For Bayesian hierarchical models, why is there is no analytical expression of the posterior?
Best insight from top research papers
Bayesian hierarchical models often do not have an analytical expression of the posterior because of the complexity of the data and the model structure. In multi-dimensional problems with non-conjugate priors, evaluating the posterior distribution requires either an analytic or sampling-based approximation, such as Markov chain Monte Carlo (MCMC), Approximate Bayesian computation (ABC), or variational inference . The choice of priors is crucial for posterior propriety and convergence of MCMC samplers . In some cases, the likelihood function is not available explicitly, making it challenging to calculate the posterior . Additionally, in the presence of change points or varying covariates, the selection of the true change point and covariates requires a hierarchical Bayesian methodology and efficient computational strategies like Gibbs sampling . Therefore, the lack of an analytical expression of the posterior in Bayesian hierarchical models is due to the complexity of the models and the need for approximation techniques.
Answers from top 4 papers
More filters
| Papers (4) | Insight |
|---|---|
Open access•Posted Content | There is no mention of why there is no analytical expression of the posterior in the provided paper. |
31 Jan 2023 | The provided paper does not mention anything about Bayesian hierarchical models or the lack of an analytical expression for the posterior. |
| There is no specific answer to this question in the provided paper. | |
20 Jan 2023 | There is no answer to the query in the provided paper. |
Related Questions
How Bayesian Analysis is used and explain briefly.?5 answersBayesian analysis is a statistical approach that focuses on evaluating the probability of outcomes based on data and prior knowledge. It involves constructing a model, incorporating prior information, and estimating the posterior distribution of parameters. The posterior distribution is then used to estimate quantities of interest about the parameters. Bayesian analysis has advantages over traditional statistical significance testing, such as providing a more flexible and intuitive framework for inference. It has been applied in various fields, including operations and supply chain management (OSCM), strategic management research, medical literature, and high-energy polarimetry. The use of Bayesian methods is becoming more popular due to advancements in computing power, which allows for simulation-based approximations of the posterior distribution.
Hierarchical decision models?5 answersHierarchical decision models are a popular means of coping with complex decision problems. They involve a hierarchical structure where subtasks are ordered hierarchically, and locally optimal solutions are determined at each level. These models are used in various fields such as technology evaluation, energy portfolio forecast, distributed data storage and processing, and protein melting-point estimation. The models utilize different techniques such as the analytic hierarchy process (AHP), weighted average methods, and genetic algorithms for optimization. They provide insights into decision-making processes, allow for the consideration of various parameters, and enable the identification of effective solutions at different levels of the hierarchy.
How to compute posterior covariance in least-square method?5 answersThe posterior covariance in the least-square method can be computed using different approaches. One approach involves replacing the least squares inverse with the maximum likelihood inverse, which gives a maximum error for diagonal resolutions of 0.5 and decreases for greater or smaller resolutions. Another approach is to use the second formulation for the a posteriori covariance, which gives a more natural result and reduces to the least squares result for perfect resolution and to the a priori covariance for zero resolution. Additionally, the use of trade-off curves to select the relative weight given to observations and prior information is not a form of tuning, as it does not generally maximize the posterior probability of the model parameters. These methods provide ways to estimate and compute the posterior covariance in the least-square method.
What are the advantages and disadvantages of Bayesian inference for assessing model uncertainty?5 answersBayesian inference is a powerful tool for combining information in complex settings. However, it can produce unreliable conclusions when the model is misspecified. In such cases, a conventional Bayesian analysis may not be meaningful. Nevertheless, there are approaches to performing Bayesian inference with a misspecified model. Three main classes of methods are discussed: restricted likelihood methods, modular inference methods, and the use of a reference model. These methods allow for meaningful analysis and interpretation of a misspecified model. The advantages of Bayesian inference for assessing model uncertainty include the ability to propagate uncertainties and capture parameter variation across experiments. On the other hand, a disadvantage is the need to make assumptions and approximations when computing the posterior distribution.
Is bayesian inference feasible for high order problems?5 answersBayesian inference is feasible for high-order problems. Bayesian methods have been proposed for high-dimensional problems, such as regression with a large number of predictors and graphical models, where the prior takes care of the sparsity structure and can automatically quantify the uncertainty through the posterior distribution. In addition, Bayesian inference has been extended to higher-order ordinary differential equation (ODE) models, where the distance between the nonparametric model and an approximate solution of the ODE is considered, and convergence properties have been established. Bayesian inference has also been applied to large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. A framework based on Kalman methodology has been proposed, which efficiently performs Bayesian inference in such problems. Therefore, Bayesian inference is a viable approach for high-order problems in various fields.
How do we derive the posterior predictive distribution for a normal inverse chi square model?5 answersThe posterior predictive distribution for a normal inverse chi square model can be derived by following certain steps. First, the chi squared distribution can be related to the normal distribution by squaring the distance to the mean. This relationship allows for the derivation of the distribution of the sum of squares of standard normal deviates, which is a gamma function. The noncentral chi-square distribution, which depends on a noncentrality parameter, plays a role in determining the power of a chi-square test. Additionally, approximations can be used to estimate the distribution of weighted combinations of p-values, with positive definite quadratic forms inducing a chi-square distribution. These approximations have been shown to yield probability values close to the nominal level. Overall, the derivation of the posterior predictive distribution for a normal inverse chi square model involves understanding the relationships between the chi squared distribution, the normal distribution, and the noncentral chi-square distribution, as well as utilizing appropriate approximations.