) Use the experiment with the most evidence for it (most decimal places) to establish another limit, $`P_1`$.
3) Decide on a factor for the difference in evidence between the two experiments using the decimal place of the reported probabilities. An 80.3% probability vs. 73% has one more decimal place and so gives the 80.3% opinion 10 times the weight of the 73% opinion. This leads to the equation $`f = 10^{Nd_1-Nd_2}`$ where $`Nd_1`$ is the number of decimal places of the most accurate probability. Of course, if the probability comes with the sample size used to calculate it we can just divide the sample sizes to obtain the factor.
2) Use the experiment with the most evidence for it (most decimal places) to establish another limit, $P_1$.
3) Decide on a factor for the difference in evidence between the two experiments using the decimal place of the reported probabilities. An 80.3% probability vs. 73% has one more decimal place and so gives the 80.3% opinion 10 times the weight of the 73% opinion. This leads to the equation $f = 10^{Nd_1-Nd_2}$ where $`Nd_1`$ is the number of decimal places of the most accurate probability. Of course, if the probability comes with the sample size used to calculate it we can just divide the sample sizes to obtain the factor.
4) Use the factor just calculated as a weighting factor in establishing the combined opinion between the lower and upper limit.
For example, using $`P_1 = 0.803`$ and $`P_2 = 0.73`$,
......@@ -107,3 +107,4 @@ P = 9/11 = 0.818181818181
but this would be to represent the result with far too many decimal places, implying that many more experiments had been done to confirm it. The correct decimal representation is 0.8, implying that about 10 experiments have taken place, leading to one decimal place.
Therefore we need to watch when reports are made with a suspiciously large number of decimal places. Sources should be encouraged to report their experimental results as fractions where we can see openly in the denominator how many experiments were conducted.