Bayesian hierarchical modeling
Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the parameters of the posterior distribution using the Bayesian method.
[1] The sub-models combine to form the hierarchical model, and Bayes' theorem 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 prior, providing an updated probability estimate.
Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters.
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 robust, 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.
[3] In decline curve analysis 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).
[4] Hierarchical modeling is used to devise computatation based strategies for multiparameter problems.
[5] Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters.
[6] Individual degrees of belief, expressed in the form of probabilities, come with uncertainty.
As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, “The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality.” These subjective probabilities are more directly involved in the mind rather than the physical probabilities.
[7] Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event.
[8] The assumed occurrence of a real-world event will typically modify preferences between certain options.
This is done by modifying the degrees of belief attached, by an individual, to the events defining the options.
[9] Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability
, given the occurrence of event y, we must begin with a model providing a joint probability distribution for
This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability,
[9] The usual starting point of a statistical analysis is the assumption that the n values
are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true.
are generated exchangeably from a common population, with distribution governed by a hyperparameter
Thus, the posterior distribution is proportional to: As an example, a teacher wants to estimate how well a student did on the SAT.
The teacher uses the current grade point average (GPA) of the student for an estimate.
The SAT score is viewed as a sample coming from a common population distribution indexed by another parameter
, which is the high school grade of the student (freshman, sophomore, junior or senior).
These relationships can be used to calculate the likelihood of a specific SAT score relative to a particular GPA: All information in the problem will be used to solve for the posterior distribution.
Instead of solving only using the prior distribution and the likelihood function, using hyperpriors allows a more nuanced distinction of relationships between given variables.
If the prior is not considered, the relationship reduces to a frequentist nonlinear mixed-effect model.
A central task in the application of the Bayesian nonlinear mixed-effect models is to evaluate posterior density:
Bayesian-specific workflow stratifies this approach to include three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function
