Bayes-and-Certainty.md

@@ -84,17 +84,17 @@ 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.
+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 $`P_2 = 0.73`$,
+For example, using $P_1 = 0.803$ and $P_2 = 0.73$,
 
-1) $`P\_{bay} = {0.803(0.73)\over {0.803(0.73) + 0.197(0.27)}} = 0.917`$
-2) $`P\_{1} = 0.803`$
-3) $`f=10^{(Nd_1 - Nd_2)} = 10^{3-2} = 10`$
-4) $`P_{comb,w} = 0.803 + {(0.917 - 0.803)\over 10} = 0.8144`$
+1) $P\_{bay} = {0.803(0.73)\over {0.803(0.73) + 0.197(0.27)}} = 0.917$
+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.
+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.