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Say I have the following empirically measured data describing a parameter of interest:

x = c(12, 4, 6, 8, 5, 12, 12, 10, 6)

where x is a continuous random variable.

My goal is to estimate the median value and a 90% credible interval for the parameter being measured.

I would like to be able to report that "there is a 90% chance that this parameter does not exceed U" where U is the upper bound of the 90% credible interval.

Let's say I have very little prior knowledge about this parameter of interest, other than that it cannot be less than or equal to 0 and likely does not exceed 50.

Past research relevant to this question:

I read through How to find 95% credible interval?, but I don't seem to have the px in that scenario (the density estimates). I'm also a bayesian neophyte, so I may well have misunderstood that question/answer. However, that makes me think that what I'd like to do is not feasible, or, at least, I don't know how to get density estimates in this case. It seems like once I got density estimates, I could follow Tim's answer in that question. Is that true?

Current attempt to solve this problem Could I simply use something like this (R code)?

 x = c(12, 4, 6, 8, 5, 12, 12, 10, 6)
px = density(x)

and then follow instructions from the answer linked above? This seems like a naive solution: it seems like I could easily go wrong using the wrong arguments (or just relying on defaults) for the density() function. If this is the recommended solution, is there a recommended way to select the right kernel and bandwidth?

Dan Scott
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  • What is the support of x? Is x a discrete random variable? – Carlos Cinelli May 06 '21 at 22:47
  • Also, you don’t specify what is the parameter you are interested in. Say you had the full distribution of X with no uncertainty. How would you compute that parameter? Are you interested in the population mean of X, the population median of X, etc. – Carlos Cinelli May 06 '21 at 22:54
  • @CarlosCinelli x is a continuous random variable (edited question to specify this). As stated in the question, I'm interested in the median value for this variable. Given the full distribution of X, I would simply compute the median value with no uncertainty. Does that answer your question? – Dan Scott May 06 '21 at 22:58
  • It does. So you say the median of X is above 0 and very likely below 50. Do you have any other information about the distribution of X beyond that? – Carlos Cinelli May 06 '21 at 23:17
  • Not really. At least, not that I feel confident in. Theory would predict that X has a heavy right tail and likely a peak between 2 and 10, but that information is very uncertain. – Dan Scott May 06 '21 at 23:26

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