# Detecting extrema under uncertainty?

An old version of this question was poorly articulated. Here is another go:

I have fifty objects. With a different, independent, unbiased scale for each object, I measure their weights 100 times each (5000 measurements in total). I can compute the variance $$\sigma^2_i$$ of each of the fifty scales based on the distribution of the measurements for each object.

Given my measurements and my $$\sigma^2_i$$, what is the probability for each of the objects being, in fact, the heaviest? (The prior probability is a uniform assumption - each has a $$\frac{1}{50}$$ chance of being the highest.)

Example: what is the probability that term 26 is the heaviest?

• Puzzled: (1) In your attached graph it seems #26 stands out as high, not #23. (2) Can't immediately see the relevance of the link. which has many topics. (3) Not sure I understand what you mean by a group being an outlier; what is your criterion for that? – BruceET Aug 13 '19 at 17:32
• @BruceET Thank you very much for your reply - I fixed the link and the highest element I refer to. My criterion for outlier here is really finding the probability, for each term, of being the highest out of all the terms. So not necessarily an outlier per se. There could be multiple terms that are vying for being the upper extrema, and they would split the probability between them. – greendolphin Aug 13 '19 at 17:41
• Well, if you have 50 groups, then doesn't each have probability $1/50$ of being the 'highest' . Still not sure what your definition of 'highest' is: has the highest mean? has the one highest single observation of all? // What kind of data do you have that shows so many outliers on the low side of the median? – BruceET Aug 13 '19 at 17:56
• You can't get probabilities out without putting some in to begin with: until you provide (a) some prior probabilities for each group to be the highest and (b) a probabilistic mechanism to explain variation within each group, your question will be unanswerable. – whuber Aug 13 '19 at 18:10
• You need some kind of assumption about the distribution within each class. For example, maybe classes 1 - 49 have expected value near 1, but class 50 is generated by some process where one sample in every 100 takes value 100 trillion. You need to be able to say something about the prior probability of this happening. – Jonny Lomond Aug 13 '19 at 18:42