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Roughly speaking (i.e., hopefully I don't need to contextualise too much), I ran an experiment five times to find the length of some aluminum sheet. Using a Bayesian model I got five different looking histograms. Is there some recommended way I can combine all this data into one final interpretation / histogram for my length measurement. I have attached an image below to show the sorts of histograms I am talking about.

enter image description here

Here is also another set of histograms doing a similar experiment but looking at width instead of length.

enter image description here

As I said, hopefully there is some way I can combine all this data into one final histogram / interpretation. If these were all Gaussian it would be easy, but now I'm not 100% sure how to go about this properly.

EDIT:

  • These are posterior distributions

  • Length / width can be negative because I have modeled distances as having a prior mean of "0", so essentially I am looking at delta values. That is, my sheet may be 2m long, but my model for length was Gaussian with mean 0 mm, and std 5 mm, so that I am looking for perturbation around 2m rather than absolute lengths. Really the two problems are equivalent imo.

  • These histograms are a result of combining estimates from a computational model, and experimental values. So my prior was $N \sim (0,5^2) mm$ and then I passed these values through a computational model which does "something" (hence my titles, mode1, mode2 ...), and compared what the computational model gave me to the experimental data via the Likelihood function (also assumed Gaussian). But because the computational model is not linear, we do not have the simple case of Gaussian x Gaussian = Gaussian, because the variable in my prior has gone through a non-linear transformation before being placed into my Likelihood.

  • So this process is kind of like: $F \sim N(g(x)|\text{exp}_{\mu},\text{exp}_{\sigma^2})\times N(x;0,5^2)$ where exp is short for experiment.

  • Mode doesn't refer to anything in statistics. Mode in this experiment means the vibrational mode of movement of the sheet metal. So one sheet metal in this case, had five vibrational modes for me to track

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    $\begingroup$ You're going to need to say more about this. Are these posterior distributions for some parameter after Bayesian analysis? How is it that you have negative lengths & widths? $\endgroup$ – gung Sep 28 '16 at 2:42
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    $\begingroup$ These are explained in main body now. $\endgroup$ – pche8701 Sep 28 '16 at 2:56

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