- #1
ChrisVer
Gold Member
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Hi, a very basic question: what is a good intuitive way to understand the importance of a reference prior? In the context of a signal search.
Bellow, I also try to give the way I understand the approach in a Bayesian analysis (roughly):
1. You have your likelihood model [itex]L = p(x_{obs} | \lambda , b+\mu s)[/itex], with the expected events ([itex]b,s[/itex] background,signal) and several priors [itex]\pi[/itex] (for your parameter of interest [itex]\mu[/itex] and other nuisance parameters [itex]\lambda[/itex] such as the uncertainties).
2. The posterior pdf is what you need in order to study the parameter of interest [itex]\mu[/itex]. By Bayes' Theorem, this is:
[itex] p ( \mu | x_{obs} ) = \frac{ L \pi(\lambda) \pi(\mu,\lambda)}{p(x_{obs})}[/itex]
The denominator several times can be taken out as it's only making sure that the normalization is correct for a pdf. Also I consider [itex]\pi(\mu,\lambda) = \pi(\mu) \pi(\lambda)[/itex], which tells that the two parameters are independent.
3. You run several experiments, and from the outcome of each experiment you "update" your knowledge on the parameter of interest. Aka in the end you build up the posterior probability (once you integrate out the extra dimensions from the NPs):
[itex] p( \mu | x_{obs} ) \propto \int d \lambda ~ p(\mu | x_{obs} , \lambda ) \pi(\lambda) [/itex]
So far I think I understand everything, maybe with some misconceptions which could potentially be pointed out.. When though one starts speaking about reference priors I am somewhat lost. Based on a few searches, I think the main target of the reference prior is to minimize the differences between the posterior and the prior. However I don't quiet understand how is that important as:
"I can give in any prior which I like (with reasonable limitations), and it's up to the observation/experiment to tell me how it evolves with the extra information. By reasonable I mean that for example it can't be 0 at ranges where the posterior is non-zero (as Bayes' theorem would result in 0 for the posterior)."
How could different distributions of a prior end up in updating it to different distributions for the posterior?
Bellow, I also try to give the way I understand the approach in a Bayesian analysis (roughly):
1. You have your likelihood model [itex]L = p(x_{obs} | \lambda , b+\mu s)[/itex], with the expected events ([itex]b,s[/itex] background,signal) and several priors [itex]\pi[/itex] (for your parameter of interest [itex]\mu[/itex] and other nuisance parameters [itex]\lambda[/itex] such as the uncertainties).
2. The posterior pdf is what you need in order to study the parameter of interest [itex]\mu[/itex]. By Bayes' Theorem, this is:
[itex] p ( \mu | x_{obs} ) = \frac{ L \pi(\lambda) \pi(\mu,\lambda)}{p(x_{obs})}[/itex]
The denominator several times can be taken out as it's only making sure that the normalization is correct for a pdf. Also I consider [itex]\pi(\mu,\lambda) = \pi(\mu) \pi(\lambda)[/itex], which tells that the two parameters are independent.
3. You run several experiments, and from the outcome of each experiment you "update" your knowledge on the parameter of interest. Aka in the end you build up the posterior probability (once you integrate out the extra dimensions from the NPs):
[itex] p( \mu | x_{obs} ) \propto \int d \lambda ~ p(\mu | x_{obs} , \lambda ) \pi(\lambda) [/itex]
So far I think I understand everything, maybe with some misconceptions which could potentially be pointed out.. When though one starts speaking about reference priors I am somewhat lost. Based on a few searches, I think the main target of the reference prior is to minimize the differences between the posterior and the prior. However I don't quiet understand how is that important as:
"I can give in any prior which I like (with reasonable limitations), and it's up to the observation/experiment to tell me how it evolves with the extra information. By reasonable I mean that for example it can't be 0 at ranges where the posterior is non-zero (as Bayes' theorem would result in 0 for the posterior)."
How could different distributions of a prior end up in updating it to different distributions for the posterior?