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I am trying to fit informed prior distributions to data using MLE, and F occasionally provides a best fit (lowest AIC value).

I am starting with only very basic knowledge of probability theory, so I am not even confident that it is appropriate to fit an F in these cases. One of the interesting aspects of probability theory is that distributions often represent specific data generating processes - and are therefore useful for specific data types. However, I can not find any such justification for the F distribution. For example, the NIST handbook states:

Since the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.

However, relationships between the F and Beta distributions suggest that F might be appropriate in general for ratios.

Are there theoretical (not computational) reasons to use or not use the F distribution as an informed prior distribution, or is having a continuous parameter with values > 0 sufficient justification?

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There is no computational nor theoretical reason not to use an F distribution as a prior distribution. If it fits your belief or the preliminary experiment well enough, go for it. The fact that there is no clear physical data generating process should not be seen as a drawback. All in all, priors are very rarely exactly defined! And the data should bring the most informative part in the inference.

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