Looking for a predictive prior for what I thought to be Bayesian Analysis I need a prior with 2 parameters location (μ) and scale (σ)
(a transformation of mean and standard deviation).
My distribution is over discrete values, equally spaced with distance $c$ between consecutive values. Only values between a lower limit $a$ and and upper limit $b$ are considered.
For $c\to0$, the distribution becomes continous and when $a\to-\infty$ $\lor$ $b\to\infty$ a truncation point (one or both) is lifted.
A distribution that kind of meet my criteria is a discrete and truncated normal.
$P(X = x\vert \mu,\sigma,a,b,c) = \frac{e^{-\frac{(x-\mu)^2}{2\sigma^2}}}{\sum_{y \in \Omega} e^{-\frac{(y-\mu)^2}{2\sigma^2}}}$
Actually, I would prefer fatter tails. I want an uninformative/flat prior (so I can conform to the principle of indifference).
Can I generalise the (discrete) uniform distribution and change the parameters (giving a close form but allowing for unevaluated sums and integrals or other special functions)?
My problem with $\mathrm{Unif}  \lbrace a,a+c,a+2c,\dots, b-c,b \rbrace$
is that it doesn't contain μ and σ so I can't calculate my posteriors for the scale and location parameters (I would prefer to calculate directly the posteriors for mean and standard deviation i.e the transformations of my parameters) and calculating a predictive posterior (for the individual scores) with a prior that has different parameters is undesirable.
The values of $a$, $b$ and $c$ will be known every time and are not parameters of interest (they are constants just with different values for each experiment but known beforehand they would be selected/chosen/set by the very researchers I just want the general answer before choosing my constants).
Such a prior would be chosen for example for the individual scores of students in a class (the testmakers decide the minimum grade $a$, the maximum grade $b$ and the granularity $c$) or some other test where the one administering and overseeing sets minimum, maximum and granularity. The individual scores are also believed to be positively correlated (but it is unknown how exactly). For example self-selection, ease of test, good teachers, good time period (e.g covid/no-covid).
I would want a uniform with changed parameters but I am also open to suggestions for alternative priors or methods for conducting my Bayesian analysis.
The best I have got to trying to solve the system of equations for the mean and the variance of the (discrete) uniform is as a PMF
$\dfrac{1}{c\sqrt{12{\sigma}^2+1}-c}$ but I can't get rid of the grandularity ot get the mean I so much want there.
Maybe the problem is that I am trying with an inherently non-parametric distribution.
Edit:
I need to clarify (confess) that I only took 1 cource of Bayesian Statistics (and even that I only passed by the skin of my teeth probably without even having a more than rudimentary grasp of the content out of pity since I was from the School of Economics and I was not studying statistics per se).
I have the belief (maybe false) that it doesn't make sense to choose a uniform prior when you actually know the values of $a,a+c,a+2c,\dots, b-c,b$ because you are the one designing the test. if you know (for certain) a, b and c (you choose them) and "believe" the prior to be $\mathrm{Unif}  \lbrace a,a+c,a+2c,\dots, b-c,b \rbrace$ there is no uncertanty left as you know everything.
So when I am not entirely sure that all the items are equiprobable and I don't know the mean nor the variance can I use the uniform as a prior (I only like the uniform because it is what conforms in my view with the principle of indiference)? What exactly is an uninformative prior? How is an uninformative prior different from a uniform prior?
What is the closest thing to an uninformative prior (predictive) supported over the known terms of an arithmetic sequence and with location and scale parametres? Is $\mathrm{Unif} \lbrace 1,2,3,\dots,24,25 \rbrace$ (α=1, b=25, c=1 are known by designing the test) a proper prior, where are μ and σ? Is there any way on making the discrete and truncated normal more uniform maybe try to change the parametres of a student t distribution? I would like to minimise the variance of the y (probability or density) values throught the support (the uniform has equal y for everything so 0 variance there) while keeping the support ($a,a+c,a+2c,\dots, b-c,b$) and the parametres (μ and σ) the same.
 A: I suspect you have slightly misunderstood how prior distributions work. I'll work through a simplified example.
Let us assume you want to have a uniform (prior) distribution on a parameter $\mu$; the prior distribution looks like:
$$p(\mu; a, b) = {1 \over b-a}$$
This doesn't "contain" $\mu$ because it returns the same value for every input value of $\mu$; that's what "uniform" means.  Now, let's show how to calculate a posterior distribution.  The likelihood function is Gaussian, and we assume $\sigma$ known:
$$\mathcal{L}(\mu; x, \sigma) = \exp\left\{-{(x-\mu)^2\over 2\sigma^2}\right\}$$
leading to a posterior that looks like:
$$p(\mu; x, \sigma, a, b) \propto {\exp\left\{-{(x-\mu)^2\over 2\sigma^2}\right\} \over b-a}1(a \leq \mu \leq b)$$
where $1(a)$ is the indicator function that takes on the value $1$ when $a$ is true, $0$ otherwise.  (You can think of it as a mechanism for reminding ourselves to honor the constraints on $\mu$ imposed by the prior.)
This of course simplifies to:
$$p(\mu; x, \sigma, a, b) \propto \exp\left\{-{(x-\mu)^2\over 2\sigma^2}\right\}1(a \leq \mu \leq b)$$
a truncated Gaussian distribution.
It should be clear how to extend this to the specifics of your problem; you'll just have a discrete uniform instead of the continuous one above, and you'll actually need two of them - one for $\mu$ and one for $\sigma$, I suspect, with different upper and lower limits and different grid step sizes.
