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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.

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    $\begingroup$ 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. $\endgroup$
    – AdamO
    Commented Jan 30, 2023 at 16:26

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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.

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

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