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Questions tagged [hierarchical-bayesian]

Hierarchical Bayesian models specify priors on parameters and hyperpriors on the parameters of the prior distributions

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two-step gibbs sampling vs block gibbs sampling

While reading Bayesian-related technical articles, I can see algorithms such as two-step Gibbs sampling and block gibbs sampling ...
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What is the difference between hierarchical modeling and setting a (fixed) prior on a parameter?

I was reading through Chapter 11 of Data Analysis using Regression & Multilevel Models, and was confused by a slight variation of a simple hierarchical model posed in the text. Lets say I have a ...
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For a separable covariance in a Gaussian process, is an inverse Wishart prior conjugate?

Suppose we have a GP for the vector $\mathbf{y}\sim\text{GP}(\boldsymbol{0},\Sigma_y)$, where $\Sigma_y=\Sigma_r\otimes\Sigma_f$ is a separable covariance matrix. Assume $\Sigma_f$ is fixed and an ...
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Hierarchical models and conditional independence

Suppose that we have a hierarchical model given by (this is Example 4.4.5 of Berger and Casella(2002)) \begin{align*} X\mid Y&\sim\text{binomial}(Y,p),\\ Y\mid\Lambda&\sim\text{Poisson}(\...
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Conditional independence in BUGs/JAGs?

I am trying to create a hierarchical model in BUGs. I am actually attempting to implement this is Nimble, but I suspect that a JAGs implementation will be informative. To attempt to reduce my problem ...
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Sequential updating vs Marginalized updating

Suppose I need to sample a posterior $\pi(\theta|D)$, whose analytic form is not tractable (not even up to a normalizing constant). However, I somehow manage to obtain an augmented posterior $\pi(\...
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Independent statements in model definition and then DAG

In this paper in Section 3.1, they give a Baysian linear regression model and then a DAG, which I show below. From my understanding a DAG tells us how the joint distribution can be factorised. But in ...
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Bayesian network extracting further conditional independence statements then just from d-separation theorem

Given a Bayesian network $(p,\mathcal{G})$, where $p$ is our joint distribution, and $\mathcal{G}$ is a DAG. Then by the d-separation theorem we can deduce conditional independence statements, in ...
Dylan Dijk's user avatar
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How to decompose the conditional posterior prob? [closed]

I am learning bayesian inference now. A problem I encountered a lot of time is, when I need to calculate or simplify the posterior prob., I don't know how should I begin, according to what I have. For ...
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Using PCA to check if parameters simulated from a hierarchical Bayesian model are close to real parameters

I have a hierarchical Bayesian model that learns a 5-parameter function for each of the N participants. The priors on each of the 5 parameters are parameterized by a scale parameter, so, it also ...
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Proposal parameterization accuracy for Importance Sampling

Suppose I am fitting a Bayesian mixture model that's structured as follows: $$ Y_i | (z_i = k) \sim \mathcal{N}(\mu_k, \sigma_k^2), \quad k = 1, \cdots, K $$ $$ z_i \sim \text{Mult}(1; w_{i1}, \cdots, ...
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E(X1 | X2 > X3) for (X1,X2,X3) multivariate normal

I'd like a closed form solution for $E(X_1 \mid X_2 > X_3)$ where $(X_1, X_2, X_3)$ is multivariate normal with possibly arbitrary mean vector and covariance matrix. The conditional distribution $f(...
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Zero inflated and right skewed dependent variable – is the Tweedie distribution a good solution?

We are conducting a variance decomposition using a hierarchical linear random effects Bayesian model to investigate the variance in a DV that is affected by three nested layers. Because the DV is ...
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Width of Confidence Intervals for Variance Estimates in Contrast to Point Estimates

We are conducting a variance decomposition using a hierarchical linear random effects Bayesian model to investigate the variance in a DV that is affected by three nested layers. We estimate credible (...
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Prior BPN based on Multi Linear Regression Model Output and Monte Carlo Simulations

On page 286 in the Prediction of road accidents: A Bayesian hierarchical approach paper. The passage describes the construction and parameter learning of Bayesian Belief Networks (BPNs), specifically ...
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Using MCMC-derived posterior to design variational approximation function

I am trying to fit a hierarchical model that estimates the covariance of some parameters, using the probabilistic programming language pyro. In simulation experiments, I saw that the MCMC generates ...
David Shor's user avatar
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How to interpret the population parameters of a Bayesian Hierarchical model?

This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model. This is incredibly ...
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Comparing Bayesian hierarchical models with different sample sizes

I have observation data covering a certain period of time. I follow a block-maxima approach where the data are segmented into equal time intervals .My goal is to first develop a Bayesian Hierarchical ...
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Bayesian Hierarchical Clustering prior update

I am working through Heller and Ghahramani's "Bayesian Hierarchical Clustering" paper (https://www2.stat.duke.edu/~kheller/bhc.pdf) and things aren't quite working out the way I expect with ...
dataphile8's user avatar
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Posterior Distribution in a Bayesian Multivariate Normal Model

I am currently working on a Bayesian inference problem and would appreciate some help on computing the posterior distribution of a hyperparameter within a specific multivariate normal model. Below, I ...
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Bayesian hypothesis testing using posterior samples of estimated parameter

I'm modeling recruitment curves using a Hierarchical Bayesian model. There is a key parameter in my recruitment curve, let's call it $P$. I have two groups (A and B) of participants of respective size ...
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Difficulties with estimation and strange fitted values for BVAR (BVAR R package)

I'm using the BVAR package in R to estimate a Bayesian vector autoregression involving the following monthly variables: US Capacity utilization, US Total Employees, US PCE index, and 5,10,20,30 year ...
Diego De Vivero's user avatar
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Determining the number of interactions between the independent variables

I am trying to use GLMMs models to analysis the morbidity status of child (yes or no) with mother’s demographic and environmental factors like Wealth with factors ("Lower quartiles”,"...
Sofonias Derso's user avatar
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The "Multiple Error Terms" notation for hierarchical models

I'm seeking clarification regarding notation for Bayesian hierarchical models, specifically the mixed effects model. Consider the following hierarchical model for the outcome of unit $i \in N$ in ...
socialscientist's user avatar
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How can a Bayesian linear hierarchical random intercept model with normally distributed priors for coefficients represent a non-normally DV?

Suppose you have a hierarchical random intercept model with a dependent variable that is zero inflated. The link function is linear and the priors for the coefficients are normally distributed. In ...
james_westfield's user avatar
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Does a variance decomposition make sense with a non-linear link function?

I am doing a variance decomposition, with a hierarchal random intercept model like the one below (BRMS R Code): ...
james_westfield's user avatar
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Alternatives to spatial and temporal aggregation of time series to discover more learnable patterns

Given taxi demand time series of towns in a country. I would like to do demand forecasting. I noticed that when the town's time series is zero inflated the prediction is poor. However, when these ...
Jose_Peeterson's user avatar
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Using Bayesian statistics in time series forecasting

I would like to forecast demand count time series of taxi fleets at different locations on the map at different points in time. I.e. multivariate demand Time series forecasting. Given hierarchinal ...
Jose_Peeterson's user avatar
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Bayesian hierarchical exchangeability assumptions reasonable with a check treatment?

This is information I believe to be true A practical feature of hierarchical Bayesian models is that partial pooling reduces (eliminates?) the need of adjusting for multiple comparisons when ...
Brendan Alexander's user avatar
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Proper analysis of completely crossed design with subjects and items as random effects (brms)

I have the following study design: stimuli: 240 pictures: 6 pictures of 40 students each (each student fixated one of six points and during each fixation one picture was taken) each stimulus was ...
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Calculate posterior distribution and full conditional of a HMM

Set up a Bayesian analysis of an hidden Markov model and calculate the posterior distribution and the full conditionals, given this assumptions: The state space of the hidden process has size m $Z_t|...
Elbarbons's user avatar
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Doing empirical Bayes with improper prior - marginals that do not exist?

I am considering a Bayesian linear model for which the prior is not proper. The model is as usual $y = X \theta + w$ where $w \sim N(0, \sigma^2)$, and $\theta, \sigma^2$ are unknown. The distribution ...
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Criterion to assign individuals to clusters in bayesian mixed model with distribution of probabilities

I have a dataset with a set of individuals indexed by $i = \{ 1, ..., N \}$, and I make a number of measuremenets under two conditions for each individual to measure the effect $\beta$ of my ...
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Best way to show one Bayesian model is more certain and accurate than another, based on simulated data?

I'm trying to compare performance of two bayesian models $A$ and $B$ on simulated data. It's a recruitment curve fitting problem and I'm interested in how accurate these models are in estimating only ...
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Inference of Beta-Bernoulli Distribution

Assume $x_1, x_2, \cdots, x_n$ follows a $Bern(\pi_0)$, Let $y_{ik}$ follows $Beta(\alpha,\beta)$, $i\in \{1,\cdots, n\}$, and $k\in \{1,\cdots, K\}$. Let $z_k$ follows a Bernoulli Distribution with a ...
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Hierarchical models: Estimating variance and combining two estimators

Assume that $y_i \sim N(50,10)$. I observe a signal with additive Gaussian noise $s_i \sim N(y_i, \sigma_d^2)$ I observe $n$ such signals, each corresponding to a different $y_i$. I want to estimate $\...
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In Bayesian modelling how to interpret hierarchical hyperparameters with regards to "borrowing"?

With regards to hierarchical models I often see these referred to as groups borrowing information from each other e.g. It will be seen that the hierarchical model posterior estimates for one school ...
gowerc's user avatar
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In my mixed effects model, are the confounded MCMC chains between my random intercepts and my global intercept problematic?

I implemented an MCMC algorithm for the following regression model: $$y_i \sim N(\mathbf{x}_i'\boldsymbol{\beta} + \eta(\mathbf{s}_i) + \theta_i,\sigma^2),$$ $$\boldsymbol{\beta}\sim N(\boldsymbol{0},...
Ron Snow's user avatar
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Is it possible to get the statistical significance of the mean of a distribution inferred through a Bayesian Approach?

I am new to Bayesian inference and I am not sure if this problem and question are well-posed. When estimating the coefficients of a linear regression we can evaluate the statistical significance of ...
Andrea Ciufo's user avatar
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Is it okay to merge seperate MCMC chains, using different seed value? [duplicate]

I'm new to Bayesian analysis. I'm trying to estimate species's abundance. As I know, when using MCMC sampling, it is recommended to make more than three chains. However, the function I use can make ...
pineapple159's user avatar
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56 views

Bayesian estimation for a ranking outcome variable

I'm interested in modeling how a ranking depends on a continuous feature. I have many related groups of these rankings, so I want to use partial pooling with the usual Bayesian machinery, but I'm ...
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A hierarchical model with conjugate hyperprior

I have a modeling problem that I am trying to formulate in a Bayesian manner to do inference. Basically, I have a prior where the variance is unknown, and we want to treat it as uncertain (though with ...
smallStackBigFlow's user avatar
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How to solve for an unkown probability distribution within a hierarchical model?

The Problem Given probability distributions $P(\theta)$ and $P(X)$, and given an inverse function $Y=f^{-1}(X,\theta)$ that returns a unique $Y$. How can one estimate the unkown distribution $P(Y)$ in ...
ellabella's user avatar
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How to structure a multi-level model for a five-a-side football problem

I'm working in an unfamiliar Bayesian context here, so apologies if my terminology isn't entirely correct! Imagine I'm trying to predict the performance of players on a of a five-a-side football team. ...
nwrdobrn's user avatar
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Advice for modeling variance in a hierarchical linear model

I have a dataset of longitudinal measurements for different sample individuals, with some covariates such as age, sex, time period, etc. The number of measurements taken for each individual varies. I ...
Mir Henglin's user avatar
2 votes
1 answer
60 views

How can I marginalize $\boldsymbol{\alpha}$ out of my hierarchical model?

Suppose I have the following hierarchical distribution: $$\mathbf{y} \sim \text{Normal}(\mathbf{X}\boldsymbol{\beta} + \mathbf{K}\boldsymbol{\alpha}, \sigma^2\boldsymbol{\Sigma}_y),$$ $$\boldsymbol{\...
Ron Snow's user avatar
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2 votes
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174 views

Bayesian meta-analysis: Why and how to weight individual study's contribution to overall effect?

I'm interested in performing a Bayesian meta-analysis, specifically, using a random-effects hierarchical model (as described here). Briefly, in this model we assume that the $k$th study's reported (...
hyoda's user avatar
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Bayesian Hierarchical Regression Models for Panel Data

I am fairly comfortable with Bayesian hierarchical regression models, but I am new to panel data analysis. As someone from the social sciences, I have found that the majority of resources on panel ...
Zlo's user avatar
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brms model specification with 3 (crossed or nested?) levels

I have a data set that looks like this toy data ...
lilla's user avatar
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Nonconvergence of some parameters in MCMC of Hierarchical Bayesian Model

In short: MCMC is used to construct posterior distributions for parameters of central tendency and all parameters used in the formula for this central tendency. I only care about the parameters of ...
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