Bayes statistics - sensitivity to priors

[rogojel]

Hi,
I came across a statement in a review of a research field, where they were determining an unknown parameter using Bayesian statistics. The reviewer said that using Bayes statistics to estimate the parameter is uncontroversial, but many of the papers dealing with this estimation are weak because of their choice of the prior probability.

So, my questions would be: is there a more or less objective way of chosing the priors? Are the results sensitive to this choice (I guess they are) and if that is the case how can any statement be made about the real world as opposed to a statement about the beliefs of the author?

I am a complete beginner with the Bayesian methods, so hopefully these are well known and easily answereable questions.

regards
rogojel

 

[TheEcologist]

Hi,
So, my questions would be: is there a more or less objective way of chosing the priors? Are the results sensitive to this choice (I guess they are) and if that is the case how can any statement be made about the real world as opposed to a statement about the beliefs of the author?

1) There isn't a single objective way. There are more or less objective priors called "flat priors" and these can be objective in most cases but even then, an unintended transformation can make them informative again. Now whether this is bad is another issue completely, and it depends on the influence of an informative prior on the posterior.

2) The results can be sensitive to this, in general however as n increases, the influence of the prior mean diminishes. So this is mostly a problem with sparse data. However where the prior probability is zero, no amount of data will change that - and the prior's influence is absolute. This is why Dason loves point mass priors so much . One way to asses this (influence of the prior) is to look at prior-posteriors plots (see my graphic here), if the mass of the posterior is at one end of the prior then the prior is badly chosen. Another indication that COULD indicate a strong prior influence is the distance between the posterior mean, prior mean and likelihood MLE - if the posterior mean is closer to the prior mean than towards the likelihood mean it indicates a strong prior influence.

3) Well this goes to the heart of Bayesian stats, the posterior basically tells us how much of the author's original beliefs remain after comparisons with the data. As long as the prior is fairly chosen, this can be very informative.

Prior choice can be very subjective, but Bayesian stats are not truly more prone to misuse than any other statistic.

"Lies, Grand Lies and Statistics"

 

[Dason]

This is why Dason loves point mass priors so much

Don't be hating on point mass priors. I'm sure you've used them before.

 

[jamajor]

There is a methodology called "robust Bayes" that deliberately uses multiple priors to see their influence and to make decisions that are good (in some sense) over the entire set of priors.

 

[hlsmith]

I believe that if you do not have a prior you can use the prevalence rate for the population or comparable sample (empirical prior), which would sort of be objective.

 

[rogojel]

so,
presenting the results correctly would require a very careful discussion of the choice of the prior, right?

The other question that occured to me, the results for a parameter estimation are given in the form of a probability distribution for the parameter value, with the mode being the point estimate.

What would be the right interpretation of this probability? I guess it is something quite different from what one would naively suppose.


regards
rogojel