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Definition: A sequence $\epsilon_n$ is a posterior contraction rate at the parameter $θ_0$ if $$\Pi_n(θ: d(θ, θ_0) ≥ M_n \epsilon_n| X^{(n)}) → 0$$ in $P^{(n)}_{θ_0}$-probability, for every $M_n → ∞$.

I am studying the subject of posterior consistency. While this subject seems straight forward there are few keywords used which I can't find any definition for. Most of these terms are used in the literature and published papers where the author assumes prior knowledge of this subject by the reader. From farther reading, it seems that these "rates" are related to the contraction rate and seem to originate from the Minimax theory/criterion however I still haven't found any definition for them or their relationship to the contraction rate $\epsilon_n$.

Could you please provide me with any helpful definition for these terms or direct me to a source where they are explained?

Some papers:

https://arxiv.org/pdf/1712.08964.pdf

https://arxiv.org/pdf/1811.06198.pdf

https://arxiv.org/pdf/0910.2042.pdf

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  • $\begingroup$ There terms may be too advanced for this site... I hope you find what you looking for though. $\endgroup$ Sep 25, 2022 at 22:16
  • $\begingroup$ @user3741635, I don't think so. These terms are widely used in the literature, Thank you anyway. $\endgroup$ Sep 25, 2022 at 22:19
  • $\begingroup$ Could you give links to papers that use those terms? $\endgroup$
    – frank
    Sep 29, 2022 at 7:05
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    $\begingroup$ @frank arxiv.org/pdf/1712.08964.pdf and arxiv.org/pdf/1811.06198.pdf and arxiv.org/pdf/0910.2042.pdf and several others.. $\endgroup$ Sep 29, 2022 at 8:06

1 Answer 1

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Can't comment due to lack of rep, but think I may be able to add some useful input if you haven't figured it out already.

I believe what you are missing here is that the speed of convergence of the mass of the posterior on the true value is an important factor and that this is likely what is being referred to as the rate (it would be useful if you gave a reference to the literature you're talking about).

Showing that updates to the posterior are a contraction about the true value of the parameter demonstrates consistency.

Then placing bounds on that contraction rate (via minmax) will allow you to guarantee some rate of convergence on the true value.

So convergence and contraction are sort of related concepts here. The Terminology might be more usefully explained in texts on real analysis, dynamical systems, optimisation etc. than statistics.

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  • $\begingroup$ But what is the difference between each of these rates listed in the title? $\endgroup$ Sep 29, 2022 at 5:52
  • $\begingroup$ Someone has clearly found a theoretically optimal rate of convergence (which I don't fancy digging around for because it's probably incomprehensible to me anyway) and the authors have compared their bounds on the rate to this theoretical optimum and decided they're close: I dont think there's a hard definition of near-optimal in optimisation, but do have a look if you don't believe me! If you search for info in the context of posterior consistency I think you'll find nothing, because they're concepts in prerequisite other etc. as suggested in the answer (optimisation added to answer) $\endgroup$ Sep 29, 2022 at 17:17
  • $\begingroup$ I don't think this answers my question. $\endgroup$ Sep 30, 2022 at 20:44

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