Priors that do not become irrelevant with large sample sizes This may be a weird question. My colleagues and I are working on a medical estimation problem, where relevant prior knowledge regarding plausible values of some physiological parameters exists. In addition, these parameters can be estimated using time series data, which often have some tens of thousands of samples. What often happens is that, due to model imperfections, MAP estimation converges towards implausible solutions (e.g., very small parameter values). The priors are essentially ignored because they become irrelevant given the large amount of available (and informative) measurements.
Now I am well aware that one way (probably the preferred one) to solve this problem is to improve the time series model and try to fix its imperfections. This is proving to be really hard, however, for a number of reasons.
Another (partial) remedy might be to somehow "fix" the influence of the prior such that it does not become irrelevant. This could be done, e.g., by choosing the prior's covariance as a function of the sample size or simply assigning the prior and the likelihood fixed weights when determining final parameter estimates. That got me wondering: is this "a thing"? Is there a name for doing something like this, i.e., a "fixed-influence prior"? Surely much more knowledgeable people than me have thought about this problem.
 A: I do agree with the previous answers, but if you really want to "fix" the influence of the prior, here are some ideas.

*

*If your prior is based on historical data, you can use a power prior [1] to control the relative influence of your prior on the posterior obtained with new data.


*Alternatively, you can also consider weighing the likelihood (power scaling) so that the relative influence of your prior is increased. However, if, for example, you have a Gaussian model, this would be equivalent to increasing the standard deviation of the Gaussian. So in the end, maybe you do need to change your model.
You may also be interesting in reading [2] which uses power scaling as a way to diagnose prior sensitivity.
[1] Ibrahim, J. G., Chen, M. H., Gwon, Y., & Chen, F. (2015). The power prior: Theory and applications. Statistics in Medicine, 34(28), 3724–3749. https://doi.org/10.1002/sim.6728
[2] Kallioinen, N., Paananen, T., Bürkner, P.-C., & Vehtari, A. (2021). Detecting and diagnosing prior and likelihood sensitivity with power-scaling. https://arxiv.org/abs/2107.14054v1
A: I dispute your main premise.
A prior distribution is your guess (hopefully a good guess, but still a guess).
Then you observe data and see what really happens.
When you have enough observations that contradict your original guess, it is reasonable to change your mind.
What you’re observing strikes me as a feature, not a bug, of Bayesian inference.
A: The answer to this question centers on its false premise. If I can sum up your question, you are saying the posterior is really far from your prior, but rather than acknowledging that either your prior is wrong or that your likelihood is misspecified, you instead want to know how you can just use a stronger prior to enforce that the posterior is not "too far" from prior... at which point why even use data? Just start with your prior, flip a coin and roll some dice, move your prior by that amount in that direction, and then call it your posterior. From your question it sounds like if you had 2x or 10x the data you would just be asking how to make your prior 2x or 10x stronger to cancel out the data and get the posterior you want. Therefore please fix your model (or acknowledge that currently it is not possible to model this data well enough), but please do not just change your prior to get a predetermined outcome.
A: Have you considered that your expectation is simply wrong, perhaps because of publication bias?
Alternatively, if you are so confident in your beliefs that you're willing to discount the posterior after an analysis of tens of thousands of data points, it seems to me that your specified prior does not truly reflect the strength of your belief. You should probably specify a much stronger prior - if it's strong enough, the posterior wouldn't be dominated by the data.
