How to specify an informed prior in JAGS/BUGS

An emailer asks about how to specify an informed prior in JAGS/BUGS. Here are some condensed excerpts of the question and my reply. Of course, while the discussion here refers to JAGS/BUGS, the same considerations apply to Stan.

Dear Prof. Kruschke,
 
I'm [on the faculty at ... university]. My name is [...]. I'm learning Bayesian method by reading your book, [1st edition]. I'm writing to you for asking some questions on determination of the prior for parameters.

I try to use the method in your book to estimate parameters in [an application that involves linear regression on logarithmically scaled data].

lnW=lna+b*lnL is similar to y=beta0+beta1*x. Denote lnW as y, lnL as x, lna as beta0 and b as beta1. I try to use the linear model to estimate parameters. 

According to P.347 of your book, it's difficult for sampling algorithms to explore when values of the slope and intercept parameters tend to be tightly correlated. In fact, parameter a and b have negative correlation. So, I try to standardize the data x and y according to the method in your book. I get the model zy=zbeta0+zbeta1*zx.  beta0=zbeta0*SDy+My-zbeta1*

SDy*Mx/SDx, beta1=zbeta1*SDy/SDx.

I have some questions on determination of the prior for zbeta0 and zbeta1. In fact, I know the prior of parameter a and b. For example, a~dunif(0.0001, 0.273), b~dunif(1.96,3.94). Can I use the next code to determine the prior for zeta0 and zeta1?
  zbeta1<-b*SDx/SDy
  zbeta0<-(zbeta1*SDy*Mx/SDx-My+lna)/SDy
  b~dunif(1.96, 3.94)
  a~dunif(0.0001,0.273)

Today I read chapter 23 of the book. I learned the method about reparameterization of probability distribution on P.516 in sector 23.4.  I am confused if I need to use the method on P.516 to transform the prior of parameter a and b to the prior of parameter zbeta0 and zbeta1.

Would you like to give me some instruction about these questions?
Thanks for your consideration.

With best wishes for Christmas and New Year.

 

My reply: 

Dear Dr. [...]:

 

Thank you for your message and for your interest in the book.

 

Please accept my apologies for the delay in my reply. [See https://sites.google.com/site/doingbayesiandataanalysis/contact]

 

Your questions do not have quick answers, but at least here are some ideas.

 

There are different aspects involved in your application. First, there is the issue of how to specify a prior based on previous data. Second, there is the issue of reparameterizing the prior when the data scales are transformed. I will focus on the first issue.

 

Suppose we are doing simple linear regression, with parameters b0 (intercept), b1 (slope), and sigma (standard deviation of normally distributed noise). Suppose that previous research provided data set D1, and that we do Bayesian estimation of parameters starting with a vague prior, yielding posterior distribution PD1 = p(b0,b1,sigma|D1). This distribution, PD1, should be the prior for subsequent data.
 

We would like to express PD1 in JAGS/BUGS. Unfortunately, in general, this cannot be done exactly. There may be some models for which the posterior takes a simple mathematical form that can be exactly expressed in JAGS/BUGS, but in general the posterior is a complicated distribution, which is why we use MCMC methods in the first place! Therefore we must express PD1 only approximately.

 

One very rough approach would be to consider only the marginal distributions of the individual parameters in PD1, and approximate each marginal distribution with some reasonable built in distribution in JAGS/BUGS. For example, consider b0. The MCMC sample of PD1 has provided several thousand representative credible values of b0, and a histogram of the marginal distribution of b0 suggests that a normal distribution might be a reasonable approximation (of the marginal distribution of b0). So we find the mean, mub0, and standard deviation, sb0, of b0 in the MCMC sample of PD1, and specify the prior in JAGS as

b0 ~ dnorm( mub0 , 1/sb0^2 )

We do the analogous process for the marginals of the other parameters. For sigma, we would not use a normal, but instead use perhaps a gamma because its support is nonnegative and it is skewed like the marginal posterior on sigma.

 

Notice that the expression of the prior should not, typically, be uniform on a limited range. All that would do is cut off the range of the subsequent posterior. If the data "want" parameter values near the limit of the range, then the posterior would pile up against the limit, like a heap of mulch against a fence. More fundamentally, a uniform distribution does not reasonably approximate the posterior from any conceivable previous data (assuming the "protoprior" for the initial data is vague). 

So far, we have addressed only the marginal distributions, and assumed there is no correlational structure among the parameters in PD1. Although some models and data might yield posterior distributions with little correlational structure, this is not usually true. For example, in linear regression (when using non-standardized data) there is usually a strong correlation between b0 and b1. As another example, in "robust" models that use a t distribution to accommodate outliers, there can be a strong correlation between the scale parameter (sigma) and the normality (df) parameter. Therefore, a more accurate approximation should also express the correlations among parameters.

 

One way to express correlations among parameters is to approximate their joint distribution by a multivariate normal. For example, in linear regression, we might consider the joint distribution of b0 and b1 in the MCMC sample of PD1, and approximate it with a bivariate normal. The bivariate normal has a mean vector with 2 means, and a covariance matrix with 3 distinct components (i.e., the variance of b0, the variance of b1, and the covariance of b0 and b1). With those 5 values estimated from the MCMC sample of PD1, we would create a prior in JAGS by putting b0 and b1 in a vector, and specifying a prior on the vector as a multivariate normal with the means and covariance matrix obtained from the prior MCMC sample.

 

As more parameters are involved, it becomes difficult to write down all the considerations of constructing a good mathematical approximation to a high-dimensional MCMC distribution. You have to take a good look at the specific prior you are trying to emulate, and make a good argument that you have captured its essentials.

 

There is a way to bypass all of the worries about approximating a previous MCMC sample. Instead of using the posterior from the previous data, just combine the previous data with your new data, and do the analysis on the combined data. This will give you an exact answer, but at the cost of running on a large data set (and requiring all the previous data).

 

Hope this helps. Thanks again for your interest.

John