Bayes-and-Certainty.md

@@ -83,3 +83,17 @@ For example, using $P_1 = 0.803$ and $P_2 = 0.73$,
 2) $P\_{1} = 0.803$
 3) $f=10^{(Nd_1 - Nd_2)} = 10^{3-2} = 10$
 4) $P_{comb,w} = 0.803 + {(0.917 - 0.803)\over 10} = 0.8144$
+
+Given how close the result is to $P_1$ we might just consider skipping the calculation and using $P_1$ in situations where an order of magnitude or more separates the two experiments. The calculation might be more useful in cases where we can't see any difference in decimal place accuracy but know that significantly different sample sizes were used.
+
+This idea is also useful for evaluating the relative merits of two experiments without necessarily combining them via Bayes. If two scientists disagree and one has significantly more experimental evidence, then we could use the above idea to perform a weighted average of their opinions.
+
+This discussion assumes that probabilities will be reported to the correct number of decimal places. If we do 11 experiments and 9 succeed we could claim
+
+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 done. 
+
+ 
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