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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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Bayesian hierarchical coin flip model

My question is: what is the marginal probability $P(x_1, x_2, \dots, x_n | y_1, y_2, \dots, y_n, \alpha, \beta)$ or $P(X|Y, \alpha, \beta)$? in the following model: $\phi \sim \text{Beta}(\alpha, \...
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Hierachical Bayesian Linear Regression using PyMC3 is super slow [migrated]

I am trying to write some code for implementing HBM in the case of logistic regression using the adults dataset from the UCI repository. I have already written the code, but sampling is super slow, ...
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DIC calculation returning zero penalty for linear regression model

I am trying to extract the DIC from a model in JAGS but it always returns zero penalty. Why is this please? For example, with the following linear model, $y_{i} \sim \alpha + \beta x_{i} + \epsilon_{...
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1answer
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Determine hyper-prior for gaussian distribution from existing data

Not sure how to determine hyper-prior for prior distributions, specifically using historical data. First what I am doing: I want to estimate parameters for a normal likelihood function using Bayesian ...
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30 views

Normal-Gamma: Metropolis-Hastings on log-scale, but no Jacobian?

I am reading the paper by Griffin and Brown (2010) where at one step in their MCMC procedure they need to sample from the following conditional posterior: $$ p(\lambda|\gamma, \Psi)\propto \pi(\...
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18 views

joint model and multivariate model

I have a crash data set which provides information about the frequency of crash by severity level on each intersection. I want to develop a joint model for frequency by severity. I am new in this ...
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DIC to compare models with different numbers of parameter?

I am interested in comparing hierarchical Bayesian models based on the same dataset but differing in their spatial and temporal resolution. In short, I am looking at animal population changes over ...
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41 views

Gaussian process regression model for comparing two groups

I have a data set consisting of functional observations, where $Y_{mi}$ is the response of the $m^{th}$ functional observation from the $i^{th}$ group, $m=1,...,M$ and $i=1,2,$. The observations are ...
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MCMC Metropolis-Hastings sampler - Estimation of multiple parameters

First time that I ask a question on this platform! Here I go... I have a dataset with two random variables X1 and X2 and an output Y which comes from a discrete Weibull distribution. I've been trying ...
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1answer
6 views

In hierarchical model, how to choose groups that meet exchangeability, automatically?

In hierarchical model, we assume exchangeability. For example, y[i] ~ Norm(b0 + b1[groups[i]], sigma) and b1 ~ Norm(mu_b1, sd_b1) above, all groups are assumed exchangeable. But, it might be better ...
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Using marginal likelihood for weighting in bayesian hierarchical model?

I have data from a series of experiments. I have a simple model for generating the data these experiments which allows me to estimate a parameter. Some experiments do not conform to my model and ...
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1answer
32 views

What is the difference between random-effects models, multilevel models and hierarchical models?

In the Bayesian paradigm, I have found examples of models that could be called any of the following: random-effects models multilevel models hierarchical models. Each of these categories even has ...
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1answer
31 views

deriving likelihood function for hierarchical bayesian model

I'm struggling with hierarchical bayesian modeling. I need to derive a full likelihood function for the given hierarchical structure of the model. $a_{it}|\lambda_i\sim TN(\lambda_i,\beta)$ $x_{it}|\...
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what is the difference between a multilayered autoencoder and a hierarchical latent variable model?

I have been trying to understand how hierarchical latent variable models are different from multilayered autoencoders and in specific the argument below Autoencoder networks resemble in many ways ...
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24 views

Bayesian chi-squared tests

I have a dataset with two groups of participants. Each participant performed a repeated measures task on which three types of errors could be made. I want to measure the difference in distributions of ...
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25 views

Finding the mode of the posterior distribution

I have the following hierachical bayesian model - $\mathbf{x}|\mathbf{c},\sigma^2 \sim \mathcal{N}(\mathbf{x}|\mathbf{c},\sigma^2)$ $\mathbf{c}|\mathbf{c}_1,\sigma^2_2 \sim \mathcal{N}(\mathbf{c}|\...
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1answer
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How to choose the best method to generate random values [closed]

In my specific case, I have a pdf that has no closed form, and I want to generate random values ​​of this distribution. It depends on a summation that goes to infinity (coming from a poisson process) ...
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Complete a Bayesian Network by specifying the probability distributions

I have a hierarchical Bayesian Network like this: Here: $R≡$ log level of poisonous gas (radon) in a house $B≡$ type of house (With a basement or without) $C≡$ a county in Minnesota where the ...
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2answers
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Formal Bayesian justification of conditional modelling

I'm having some trouble following the logic of this passage from Chapter 14 in Bayesian Data Analysis, A. Gelman: The numerical 'data' in a regression problem includes both $X$ and $y$. Thus, a ...
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1answer
29 views

How do Bayesian hierarchical models adaptively learn the prior?

It seems the main difference between a hierarchical and a non hierarchical model is that the hierarchical model learns the prior. That is it adaptively comes up with a regularizing prior to be applied ...
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31 views

Bayesian estimation in 2x2 mixed design study

I'm trying to correctly set up Bayesian parameter estimation for a mixed-design study with one 2-level between-groups independent variable and one 2-level within-subjects independent variable. The ...
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11 views

Can Deviance Information Criterion be used for model comparison when the response variable has Poisson distribution?

I just constructed a Bayesian Hierarchical Model for my response variable Y that follows Poisson distribution with the parameter $\lambda$. In my model, I have modelled $log(\lambda)$ as a linear ...
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How to evaluate double Integral with importance sampling

I am trying to recreate the Bayesian Hierarchical Clustering algorithm using Python. The example in section two requires evaluating the following double integral (univariate case): \begin{align} p(...
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1answer
30 views

Am I doing hierarchical bayesian regression?

I'm doing a Bayesian logistic regression to predict the probability of my dependent variable Y with two predictors, one continuous (X) and the other categorical (C). I deal with C by building 3 models ...
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Marginal Distribution of Hierarchal Model Normal distribution with unknown mean and precision

I am trying to use a Hierarchical model where there I have a normal distribution with random mean and precision: $$ y \sim N(\mu, \tau)\\ \mu \sim N(M, T)\\ \tau \sim Gamma(\alpha, \beta) $$ I'm ...
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In a Hierarchical Bayesian Model, how can we sample and see how a prior distribution looks like if it contains hyperparameters with hyperpriors?

I have a Bayesian Hierarchical Model that looks like: \begin{equation} Y_i \sim N(\mu, \sigma^2) \\ \mu \sim N(\mu_0, \sigma_0^2) \\ \sigma^2 \sim Gamma(1,1) \\ \mu_0 \sim N(0,1) \\ \sigma_0^2 \sim ...
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1answer
36 views

Out-of-sample predictions for mixed model are the same as naive model (ignoring the random effects)

I have a dataset that consists of subjects coming into the clinic (for treatment of another disease) and they are screened for Tuberclosis (as they are a high risk population). Every time they are ...
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In a Multi-level Bayesian Hierarchical Model, would higher level parameters be affected by how they are jointly modeled in lower levels?

Suppose we have a Multi-level Hierarchical Model where: $$ \begin{equation} Y_{0i} \sim Bin(\theta_{0i}, n_{0i}) \\ Y_{1i} \sim Bin(\theta_{1i}, n_{1i}) \\ \theta_{0i} \sim Unif(0,1) \\ log\left(\...
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1answer
54 views

Is the methodology for my undergrad dissertation sufficient - should I use a hierarchical negative binomial model instead, despite beginner ability?

As said in the title, I know almost nothing about statistics. My hypothesis for my dissertation is that UK Members of Parliament with a larger margin of victory will do less work than those with a ...
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Bayesian concepts : multivariate mixed models

I am working with a multilevel multivariate mixed mode with 4 outcomes. I am having difficulties extracting the variances, coefficients of variations. Could anyone advise? I am new to bayesian ...
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1answer
73 views

computing the distribution over the latent function values with the form of a GP predictive

If we have a latent state space $\mathbf{X}$ and the observations $\mathbf{Y}$ and the transition function between two states $\mathbf{x}_{t-1}$ and $\mathbf{x}_{t}$ is given by $\mathbf{f}$ which is ...
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A node in Bayesian network model with a hybrid parents (i.e., contentious and discrete Parents)

I have a Bayesian network model that has a node with hybrid parents. Assume that the discrete nodes are beta distributed and the continuous is CLG (continuous linear Gaussian) distributed. If I ...
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1answer
48 views

Bayesian Hierarchical Clustering: How to calculate probability of Data under $H_1$?

I am working on implementing the Bayesian hierarchical clustering algorithm found here from scratch as a way to practice and learn the algorithm. However, I have hit a snag in calculating the quantity ...
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Which gradient to compute in a hierarchical model for M-H MCMC?

We have the following model: $$y_t=Mx_t+\epsilon_t$$ with $M$ being a matrix such that $M\sim F_{\lambda}$(assume it's a conjugate prior). The $\lambda$ does not appear in $M$, only in its ...
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Hierarchical Choice based Conjoint analysis on transaction data

I want to get the utility of each level of each attribute for every transaction(at a customer level). How can I do that in python/R(could find a good way to do this in python so I tried using ...
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Multilevel modeling, constraint for positive values

I'm currently trying to fit a shifted inverse gaussian to reaction time data (always postive). My paramterization of the model includes 3 parameters, alpha, gamma and tau, which must always be ...
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36 views

Finding mode of posterior using Newton method in R

I am attempting to approximate the posterior $\tilde{\pi_{G}}(z|\theta,Y)$ which is the Gaussian approximation to the full conditional of $z$, and in order to do this I need to find the mode $z^{*} \...
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Posterior predictive check for capture-recapture data using jagsUI wrapper

The R package jagsUI is a wrapper for JAGS that has some awesome functions, including a posterior predictive check. As discussed here, you simulate the new data for the parameters based on the ...
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78 views

Posterior predictive distributions and predictive intervals

I'm confused about the role of posterior predictive distributions in Bayesian inference and predictive inference. As I understand it, the frequentist approach would typically involve fitting the MLE,...
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41 views

Hierarchical bayesian model: should I account for lack independence?

I am working with vegetation surveys that were conducted in several river networks. See the attached image that shows one of the those river basins/networks. I am interested in analyzing how the ...
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2answers
380 views

Differences between prior distribution and prior predictive distribution?

While studying Bayesian statistics, somehow I am facing a problem to understand the differences between prior distribution and prior predictive distribution. Prior distribution is sort of fine to ...
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0answers
38 views

Metropolis-Hastings Algorithm for Bayesian Hierarchical model

I have developed a Metropolis-Hastings Algorithm for a double sigmoidal model, but now the aim is to create a Bayesian Hierarchical model that depends on incoming temperature data. For example, the ...
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1answer
57 views

Generalization performance in Bayesian errors-in-covariates model

I'm working on a model with this basic structure: The square nodes are data, and the round nodes are parameters and/or latent variables. The left plate represents the "training observations" we ...
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0answers
36 views

Displaying three-level multilevel model in vector notation

I estimated a three-level multilevel model in stan but I have some trouble writing it correctly in terms of vectors and matrices. Now, I have the following: $\begin{equation} \begin{gathered} y_{ijt} ...
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35 views

Interpreting Hierarchical Bayesian Model, how to do paired t-test?

I have two hierarchical models. Both models include 80 participants and output both a group-level posterior distribution and 80 individual-level posterior distributions for my variable of interest. ...
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90 views

A hierarchical Bayesian model in pymc3

Suppose we have the following model: $X$ unobserved $Y$ such that $Y|X \sim \mathcal{N}(X,\sigma^2)$, observed $Z$ such that $Z|X \sim \mathcal{B}(1,X)$, observed and suppose, given observed data $...
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Hierarchical time series using DLM

I am developing a forecasting solution using R's dlm package and it is proving to be very useful for most of our requirements. However, I am also keen on sharing information among different time ...
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16 views

Spatio-tempral Bayesian Poisson model convergence investigation

I am fitting a spatio-temporal Bayesian Poisson model with 22 explanatory variables, an offset variable, 2200 observations and non-informative priors. I am using the package ...
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1answer
71 views

bayesian predictions in multilevel model for panel data

I want to make predictions for a bayesian multilevel which basically looks like: $y_{it} = \alpha_{i} + x_{it}\beta$. I was told that I could make predictions by using (in the case that $y_{it}$ is ...
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17 views

Bayesian hierarchical temporal models

I have been asked recently to transform an already existing Bayesian hierarchical model into an non-stationary model by making the input and the latent variable time dependent(or non stationary). let ...