# Possibility priors in Bayesian analysis?

A couple of trains of thought have come together for a model I am designing. Let's start with the first part: Bayesian inference doesn't update strongly enough.

One of the parameters $$\theta$$ is an angle on a unit circle. For a standard Bayesian treatment I would pick a weakly-informative prior over $$[0,2\pi)$$, but I think I should be able to do better than that in this case. The system I am studying has a physical constraint in which the possible angles will always fall within an interval $$\theta \in [a,b]$$ where $$b-a = \pi$$. I could include a parameter representing a translation of $$[0+\tau, \pi + \tau]$$, but the Bayesian update is not aggressive enough. There will be a very small uncertainty in $$a$$ and $$b$$ due to measurement error, and anything outside of this interval is (in the context of this system) obviously a physical impossibility.

On the one hand, I actually don't know ahead of time which interval $$[a,b]$$ it will be. But once I take measurements of $$[a,b]$$ it becomes quite obvious that either of the above choices do not adequately account for the obvious impossibility of the parameter being outside of $$[a,b]$$.

This brings us to the second part: prior possibility. Ben's post introduced me to the idea of possibility measure. In this question I would like to consider a classic possibility measure. Analogous to putting scientifically justified prior probabilities on parameters in a Bayesian context, I am hoping to choose priors on possibilities. Declaring before seeing data that there will be impossible values for $$\theta$$ is what I want to argue to justify the following approach:

I would like to use a truncated normal distribution for $$\theta$$ and priors on the interval $$[a,b]$$ representing a teeny-tiny measurement error ($$\epsilon_c \sim \mathcal{N}(\mu_c, \sigma_c)$$ where $$c\in\{a,b\}$$) in those boundaries. A subtly here is that it really is impossible for $$\theta$$ to be outside of $$[a,b]$$, but $$[a,b]$$ are not known perfectly. The less standard aspect of my thinking here is to directly set $$\mu_a$$ and $$\mu_b$$ to be set to the observed values of $$a$$ and $$b$$. This is kind of an empirical Bayes method, but I am wondering if there is a way to formalize this choice explicitly in terms of a prior classic possibility measure.

• One big picture question that comes to mind before all this: have you adequately explored the space of priors? Lindley's comments about Bayesian inference and choice of prior already reflects "belief" or "possibility" - an intuitive understanding of Bayesian probability. We already understand priors do not need to be formal probability models, such as in the cases of a Jeffrey's "flat" prior - so Ben's formal treatment of the probability space seems irrelevant to choice of prior. Commented Jan 30, 2023 at 16:26

For this kind of situation, I think what you really need to do is to set a Bayesian model that captures the possible outcomes and the measurement error. This could be framed as a two parameter model where the centre of the interval is $$0 \leqslant \theta < 2\pi$$ and the variance of the measurement error is $$\sigma^2 \geqslant 0$$. To facilitate the analysis, define the interval centered at $$0 \leqslant \theta < 2\pi$$ and having length $$\pi$$ (but expressed in the space $$[0, 2 \pi)$$) as $$\mathscr{I}(\theta)$$, which is given by:

$$\mathscr{I}(\theta) = \begin{cases} [0, \theta+\tfrac{\pi}{2}] \ \cup \ [\tfrac{3\pi}{2} +\theta, 2 \pi) & & & \text{if } \theta < \tfrac{\pi}{2}, \\[6pt] [\theta - \tfrac{\pi}{2}, \theta + \tfrac{\pi}{2}) & & & \text{if } \tfrac{\pi}{2} \leqslant \theta \leqslant \tfrac{3 \pi}{2}, \\[6pt] [0, \theta-\tfrac{3\pi}{2}] \ \cup \ [\theta-\tfrac{\pi}{2}, 2 \pi) & & & \text{if } \theta > \tfrac{3 \pi}{2}. \\[6pt] \end{cases}$$

Your model could be framed as a latent value model where you observe (conditionally) independent values $$Y_1,...,Y_n$$ that are based on unobserved values $$X_1,...,X_n$$ but subject to measurement error. If you have uniform values on the interval but then normally distributed measurement error (for some small $$\sigma$$) then your model might look like this:

\begin{align} X_i | \theta, \sigma &\sim \text{U}(\mathscr{I}(\theta) ), \\[6pt] Y_i | X_i, \theta, \sigma &\sim \text{N}(X_i, \sigma^2) \ \text{mod} \ 2 \pi. \\[6pt] \end{align}

You could then assign a prior for the parameter $$(\theta, \sigma)$$ and derive the corresponding posterior distribution. With a reasonable amount of data you ought to be able to make a good inference for both parameters, which will allow you to infer the location of the true half-circle support and the variance of the measurement error. This will not be a simple model, owing to the modular properties of circular representation of coordinates, but it ought not be too complicated to implement.

• This is interesting (+1). I need to think a little more about the definition of $\mathcal{F}$. Commented Feb 13, 2023 at 15:55
• At a glance I thought $U(\mathcal{F}(\theta))$ might be a probability box but I see now that is unnecessary (if not wrong). Commented Feb 13, 2023 at 15:57
• Notation I have used is \mathscr{I}(\theta). This is just representing the half-circle with centre $\theta$, shown on the angular set $[0,2 \pi)$.
– Ben
Commented Feb 13, 2023 at 20:37