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In my Bayesian class, we are always required to specify vague (or noninformative) priors for bayesian modeling. I am quite confused about that.

If I understand correctly, the main advantage of Bayesian over frequentist is that Bayesian approach provides a formal and scientific way to incorporate prior useful information into models.

My questions are:

(1) Generally, when and why are vague or non-informative priors used in Bayesian modeling?

(2) If the prior knowledge is rather rare, then what is the point of using bayesian modeling?

For example, consider a linear mixed-effects model, $$\mathbf{y} = \mathbf{X} \boldsymbol \beta + \mathbf{Z} \boldsymbol \gamma + \boldsymbol \epsilon$$ where $\boldsymbol \beta$ is fixed effect vector, $\boldsymbol \gamma \ \sim \ N(\mathbf{0}, \sigma_{\gamma}^2 \mathbf{I})$, and $\boldsymbol \epsilon \ \sim \ N(\mathbf{0}, \sigma_{\epsilon}^2 \mathbf{I})$.

Then with Bayesian approach, we'd specify some vague or noninformative priors on parameters $\boldsymbol \beta, \ \sigma_{\gamma}^2, \ \sigma_{\epsilon}^2$.

I just don't understand if the priors are non-informative, why don't we just fit the linear mixed-effects model in the frequentist way.

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marked as duplicate by Xi'an, Scortchi - Reinstate Monica, Nick Cox, Peter Flom - Reinstate Monica Mar 17 '15 at 10:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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  1. Vague and noninformative priors are often used if researchers cannot agree on what prior information they have, or if they legitimately do not feel comfortable specifying a distribution. One of my professors always said that the biggest problem for Bayesian statistics is that if you shove 15 statisticians into a room and say "Do a Bayesian analysis of this problem", you'll get 15 different priors. Saying "we really don't know" is a good way to alleviate that problem and to avoid accusations of bias.

  2. Even though prior knowledge may be rare, sometimes you still want to take advantage of the Bayesian interpretation. As you probably know, a common critique of frequentist methods is that they are often difficult to interpret. For example, confidence intervals treat the bounds as random and the parameter as fixed, so the statement "The probability that this mean is in the interval" don't have a meaningful interpretation because the interval either captures the mean or it doesn't. Using a noninformative prior allows you access to a Bayesian interpretation without having to incorporate unavailable or not-agreed-upon prior knowledge.

For more information, read this older question where someone asked a very similar question.

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