Weakly informative prior distributions for scale parameters

Question

I have been using log normal distributions as prior distributions for scale parameters (for normal distributions, t distributions etc.) when I have a rough idea about what the scale should be, but want to err on the side of saying I don't know much about it. I use it because the that use makes intuitive sense to me, but I haven't seen others use it. Are there any hidden dangers to this?

Answer 1

I would recommend using a "Beta distribution of the second kind" ($Beta_2$ for short) for a mildly informative distribution, and to use the conjugate inverse gamma distribution if you have strong prior beliefs. The reason I say this is that the conjugate prior is non-robust in the sense that, if the prior and data conflict, the prior has an unbounded influence on the posterior distribution. Such behaviour is what I would call "dogmatic", and not justified by mild prior information.

The property which determines robustness is the tail-behaviour of the prior and of the likelihood. A very good article outlining the technical details is here. For example, a likelihood can be chosen (say a t-distribution) such that as an observation $y_i \rightarrow \infty$ (i.e. becomes arbitrarily large) it is discarded from the analysis of a location parameter (much in the same way that you would intuitively do with such an observation). The rate of "discarding" depends on how heavy the tails of the distribution are.

Some slides which show an application in the heirarchical modelling context can be found here (shows the mathematical form of the $Beta_2$ distribution), with a paper here.

If you are not in the hierarchical modeling context, then I would suggest comparing the posterior (or whatever results you are creating) but use the Jeffreys prior for a scale parameter, which is given by $p(\sigma)\propto\frac{1}{\sigma}$. This can be created as a limit of the $Beta_2$ density as both its parameters converge to zero. For an approximation you could use small values. But I would try to work out the solution analytically if at all possible (and if not a complete analytical solution, get the analytical solution as far progressed as you possibly can), because you will not only save yourself some computational time, but you are also likely to understand what is happening in your model better.

A further alternative is to specify your prior information in the form of constraints (mean equal to $M$, variance equal to $V$, IQR equal to $IQR$, etc. with the values of $M,V,IQR$ specified by yourself), and then use the maximum entropy distribution (search any work by Edwin Jaynes or Larry Bretthorst for a good explanation of what Maximum Entropy is and what it is not) with respect to the jeffrey's "invariant measure" $m(\sigma)=\frac{1}{\sigma}$.

MaxEnt is the "rolls royce" version, while the $Beta_2$ is more a "sedan" version. The reason for this is that the MaxEnt distribution "assumes the least" subject to the constraints you have put into it (e.g., no constraints means you just get the jeffreys prior), whereas the $Beta_2$ distribution may contain some "hidden" features which may or may not be desirable in your specific case (e.g., if the prior information is more reliable than the data, then $Beta_2$ is bad).

The other nice property of MaxEnt distribution is that if there are no unspecified constraints operating in the data generating mechanism then the MaxEnt distribution is overwhelmingly the most likely distribution that you will see (we're talking odds way over billions and trillions to one). Therefore, if the distribution you see is not the MaxEnt one, then there is likely additional constraints which you have not specified operating on the true process, and the observed values can provide a clue as to what that constraint might be.

Answer 2

The following paper by Daniels compares a variety of shrinkage priors for the variance. These are proper priors but I am not sure how many could be called non-informative if any. But, he also provides a list of noninformative priors (not all proper). Below is the reference.

M. J. Daniels (1999), A prior for the variance in hierarchical models, Canadian J. Stat., vol. 27, no. 3, pp. 567–578.

Priors

1. Flat: $K$ (constant)
2. Location-scale: $\tau^{-2}$
3. Right-invariant Haar: $\tau^{-1}$
4. Jeffreys': $1/(\sigma^2 + \tau^2)$
5. Proper Jeffreys': $\sigma / (2(\sigma^2 + \tau^2)^{3/2})$
6. Uniform shrinkage: $\sigma^2 / (\sigma^2 + \tau^2)$
7. DuMouchel: $\sigma/(2\tau(\sigma+\tau)^2)$

Another more recent paper in a related vein is the following.

A. Gelman (2006), Prior distributions for variance parameters in hierarchical models, Bayesian Analysis, vol. 1, no. 3, pp. 515–533.

Answer 3

For hierarchical model scale parameters, I have mostly ended up using Andrew Gelman's suggestion of using a folded, noncentral t-distribution. This has worked pretty decently for me.