Non-informative priors for the AR(1) model

Question

I have a question about the AR(1) model. Expressed mathematically as:

$$ Z_{t} = \rho Z_{t-1} + \epsilon_{t}, t=1,..,T$$ $$ \epsilon_{t} \sim iid \ N(0,1) $$

My question is about the "transformation group" method of creating non-informative priors, which I believe initially was suggested by Edwin Jaynes (and is discussed in Chapter 12 of his book Probability Theory: The Logic of Science).

One possible suggestion for a transformation group is to consider "reversing" the time series and then rescaling. Thus my transformation group is the following:

$$\rho^{(1)} = \rho^{-1}$$ $$Z_{t}^{(1)} = \rho^{(1)}Z_{T-t+1}$$

Using the original AR distribution, you can show that this transformation basically just "shuffles" the $\epsilon_{t}$ terms, which by definition of the model are exchangeable. So, estimating $\rho$ using $Z_{t}$ is equivalent to estimating $\rho^{(1)}$ using $Z_{t}^{(1)}$ (i.e., the joint distribution of the noise is the same in both cases). Thus the prior for $\rho^{(1)}$ must be the probability transform of the prior for $\rho$. Or, in mathematical terms, the prior must satisfy the following functional equation:

$$f(\rho)=|{\frac{\partial \rho^{-1}}{\partial \rho}}| f(\rho^{-1})=\rho^{-2}f(\rho^{-1})$$

Unfortunately this does not describe a unique function. In fact, any function with the following form will satisfy the above functional equation:

$$ f(\rho) = (constant) \times \begin{bmatrix} \ \rho^{2b} (1-\rho^{2})^{a} & |\rho|<1 \\ \ \rho^{-2(b+a+1)} (\rho^{2}-1)^{a} & |\rho|>1 \end{bmatrix}.$$

For $a > -1$ and $b>-\frac{1}{2}$ this distribution is proper, with the normalizing constant being the reciprocal of $2\beta(b+\frac{1}{2},a+1)$ where $\beta(a,b)$ is the "beta integral". Note that this class includes the "symmetric reference prior" recommended in Berger, J. O. and Yang, R. (1994). Noninformative priors and Bayesian testing for the AR(1) model. Econometric Theory 10 461–482.

Usually specifying a transformation group makes the solution unique, so I am perplexed as to how this group of transformations does not produce a unique solution. Have I done something wrong in the process of creating the transformation group?

UPDATE:Perhaps there is no transformation group which uniquely determines the prior in this case?

Answer 1

This transformation group is discrete and finite: it contains exactly two elements, the identity and inverting $\rho$. It's simply not big enough to determine a prior. In fact, you can choose any measurable function $f$ defined on $[-1,1]$ provided (a) it is integrable and (b) $\rho^{-2}f(1/\rho)d\rho$ is integrable on $[1, \infty]$. The latter restricts $f$ only in a neighborhood of $0$.

BTW, for this model to be practical you need to introduce a nuisance parameter $\sigma$: $\epsilon_t \sim N(0, \sigma^2)$.