Bayesian hierarchical modeling

Bayesian hierarchical modelling is a <a href="/facts/Statistical_model/jKkT5ftm">statistical model</a> written in multiple levels (hierarchical form) that estimates the <a href="/facts/Parametric_model/FeynAa0f">parameters</a> of the <a href="/facts/Posterior_probability/fojsMD0D">posterior distribution</a> using the <a href="/facts/Bayesian_inference/tEpK3zLx">Bayesian method</a>. The sub-models combine to form the hierarchical model, and <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a> is used to integrate them with the observed data and account for all the uncertainty that is present. The result of this integration is it allows calculation of the posterior distribution of the <a href="/facts/Prior_distribution/JQKAD4o0">prior</a>, providing an updated probability estimate.
<a href="/facts/Frequentist_statistics/UcNgebQc">Frequentist statistics</a> may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as <a href="/facts/Random_variable/TwTBXnLT">random variables</a> and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results aren't technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be <a href="/facts/Robust_statistics/RwIMvtD2">robust</a>, with the posterior distribution less sensitive to the more flexible hierarchical priors.
Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In <a href="/facts/Decline_curve_analysis/5wIdRxTV">decline curve analysis</a> to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computatation based strategies for multiparameter problems.

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Bayesian hierarchical modeling