I am relatively new to statistics and Bayesian theory but I am trying to understand it by working through a few books. There are some things I am confused about. ( As I believe my analysis and such is not up to the ideal level for some of the material)
Firstly, the author writes, "If some characteristics of the prior distribution (moments, quantiles,etc) are known, assuming that they can be written as prior expectation, $E^{\pi}[g_{k}(\theta)]=w_{k} , k=(1,2,...K)"$
I am wondering if this is just another way of talking about moments? For example for the n'th moment $g_{k}(\theta)=\theta^{n}$
I am having trouble understanding what is meant by reference priors, and especially what is meant by saying "if the reference measure $\pi_{0}$ is the Lebesgue measure on $\mathbb{R}$"
For example, the author has an example where he writes ( example 3.2.4 )
"Consider $\theta$ , a real parameter such that $E^{\pi}[\theta]=\mu$. If the reference measure $\pi_{0}$ is the Lebesgue measure on $\mathbb{R}$, the maximum entropy prior satisfies $\pi^{*}(\theta)$ $\alpha$ $exp(\lambda \theta)$ and cannot be normalised into a probability distribution, On the contrary if in addition it is known that $var(\theta)=\sigma^{2}$, the corresponding maximum entropy prior is $\pi^{*}(\theta)$ $\alpha$ $exp(\theta \lambda_{1}+\theta^{2}\lambda_{2})$, ie the normal distribution.."
So few questions. One is what I asked above about Lebesgue measure. Second about the characteristic thing. How do we know to include a $\theta^{2}$ in the final form? I get that we know it from the definition of variance of $E[\theta^{2}]-(E[\theta])^{2}$ and we have the second term. But how do we know that is the g term we use? Or is that just that we know it so we can use it? Because I am not sure if we know that the g is moment, quantile, etc.
Finally, on the same page the author writes that when "characteristics" are related with quantiles, that we will not be able to derive a distribution. Why? Just because quantiles are not enough? For example he has "when characteristics are related with .., $I_{[b_{k},\infty)}(\theta)$. I.e just an indicator of if $\theta$ lies in that interval or does not.But is that the reason that the maximum entropy wont exist? Does this mean we would know the associated probability of being in that interval and we would know $b_{k}$. I get that the expected value of it would be the probability we are in that interval. But is that enough to conclude we can not define a distribution in such a way? What if one of the intervals told us that the probability of being in such was 1. I guess because we are on $\mathbb{R}$ this couldn't happen for a distinct value?
Someone had mentioned "any minimal, summarizing sequence of statistics is recursively computable" as a reason , but I don't understand that.
Again, I apologise if there are obvious things I am missing. I am trying to catch up on my analysis and such as well, so hopefully that will make it easier for me in the future.
Thank you all for your time and patience.