Questions tagged [hierarchical-bayesian]

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

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99 views

The joint distribution for a hierarchical Bayes model

Consider the following hierarchical Bayes model. We observe random variables $X_1, X_2, ..., X_n$ conditionally independent and having Poisson distributions $X_i \sim Poiss(m_i\theta_i)$, where $m_i$ ...
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2answers
99 views

How can I apply Bayesian Statistics when the number of data that I have is 1?

I want to show why Maximum Likelihood Estimation is not the right choice when the number of data that I have only is 1. For example, if I have an observed data which is heads from a coin toss. I only ...
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1answer
24 views

Analysis of counts with changing rate of succes

I have a large number of locations, let's say they're stores. At each store, $N_{it} \sim Pois(n_i)$ people walk through the door each week. We know the $n_i$ for each location. Of the $N_{it}$, a ...
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1answer
52 views

Estimate covariance given noisy data and the mean

Basically, I'm inferring the parameters of a Gaussian, $\mu$ and $\Sigma$ given observed data $y_i$ that have uncertainties $\sigma^2_i$ associated with them. 1D example: Prior I intend to use the ...
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27 views

Does MCMC Gibbs sampling algorithm first build a steady Markov Chain, then does the sampling to build the posterior distribution?

I am currently studying MCMC Gibbs sampling and while reading this part, a question has come into my head if MCMC Gibbs sampling first build a steady Markov Chain and does the sampling or does ...
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0answers
101 views

Graphical model of the Gaussian mixture: where is n?

TL;DR: Where are the occupation numbers in the Graphical model of the GMM? I am implementing a Finite (to be adapted to infinite later) Gaussian Mixture Model. I am using the Gibbs sampler-ready ...
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3answers
160 views

Hierarchical model: does leaving out a latent variable (hierarchy level) result in an equivalent model?

Say we have a hierarchical model: $$z_i \sim \mbox{Bernoulli}(\pi_i); \mbox{logit}(\pi_i) = ... \text{(linear function of covariates for site i)}$$ $$y_{i,j} \sim \mbox{Bernoulli}(z_i \cdot p_{i,j}) ...
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1answer
39 views

What exactly is meant with “data parameters” in Bayesian hierarchical models?

Please excuse me if the question in unclear, but I am seeing much nomenclature for the first time and am somewhat confused. I am reading "Statistics for Spatio-temporal data" by Noel Cressie ...
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24 views

Classification of Bayesian probabilities [duplicate]

I have run a series of Bayesian models with flat priors in which I obtain a posterior probability distribution for my coefficient of interest. The reviewer of my paper wishes us to classify these ...
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0answers
170 views

Classification of Bayesian posterior probabilities

I have run a series of Bayesian models with flat priors in which I obtain a posterior probability distribution for my coefficient of interest. The reviewer of my paper wishes us to classify these ...
2
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1answer
126 views

Need some interpretation with plain English for a part in Bayesian Statistics with Beta proability distribution? [duplicate]

Can somebody explain why equation (6.3) and (6.4) are shown in the book and what the author is trying to say? It feels to me that I am reading the text but I don't think I getting the true meaning ...
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0answers
12 views

Effect of heteroskedasticity in hierarchical (non-)linear models

Unlike linear models estimated via OLS where heteroskedasticity lead to inconsistency of the variance estimator but not the coefficient estimates, heteroskedasticity causes inconsistency of both ...
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1answer
202 views

Admissible Empirical Bayes Examples

I would like to hear about a few simple empirical bayes estimators that are admissible for high (i.e. at least 3) dimensional parameter space. What are some textbook lollipop examples to study for ...
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1answer
106 views

Hierarchical model for A/B experiment?

I'm new to Bayesian statistics. I have a metric that has a very non-parametric distribution, which would make it very difficult to use in an A/B experiment. However, it can be broken up into ...
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0answers
66 views

Mixed model with panel data when some cases have constant responses (zero) over time

I have a panel data with about 300 units observed over a period of 4 weeks. In each week, I recorded a response that is a binary variable, y, for each unit of that week. For about 50% of the units, ...
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2answers
525 views

How to calculate the Jacobian of the transformation ( for covariance matrix)

I'm reading this Paper about a separation strategy for modeling covariance matrices with focus on Bayesian analysis. Direct decomposition of covariance matrix is as follows: $\Sigma = \text{diag}(S)\,...
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119 views

Posterior Predictive Check for Hierarchical Logistic Regression Model

I need to apply Posterior Predictive Check (PPC) on Hierarchical Logistic Regression Model (so I have binary outcome) to validate my model (to see goodness of fit of my model). I know that I need to ...
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0answers
26 views

Simple one way random effect model into Bayesian approach

the random effect model $y_{ij}=\beta +u_i+\varepsilon_{ij} \left\{\begin{array}{c} i=1,2,\ldots,k \\ j=1,2,\ldots,J \end{array}\right.$ Assumptions: $$\varepsilon_{ij}\ \mbox{is }NID(0,\...
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1answer
542 views

How can I derive mathematical posterior predictive distribution calculation steps for beta prior and binomial likelihood

I would like to know the mathematical calculation step by step processes with beta prior and binomial likelihood for posterior predictive distribution.
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0answers
107 views

Shrinkage effects in a hierarchical model

I am working on the chimpanzees dataset from Richard McElreath's text, "Statistical Rethinking", edition 2. I have built 2 simple models, one a fixed effects model and the other a hierarchical model. ...
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3answers
64 views

When do we use a hierarchical model structure in Bayesian Analysis?

I am having trouble understanding when it is advantageous or when it is rational to use a hierarchical model set up in Bayesian Analysis. Basically what kinda of data do I have or what kind of ...
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1answer
96 views

How do I factor this conditional probability?

I am having a brain freeze. Could you show the steps to get from line 1 to line 2? Thanks!
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0answers
47 views

When regularizing based on an informative prior, how to give model a little more freedom to partially reject regulariziation

I am new here I hope this question is appropriate. I am modelling a spatial domain, whereby I have repeated measures at n locations. I make a bayesian linear model at each n locations based on about ...
2
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1answer
103 views

Multi-level logistic regression - probability received from the intercept is different from the original sample

I am running multi-level logistic regression in order to perform case-mix adjustment (corrected estimates) for 100 clinics. I got some results but they are somehow suspicious to me. I noticed that ...
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0answers
167 views

Difference between Mixed Logit model and hierarchical bayesian logit?

I'm studying the discrete choice analysis; The utility of person $i$ for alternative $k$ is: $$U_{ik} = \beta_kx_{ik} + \epsilon_{ik}$$ where $\beta_k$ is the parameter of interest and with $\...
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0answers
18 views

How can parameters be modeled differently if they share hyperparameters?

In one popular example of multilevel Bayesian models (2007 Gelman et. al paper), radon exposure in a household is modeled as a function of the county and whether the house has a basement. In this ...
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20 views

Multivariate Bayesian Car Model Result

I have developed a multivariate Bayesian Car model for three crash severity level analysis. I found that the covariance for both heterogenous effects and the spatial effect is not significant for any ...
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2answers
183 views

Is appropriate to use empirical Bayes (EB) in this way?

Background. I have data from a study where participants make a series of judgments (a series of decisions with a binomial outcome, either $y=1$ or $y=0$). I have a model of the underlying decision-...
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1answer
56 views

Bayesian updating of a probability density for evidence on its cumulative distribution

Suppose that I have a continuous variable E as a result of a simulation, which has a probability distribution as in the figure below: As seen from the cumulative plot, ...
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1answer
54 views

How are higher level posteriors modeled in a hierarchical Bayesian model?

Hope the question isn't too naive. I've been playing around with examples from Doing Bayesian Data Analysis by Kruschke, and in the Therapeutic Touch data section there's this multi-level model ...
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1answer
305 views

Conditional Probability - Mixture Model

I know that the likelihood in a p-dimensional Gaussian mixture model is given by $$ p(s|\theta) = \sum_{b_1 = 0}^1\cdots\sum_{b_p = 0}^1\left[ \prod_{i=1}^pw^{1-b_i}(1-w)^{1-b_i}\right]\phi_p(s|\mu(b,...
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1answer
76 views

Bayesian Inference: Prior in Chinese Restaurant Process

For the Chinese restaurant process, as used in Dirichlet Process mixture models, we have a prior that data point i belongs to cluster j, where c is an indicator. n represents the total number of data ...
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1answer
295 views

Proposal for correlation matrix with LKJ prior

I am writing a Gibbs sampler from scratch. As recommended in various places (http://www3.stat.sinica.edu.tw/statistica/oldpdf/A10n416.pdf, and in another question Covariance matrix proposal ...
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0answers
40 views

Bayesian estimation of mixed effects models covariance matrix

For a mixed model of the form: $$Y = X\beta + Z u + \epsilon$$ I know it is usually assumed in the parametric approach that: $u \sim N(0, D)$ and $\epsilon \sim N(0, \sigma^2I)$ Where $D$ is a ...
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28 views

Derivation of a Bayesian predictive probability

Suppose a random variable $Y$ is governed by an unknown parameter $p$. From a set of observations $X$, I want to draw the probability of $Y=y$. How can I compute the Bayesian predictive probability $f(...
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1answer
54 views

Derivation of posterior for Bayesian hierarchical models

In Bayesian hierarchical models, the following posterior is used: $$p(\theta,\phi|y)\propto p(y|\theta)p(\theta|\phi)p(\phi)$$ I'm trying to derive this myself but when I use Bayes' rule, I get the ...
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1answer
29 views

What are the necessary qualifications or assumptions to say that a graph structure is a Markov Chain?

I have a graph structure and want to say it is a Markov Chain. But I am wondering what necessary assumptions or properties that my graph structure need to meet to be called a Markov Chain?
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56 views

Probability distribution of the standard deviation of a gamma distribution

I want to generate some data using a series of Gamma distributions in a Bayesian hierarchical setting. I need to generate the data for a series of contexts, but I got only 2 data points per context, ...
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2answers
105 views

Mixed Effects, Doctors & Operations: predicting on new data containing previously unobserved levels, and updating our confidence accordingly

Here's a quick sketch of a hypothetical situation. There are Doctors $\{1, \ldots, J\}$ who perform different types of operations $\{1, \ldots, K\}$. Our response variable is whether the operation ...
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1answer
140 views

Hierarchical bayesian model without packages

I'm attempting to build a hierarchical Bayesian model. For various reasons (including my own edification), I want to do this from scratch (i.e., without using the various packages and libraries ...
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0answers
43 views

Statistical Significance for Bayesian parameter estimation

I was reading a paper that estimated parameter using the Bayesian method. I am wondering how they can write the following statement based on the table below "Two lane indicator is found to be ...
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0answers
86 views

Multilevel Negative Binomial fails with MLE

I have a pretty complex multilevel neg. binomial regression that does not converge when using a regular MLE (but from what I understand, when dealing with multilevel models, MLE is not regular, per se)...
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14 views

Weight of data vs. likelihood [closed]

I'm fitting a Bayesian multi-level model with an optional quantity of data (1 year, 5 years, 10 years, etc. of observations), and I have the option to include all of the data or less, does it ever ...
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2answers
155 views

Selecting informative priors

I am questioning myself on how to chose the priors for a bayesian analysis in Rstudio. I'm trying to investigate the chances of relapse in a set of patients. These patients are all affected by a ...
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1answer
100 views

“Mean” & “median” comparison and zero variance confusions when making inferences in Bayesian model

Background: In Chapter8 of this great book, the authors build a Bayesian model and use to show the posterior distributions of the latent state $N_{t}$ and its credible intervals. The model is ...
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0answers
58 views

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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1answer
86 views

Determine hyper-prior for gaussian distribution from existing data [closed]

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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0answers
151 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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0answers
39 views

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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0answers
81 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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