I have just reproduced an example regarding a regression model for fibre strength data.
The data consisted of tensile strengths of silicon carbide fibre at four different lengths. From the data, a determination of the strength distribution of the fibres as a function of gauge length is of interest.
The data contained 50, 64, 50, and 50 tensile strength measurements for gauge lengths of 265, 25.4, 12.7, and 5.0mm respectively. An example of the data is shown below:
265mm: 0.36, 0.50, 0.57, ...
25.4mm: 1.25, 1.50, 1.57, ...
12.7mm: 1.96, 1.98, 2.06, 2.07,...
5.0mm: 2.36, 2.40, 2.54, 2.67,...
We assume that, for a given fibre length $x$, fibre strength $S_{ij} \sim Weibull(\lambda(x_i), \beta(x_i))$, $i = 1, \dots, 4$, $j = 1,\dots, n_i$, where $\mathbf{n} = (50, 64, 50, 50)$, and $\mathbf{x} = (265, 25.4, 12.7, 5)$.
Defining prior distributions for all parameters and then using MCMC to obtain draws from the joint posterior distribution I have obtained 10,000 samples for both $\lambda(x_i)$, and $\beta(x_i)$, for $i = 1,\dots, 4$.
From these samples I obtained four samples (of size 10,000) for $S_{ij}$, $i = 1,\dots,4$. I have plotted a histogram of these samples and they are my empirical distributions for the fibre strengths for the four different gauge lengths.
My questions are:
What can I do with these four sampling distributions? How can I make residual plots when I have sampling distributions and not one forecast for each tensile strength? Are residual plots necessary here?
The data do seem to be inside my $95\%$ highest posterior density intervals, but is there anything else that I can do?
Is it an option to plot the histogram of the data (for each fibre length) on top of the histograms for the empirical sampling distributions for a graphical plot? I could perhaps use Kullback-Leibler to find the distance between the empirical distributions and the data distributions?
Finally, I only used samples from one MCMC chain to make inferences (disregarding the warm-up draws). I just did this because I was trying to reproduce an example. In the future, should I combine all chains (not including warm-up draws) to make inferences in the future? Assuming that all chains converge and mix well.
