A uniform prior would be one example. That's just saying "I've got no idea what the value is".

There is a notion of an uninformative prior [0].

What would you do if you were studying a process that had not been previously studied, so you had no previous experience/data on which to base your a prior? If you wait, and choose a prior based on the data you observe then you are using an empirical Bayes method [1].

[0]: https://en.wikipedia.org/wiki/Prior_probability#Uninformativ...

[1]: https://en.wikipedia.org/wiki/Empirical_Bayes_method

It's quite common to choose a conjugate prior, which aids computation, and which is readily comprehensible within the discipline ("...and a Wishart prior for the covariance, of course..."). But which is, in effect, not at all informed by the data.

It's also common to have a complex, multi-level model setup which has a few hyperparameters (say, gamma distribution shapes) which are set basically arbitrarily. The idea being that the posteriors for the lowest levels (closest to data) will be learned/fitted, but the hyper parameters at the top-level of the model are fixed. This is very common in spatial statistics.

It's also common to make lots of (conditional) independence assumptions, just because they are convenient. Such as diagonal covariances, or conditional independence between "separate" elements of a model. But these independence assumptions are often just convenient or based on intuition, but not re-checked. Or, if they are checked and found wanting, it's "left for future work".

These practices are defensible, and attackable. But choosing convenient priors out of a bag of standard priors, without reference to the problem at hand, is very common.

EDIT: I forgot about the prevalence of uninformed priors. Thanks HN!