NB: I was gonna post on physics stack exchange, not really sure where this fits in. But I'm only a lowly Engineer so please go easy on the notation if you can
Using Student's t-distribution I can infer the parameters ($\mu,\sigma^2$) of a probability distribution based on $n$ samples of data that I assume will fit a gaussian prior. However in all the examples I've seen, the $n$ samples are all simple values. How can I infer a probability distribution based on samples of data with uncertainty; if my $n$ samples are not simple values but probability distributions themselves? What is the effect of measurement uncertainty on the shape of the inferred distribution?
I'm trying to measure how long some code takes to run on a computer. The timer is low resolution - similar order of magnitude to the duration I'm trying to measure - and so the true timestamps are quantized into 100 ms bins. Assuming a uniform rectangular probability distribution within these bins, then the time differences have a triangular probability distribution.
i.e. A task starting at $142ms$ and ending at $331 ms$ when quantised will appear to start at $100\pm50ms$ and end at $300\pm50ms$. Then the difference will be a triangular probability distribution, centered on $200ms$ and with a width of $\pm 100ms$.
I have several of these triangular timespan measurements, and I'd like to use them to determine the parameters of a distribution. As I say, I could just ignore the quantisation errors in my samples, and plug the modal (centre) values into the t-distribution, but surely those errors will increase the uncertainty ($\sigma$) of my inferred gaussian?