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. 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 the posterior distribution, also known as the updated probability estimate, as additional evidence on the prior distribution is acquired. 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. 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 the posterior distribution, also known as the updated probability estimate, as additional evidence on the prior distribution is acquired. 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. 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 robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling is used when information is available on several different levels of observational units. The hierarchical form of analysis and organization helps in the understanding of multiparameter problems and also plays an important role in developing computational strategies. Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies dependence of the joint probability model for these parameters.Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. 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. 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. 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. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {displaystyle heta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {displaystyle heta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {displaystyle heta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {displaystyle P( heta )} and the sampling distribution P ( y ∣ θ ) {displaystyle P(ymid heta )} respectively: Using the basic property of conditional probability, the posterior distribution will yield: This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to incorporate the updated belief, P ( θ ∣ y ) {displaystyle P( heta mid y)} , in appropriate and solvable ways.