Talk:Marginal likelihood

I wanted to learn from this page. Very puzzling: In the Concepts section there are equations including $$\alpha$$ but nowhere it is defined, discussed or explained. --Janlo (talk) 10:23, 7 November 2022 (UTC)

Gibbs sampling is a Markov chain Monte Carlo algorithm. As a consequence, the last sentence may need a reformulation.

x isn't defined in the first equation, nor is the use of the semicolon - I can't find another Wikipedia page on probability that uses the semicolon in this context. See for a more detailed explanation.

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 * Reference : Song Li & Mervyn J. Silvapulle & Param Silvapulle & Xibin Zhang, 2012. "Bayesian Approaches to Non-parametric Estimation of Densities on the Unit Interval," Monash Econometrics and Business Statistics Working Papers 3/12, Monash University, Department of Econometrics and Business Statistics.

ExpertIdeasBot (talk) 16:25, 19 May 2016 (UTC)

Frequentist marginal likelihood
The following passage is problematic, because it suggests that a distribution can be given for a parameter in frequentist statistics, which is not the case: In classical (frequentist) statistics, the concept of marginal likelihood occurs instead in the context of a joint parameter $$\theta = (\psi,\lambda)$$, where $$\psi$$ is the actual parameter of interest, and $$\lambda$$ is a non-interesting nuisance parameter. If there exists a probability distribution for $$\lambda$$, it is often desirable to consider the likelihood function only in terms of $$\psi$$, by marginalizing out $$\lambda$$: $$\mathcal{L}(\psi;\mathbf{X}) = p(\mathbf{X}\mid\psi) = \int_\lambda p(\mathbf{X}\mid\lambda,\psi) \, p(\lambda\mid\psi) \ \operatorname{d}\!\lambda $$ AVM2019 (talk) 17:56, 3 February 2023 (UTC)

On a related note, "marginal likelihood" means a different thing in frequentist statistics, which has to do with likelihood factorisation (google "conditional likelihood"), not integration with respect to a prior. I see in the edit history that early versions of the article made it clear that it is about the Bayesian concept. Later, someone with more confidence than knowledge came and said, I quote, "this is the same concept in frequentist and bayesian statistics", link to the edit. This is why we are having a mess right now. AVM2019 (talk) 18:25, 3 February 2023 (UTC)