Is this ok to average p values? I have seen other posts about averaging p values. I believe I have a special case where average may apply.
Finding the exact probability distribution of my data is an NP hard task, so I am using a non deterministic algorithm to estimate my p value. The algorithm was published and validated on experimental data. Because it is non deterministic, I have repeated the same test 1000 times.  In this case I think it would make sense to report the mean p value along with the standard deviation.  In most cases standard deviation is < 0.05.
For combining p values, I would use Fisher's method, but the 1000 tests are not independent.  They are just non deterministic iterations of the same test.  This is why I want to use mean p.  Reporting mean and standard deviation would be a more descriptive estimate of the p value rather than running the algorithm just one time.
Searching through the literature, I found a few things that appear to contradict the idea that a p value cannot be averaged:
Using harmonic mean p value for dependent tests
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6347718/
different ways of combining p values via averaging
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6347718/

It is quite common to assess statistical significance via resampling, bootstrap or permutation tests, all of which are randomized (see, e.g., the textbook by Good, 2006). In such cases, people will often run their procedure just a single time and report the resulting p value. More sophisticated modelers will re-run their analysis a few times to see how stable the p values are. (Unscrupulous modelers will re-run them until they get a low p value.)
Running the analysis multiple times and averaging the resulting p values seems to me to be an excellent way of getting more stable p values. Essentially, this treats the p value as an estimate of an underlying test statistic itself (which it is!), and harnesses the power of averaging. The relevant criticisms of p value averaging do not apply. (If anyone disagrees, I would be interested in their argument.)
