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I am currently working on a homework assignment and have the following question:

$\theta_1$ and $\theta_2$ are parameters of interest and $y_1$ and $y_2$ are the likelihood functions which are $\text{Bin}$~$(n_1, \theta_1)$ and $\text{Bin}$~$(n_2, \theta_2)$ respectively.

It is known that:

  1. Both $y_1$ and $y_2$ are independent.
  2. $P(\theta_1, \theta_2) \propto 1$ and is bounded.

I am currently struggling on interpreting what it means that $P(\theta_1, \theta_2) \propto 1$, especially in identifying the marginal priors for this.

My perspective is that since $P(\theta_1, \theta_2) \propto 1$ is a non-informative prior, it suggests that they do not value-add in terms of beliefs to the overall posterior distribution (i.e. the likelihood function is responsible for the update in beliefs). Since the joint prior is non-informative in this sense, the marginal prior must also be non-informative.

Shawn Hemelstrand
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ak_mng
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    $\propto 1$ is a uniform distribution. If you have a uniform distribution for the joint distribution, then the marginal distributions will be uniform as well. – Sextus Empiricus Feb 17 '24 at 09:22
  • I am not sure what the question is exactly about. "... a non-informative prior, it suggests that they do not value-add in terms of beliefs to the overall posterior distribution ... " yes that is the idea behind non-informative. ('weakly informative' might be a better term than 'non informative') – Sextus Empiricus Feb 17 '24 at 09:24
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    There is no non-informative prior: https://stats.stackexchange.com/a/27835/7224 – Xi'an Feb 17 '24 at 09:40

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