Questions tagged [hierarchical-bayesian]

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

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347 views

Pro and cons between Bayesian structural time series (BSTS) vs difference-in-differences?

Google's paper markets BSTS's benefits over DID such that "In contrast to classical difference-in-differences schemes, state-space models make it possible to (i) infer the temporal evolution of ...
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1answer
167 views

Hierarchical Version of Bayesian Change Detection Model in JAGS

I am trying to create a hierarchical changepoint detection model in JAGS, estimating group difference in changepoint based on individual changepoints in scores for an outcome variable (fictional in ...
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28 views

Bayesian modeling: likelihood function for continuous random variables, why is it not always 0? [duplicate]

For continuous random variables, evaluating p(x) for a specific value of x is always 0 as show here, here and here. So when we're calculating the likelihood for a random variable X that is represented ...
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1answer
272 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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110 views

sampling from a posterior derived from hierarchical Bayesian using HMC

I have a complex pdf based on hierarchical Bayesian formalism where x depends on the priors w'and w'', and I consider hyper-prior for the latter's as w=Php(zeta,beta) where Php stands for the hyper ...
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56 views

Dependence between parameters in Bayesian multilevel regression

I am working on a Bayesian multilevel regression model, which is specified as $$ y_{ij}=x_{ij}'\beta+\delta_j+\varepsilon_{ij}\\ \delta_j=\gamma_{\operatorname{region}(j)}+\eta_j $$ where $i$ indexes ...
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80 views

Hierarchical Bayesian Modelling: deriving a Conditional PDF relation

Background I am currently reading through the book: Data Analysis: A Bayesian Tutorial, second edition by D.S. Sivia. I am stumped by one of the relations that he introduces in the context of model ...
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397 views

Hierarchical modelling - partial pooling with correlation

I am doing a Bayesian regression. I have groups of data $(y_1 ~X_1), (y_2~X_2),...$, where each $y$ and $X$ is a vector. The subscript is regarded as group number. The completely unpooled regression ...
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29 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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275 views

Does a closed form marginal posterior exist for regression coefficients in hierarchical linear model?

Given the standard linear model $$Y=X\beta+\epsilon$$ a Gaussian likelihood function $Y|b,\sigma^2 \sim N(X\beta,\sigma^2 I)$ and a hierarchical model for the regression coefficient $\beta$ of a form ...
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1k views

Hierarchical (multilevel, random-effects) Gaussian process regression

If we have a $J$ groups of predictor, outcome (univariate) variable pairs, $$ \{(y_{j1}, x_{j1}) \ldots (y_{jn_j}, x_{jn_j})\}, \quad\text{for $j \in 1\cdots J$}, $$ a hiearchical linear regression ...
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1k views

Is this correct hierarchical Bernoulli model?

I have a question about correctness of a model that I used for a fairly simple experiment. I'm not sure if it should go to stackoverflow or crossvalidated, because I feel like my question is both ...
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1k views

Fitting a Bayesian Hierarchical Poisson Regression in R

I'm trying to fit a Bayesian hierarchical poisson regression. To do so, I'm using MCMChpoisson function from MCMCpack in R. Based on this package, the model is: $$Y_i \sim Poisson(\lambda_i)$$ $$\phi(...
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191 views

Bayes Net Parameter Learning in pymc

My goal is to infer the conditional probability tables (CPT) from the classic rain, sprinker, wet grass problem. Normally in this problem we know the CPTs and, given an observation like "the grass is ...
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70 views

Incorporating population priors into MLE fits with few/limited samples

I am fitting Beta distributions to data resulting from each of many experiments using maximum likelihood. My goal is for each experiment, given iid data $y_{1:k}$, fit a Beta distribution, and then ...
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188 views

Pymc: Does this model call for Index, and if so, how would I use it?

I'm working on speeding up the mcmc for a hierarchical pymc model that is taking .2 seconds per iteration. It's the second model from this paper, modeling a soccer league using team-specific attack ...
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176 views

Question about foundations of the uniform shrinkage prior

I am collecting papers about the uniform shrinkage prior for hierarchical Bayesian model. In "A prior for the variance in hierarchical models" of Michael J. Daniels it is stated at the end of page two ...
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429 views

A better bayesian way of modelling autoregressive mixtures

I have a JAGS hierarchical model which includes a temporal sub-model for the primary vote share between four party groups (LNP, Labor, Green, and Other). For each day in the temporal model, the vote ...
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2answers
378 views

comparing distributions - bayesian decision analysis

I am attempting to use Bayesian analysis to compare distributions to help with decision analysis - when to treat a patient based on a blood measurement X. Here you can see 1000 samples from two ...
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2answers
481 views

How to analyze this data using rjags, or any other way?

There are three groups that received different treatments and then learn a task which they are scored on over the course of 60 sessions. 1) How to choose a function to fit? -The best choice in this ...
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1answer
31 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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1answer
180 views

What problem do these trace plots indicate?

The following plots are trace plots of 3 variables for MCMC results of a hierarchical Bayes probit model. The plots are fairly linear and seem to grow (or decline) without bound. This looks like a ...
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2answers
204 views

Bayesian output vs frequentist. Which should I rely on? MLM/ RE HLM

I have 2 questions. 1)My Bayesian output is providing some trouble. I have data that will vary across 5 countries. This means my group level has a small n of 5. This results in my data hovering ...
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1answer
662 views

How to find the Likelihood Function in a Bayesian Model given some Data

How should I find the likelihood function of a Bayesian Model? For example, if I'm given a coin, I can use the Bernoulli Distribution as the likelihood function (because I know in advance that the ...
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1answer
138 views

Is there any reason to prefer a bayesian model with few variables?

I have two alternative hierachical bayesian models that were designed to the describe the same process (from a high-level point-of-view). Both model provides comparable (but not identical) inferences ...
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2answers
50 views

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
110 views

Bayesian multilevel model in practice - selection of package, specification and interpretation

I am trying to fit a Bayesian multilevel model in R and have several questions. I found two packages (brms and rstanarm) and am able to perform the analysis with both of them, so the technical part is ...
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2answers
398 views

Confusion about Cross-Validation for Hierarchical Bayesian Regression Models

I had two questions regarding model selection for a Hierarchical Bayesian (HB) Regression Model and the purpose of Cross-Validation. 1). I understand cross-validation as one way to perform model ...
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2answers
58 views

How can I use ratios to set priors on multinomial probabilities?

I have a vector, $k$, that determines allocation to five pools. I'd like to set priors on these probabilities, and I can provide informative priors on a few of the ratios, e.g.: $$ \frac{k1}{k2} \...
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1answer
318 views

Decomposing Laplace prior for a hierarchical representation of a model

I have a conditional Laplace prior: $$ \pi(\boldsymbol{\beta}|\sigma^2) = \prod\limits_{j=1}^{p}\frac{\lambda}{2\sqrt{\sigma^2}}e^{-\lambda|\beta_j|/\sqrt{\sigma^2}} $$ and a marginal prior on $\...
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2answers
100 views

controlling for clustering at id level in mixed effects model

I have one group ($n=40$) of subjects pre- and post-tested (time; coded $0$ and $1$) on a continuous variable (y). I also have a ...
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1answer
213 views

Plate notation for a hierarchical regression model (bayesian)

I've been recently studying hierarchical bayesian regressio (with pymc3), and I was wondering, how does the following example: http://twiecki.github.io/blog/2014/03/17/bayesian-glms-3/ look like ...
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1answer
175 views

Hierarchical linear modelling in R

I am trying to build a hierarchical linear model based on data structured like this dataset below. The model form I am looking to build is Purchased ~ f(price + color + more item attributes + age + ...
2
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1answer
458 views

In a Bayesian Hierarchical Model set-up, what is the definition and difference between random and fixed effects?

I understand that fixed vs. random effects have different meaning whether it be in biostatistics or econometrics. I recently came across a talk regarding fixed vs. random effects in the hierarchical ...
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1answer
280 views

Risk and posterior expectation Bayesian Statistics

Consider $x\sim B(n,\theta)$ with $n$ known a)If $\pi(\theta)\sim Beta(\sqrt{n}/2,\sqrt{n}/2)$ give the associated posterior distribution and posterior expectation $\delta^\pi(x)$ b)Show ...
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1answer
107 views

Levels of “hyperparameterization” in Hierarchical Modeling

Suppose we have observations $y$ that we wish to model as having being randomly sampled from a distribution with parameter $\theta$. General Bayesian approach assumes a prior distribution over $\theta$...
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1answer
271 views

Hierarchical Bayesian Regression, Can an Inverse-Gamma distributed Variance look Normal or t?

Using Peter Hoff's book, A First Course in Bayesian Statistical Methods, I used some of my own data to fit a Hierarchical Bayesian Regression following his example. In his book, he utilized a Gibbs ...
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1answer
736 views

Bayesian meta analysis: implementation in BUGS/JAGS/STAN

I would like to conduct a meta analysis in order to collate the information from a number of studies. The parameter of interest is a probability $\theta$. In each of the studies, the observed data $...
2
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1answer
687 views

Discrete predictors in a mixed effects model in JAGS

I'm fitting a random effects hierarchical model in JAGS and have a question regarding discrete predictors in the contexts of mixed effects models. In my data there is a group variable, which is the ...
2
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1answer
185 views

Fisher information metric for hierarchical Bayesian model is negative-definite?

I'm strugling with the computation of the Fisher information matrix for the hierarchical Bayesian model. For simplicity, consider theta following hierarchical Bayesian model: \begin{align} X|\sigma &...
2
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1answer
877 views

How to Implement an Empirical Bayes Analysis in BUGS/JAGS/Stan

My data is a set of $N$ observations $y_i$. I would like to estimate $\mu$ and $\sigma$ in the following model: $y_i \sim \mathrm{Normal}(\theta, \sigma)$ $\theta \sim \mathrm{Normal}(\mu, \frac{\...
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1answer
279 views

Multilevel model with Hierarchical Bayes

I want to apply a multilevel model (random intercept random slope model) comparing Maximum Likelihood and Bayesian estimation. I am used to Maximum Likelihood estimation, however, the Bayesian ...
2
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1answer
71 views

What is the connection between Bayesian Model Averaging and SSVS?

What exactly is the difference between Bayesian Model Averaging (BMA) and the Stochastic Search Variable Selection (SSVS) prior when we talk about linear regression models? The SSVS prior is used ...
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1answer
536 views

Difference between Bayesian Hierarchical Model and a Bayesian regression model?

Are Bayesian Hierarchical models and Bayesian regression models the same in books, papers?
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1answer
940 views

How can I re-code this hierarchical model in PyMC 3?

I wish to model data from an experiment using a hierarchical Bayesian logistic regression. The experiment involved many subjects, and many trials collected from each subject. The DV is the outcome of ...
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1answer
125 views

Computing a marginal posterior of a hierarchical model

This is my first time posting an actual homework question. Usually I have more local resources such as office hours and student peers but this time I am a little short on those. I also need to rest ...
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2answers
47 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
87 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
51 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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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}|\...