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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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Bayes Factor for linear mixed models with BayesFactor package in R [on hold]

I am trying to compute the Bayes Factor (BF) for one of the fixed effect with the BayesFactor package in R. The data has the following structure: ...
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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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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
76 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
292 views

Bayesian inference on mean of statistic from population

Suppose that a collection of time intervals $t_i$ have occurred, for $i=1,...,n$. These should be considered as samples from a population governed by some distribution. During these time intervals, ...
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1answer
200 views

Implementing a hierarchical bayesian graphical model in R

The shorter version: 1. Bayesian graphical models are new to me. 2. I want to use R to model spatial variation in county level crime using a BGN. I have been working with bnlearn, and would ideally ...
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148 views

Run MAP estimates before MCMC in most cases?

I am learning Bayesian statistics. I found that this pymc3 introduction sometimes uses MAP to estimate the parameters for the MCMC input (the regression example), but the intro doesn't run MAP for ...
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1answer
39 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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13 views

Dirichlet Process mixture model with independent features

I'm trying to construct a Dirichlet process mixture model for clustering where the samples have independent features. In other words, to evaluate the likelihood of sample $x_i$, I would compute $\...
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17 views

Bayesian hierarchical model with varying scales [closed]

I would like to have a bayesian hierarchical regression model. Suppose that we have multiple data sets, which adhere to a hierarchy. Let us call the response variable in dataset 1 and 2 as $Y_1, Y_2$. ...
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1answer
50 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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1answer
37 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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32 views

Approaches to fast estimation of new levels of a hierarchical linear model from new data

I have a hierarchical linear model I've applied to a dataset in which the effect of a factor on my outcome measure can vary for different people. Say I have a new individual for whom I have some ...
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1answer
371 views

Hierarchical bayes

I am programming in R using hierarchical bayes for a choice-based conjoint task and wondering how I code the "none of the above" option in the design matrix? The <...
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1answer
83 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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21 views

Shouldn't the distribution of alpha be included in the predictive distribution of variational linear regression?

I was reading "pattern recognition and machine learning" book and I see that in the predictive distribution of the bayesian linear regression with variatioanl inference, it uses this equation $p(t|x, ...
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222 views

Appropriate Distribution for Diagonal Covariance Matrices

Let's say I have a model like: \begin{align} X\mid\mu,\Sigma_X &\sim \mathcal{N}(\mu,\Sigma_X)\\ \mu\mid m, \Sigma_\mu &\sim \mathcal{N}(m,\Sigma_\mu) \\ \Sigma_X\mid \Psi, c &\sim \...
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1answer
141 views

How does this Sampler work for the Concentration parameter of Dirichlet Process?

I am puzzled by how this Gibbs sampler on section 6 of Escobar & West (1995) works. To put it in simple words, the aim is to sample $\alpha$. The defined terms are: $$\eta\sim \texttt{Beta}(a,b)$$ ...
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2answers
64 views

Two priors on the same parameter?

I received a text where the author was employing Stan language in order to show how to create a random walk with normally distributed parameters. His model had parameters $\mu_{t}$, ($1,2,...,T$), ...
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12 views

Gibbs Sampling attempt at a simple Coxian distribution

I have the following Coxian model for inter-arival times ($x_i$) that has $C_x^2 < 1$: $$ p(x_i\mid \lambda,\theta) = \theta \lambda^r x_i^r e^{-\lambda x_i} + (1-\theta)\lambda e^{-\lambda x_i} $...
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12 views

ChoicemodelR - How to calculate variable importance using output of ChoiceModelR?

i have built a HB choice model using ChoiceModelR. Can someone tell me how to calculate the Relative Variable Importance from the output?
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1answer
43 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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28 views

Statistical model suggestion for binary decision problem

I am looking for a statistical/machine learning model, which can describe and predict a (forced) binary decision between say A and B at any moment in time. I have input data from time series of say 3 ...
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37 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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Shape of parameters marginal posterior in hierarchical Bayes model

Consider a generic hierarchical Bayes model with data $y_i\sim p(y|\theta_i)$, dependent of parameters $\theta_i\sim p(\theta|\phi)$ and hyperparameters $\phi\sim p(\phi)$. Furthermore, assume that $\...
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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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19 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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17 views

Wrong sign of a model coefficient in Bayesian Poisson-Lognormal Car model

I am trying to develop a multivariate Poisson lognormal CAR model. One of my most important variables in the model is providing a negative sign which should be positive. However, when I develop a ...
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1answer
19 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
17 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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1answer
238 views

How can I identify market regimes with a Hidden Markov Model?

I am trying to identify market regimes (2 states: bull or bear) with percent changes in equity returns. Can you help me in the mathematicl modeling of this? So far, I thought that for each day, there ...
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2answers
70 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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27 views

Use zeros trick for modeled parameters

Can the zeros trick be applied for specifying a new distribution for modeled parameters such as a varying intercept model? I am trying to estimate a hierarchial model (like the following) where I have ...
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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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1answer
280 views

Using PyMC3, how could I force a maximum to posterior distribution?

I am pretty new to bayesian statistics and PyMC3. I am doing a hierarchical model where the output variable I am trying to predict is a percentage with a maximum of 100%. My problem is that my ...
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63 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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Pseudo T-stat in Winbugs?

I am trying to obtain the "Pseudo T-stat" in Winbugs for a Poisson Log-normal model. Any suggestion of how can I get that.
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1answer
283 views

Comparing top level group effects using a 3-level hierarchical regression

I would like to detect group effects (if any) along with statistical confidences. I have a hierarchical data set structured as follows: Drug Groups ...
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40 views

Metropolis-within-Gibbs for parametric inference of a regressive model

I have a regressive model of this form \begin{equation} y=f(\theta)+\varepsilon \end{equation} to describe observations $y$, with noise $\varepsilon$ and a parametric function $f$ with parameters $\...
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1answer
44 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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39 views

Winbugs code multivariate hierarchical Poisson Log-normal CAR model

I am looking for some example code to develop multivariate hierarchical Poisson Log-normal CAR model using Winbugs. Can anyone help me with similar reserach that added their code? Also, how can I ...
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30 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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10 views

Hierarchical simulation of patient waiting times in order to keep annual average below some threshold

I have a dataset about patient waiting times in a healthcare district. These data have 3 categorical variables: - healthcare provider; - healthcare service (eg. cardiology visit, electrocardiogram, ...
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12 views

Weight of data vs. likelihood

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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1answer
89 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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2answers
54 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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42 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
22 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
59 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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19 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 ...