I have a model for data in my experiment that states that the data has a Gumbel distribution with known location and scale.
I am then looking at observations with very high scores that I suspect to be outliers.
Is there a Bayesian approach (or otherwise) to get the probability that this observation is from the underlying Gumbel distribution?
As an end result I would like to decide whether or not I accept this observation as coming from the underlying Gumbel distribution.
I am in fact working with biological data and have demonstrated both empirically and through a set of assumptions that the distribution is in fact of Gumbel type with parameters that can be estimated from the data itself and/or from a theoretical perspective given some assumptions.
Now, while most data behaves in-line with this distribution, I would like to find the probability that observations with very large values might not be from the same Gumbel distribution but due to some other phenomenon that is not described in this distribution that I want to detect.
In this case, updating the data using the new observation would not be useful (especially considering that this observation might come from another random variable with a different distribution).
I am able to calculate the one-side probability of how extreme the observation is by using.
From my (very limited) understanding of statistics and probability, I fear that quoting this value might be erroneous. My main concern is that this is the "probability of observing more extreme observations" and NOT the "probability that this observation is not from the distribution".
Any thoughts on this?