Questions tagged [hierarchical-bayesian]

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

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81 votes
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XKCD's modified Bayes theorem: actually kinda reasonable?

I know this is from a comic famous for taking advantage of certain analytical tendencies, but it actually looks kind of reasonable after a few minutes of staring. Can anyone outline for me what this "...
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34 votes
2 answers
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What's the difference between "deep learning" and multilevel/hierarchical modeling?

Is "deep learning" just another term for multilevel/hierarchical modeling? I'm much more familiar with the latter than the former, but from what I can tell, the primary difference is not in their ...
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22 votes
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Why LKJcorr is a good prior for correlation matrix?

I´m reading the chapter 13 "Adventures in Covariance" in the (superb) book Statistical Rethinking by Richard McElreath where he presents the following hierarchical model: (...
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What prior distributions could/should be used for the variance in a hierarchical bayesisan model when the mean variance is of interest?

In his widely cited paper Prior distributions for variance parameters in hierarchical models (916 citation so far on Google Scholar) Gelman proposes that good non-informative prior distributions for ...
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18 votes
2 answers
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What is the problem with empirical priors?

In literature I sometimes stumple upon the remark, that choosing priors that depend on the data itself (for example Zellners g-prior) can be criticized from a theoretical point of view. Where exactly ...
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18 votes
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In Gelman's 8 school example, why is the standard error of the individual estimate assumed known?

Context: In Gelman's 8-school example (Bayesian Data Analysis, 3rd edition, Ch 5.5) there are eight parallel experiments in 8 schools testing the effect of coaching. Each experiment yields an ...
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17 votes
2 answers
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Bayesian estimation of $N$ of a binomial distribution

This question is a technical follow-up of this question. I have trouble understanding and replicating the model presented in Raftery (1988): Inference for the binomial $N$ parameter: a hierarchical ...
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15 votes
2 answers
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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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15 votes
3 answers
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Multinomial-Dirichlet model with hyperprior distribution on the concentration parameters

I will try to describe the problem at hand as general as possible. I am modeling observations as a categorical distribution with a parameter probability vector theta. Then, I assume the parameter ...
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14 votes
2 answers
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What are the parameters of a Wishart-Wishart posterior?

When infering the precision matrix $\boldsymbol{\Lambda}$ of a normal distribution used to generate $N$ D-dimensional vectors $\mathbf{x_1},..,\mathbf{x_N}$ \begin{align} \mathbf{x_i} &\sim \...
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14 votes
1 answer
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Hyperprior density for hierarchical Gamma-Poisson model

In a hierarchical model of data $y$ where $$y \sim \textrm{Poisson}(\lambda)$$ $$\lambda \sim \textrm{Gamma}(\alpha, \beta)$$ it appears to be typical in practice to chose values ($\alpha, \beta)$ ...
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14 votes
1 answer
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Why does adding a lag effect increase mean deviance in a Bayesian hierarchical model?

Background: I'm currently doing some work comparing various Bayesian hierarchical models. The data $y_{ij}$ are numeric measures of well-being for participant $i$ and time $j$. I have around 1000 ...
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13 votes
5 answers
3k views

What precisely does it mean to borrow information?

I often people them talk about information borrowing or information sharing in Bayesian hierarchical models. I can't seem to get a straight answer about what this actually means and if it is unique to ...
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12 votes
1 answer
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Why does the redundant mean parameterization speed up Gibbs MCMC?

In Gelman & Hill (2007)'s book (Data Analysis Using Regression and Multilevel/Hierarchical Models), the authors claim that including redundant mean parameters can help speed up MCMC. The given ...
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11 votes
2 answers
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What is a good analogy to illustrate the strengths of Hierarchical Bayesian Models?

I'm relatively new to bayesian statistics and have been using JAGS recently to build hierarchical bayesian models on different datasets. While I'm very satisfied of the results (compared to standard ...
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11 votes
2 answers
2k views

Why does Restricted maximum likelihood yield a better (unbiased) estimate of the variance?

I'm reading Doug Bates' theory paper on R's lme4 package to better understand the nitty-gritty of mixed models, and came across an intriguing result that I'd like to understand better, about using ...
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11 votes
1 answer
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Hyperprior distributions for the parameters (scale matrix and degrees of freedom) of a wishart prior to an inverse covariance matrix

I'm estimating several inverse covariance matrices of a set of measurements across different subpopulations using an wishart prior in jags/rjags/R. Instead of specifying a scale matrix and degrees ...
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11 votes
0 answers
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Hyperprior Noninformative Beta Binomial Model [closed]

I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model ...
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10 votes
2 answers
2k views

Hierarchical models for multiple comparisons - multiple outcomes context

I've just been (re-)reading Gelman's Why we (usually) don't have to worry about multiple comparisons. In particular the section "Multiple outcomes and other challenges" mentions using a hierarchical ...
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10 votes
2 answers
2k views

Bayesian inference and degrees of freedom

While learning frequentist linear regressions, one thing the professors always talked about was about the number of degrees of freedom, I never saw this expression in a bayesian book though. Perhaps ...
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10 votes
2 answers
458 views

Feature selection on a Bayesian hierarchical generalized linear model

I am looking to estimate a hierarchical GLM but with feature selection to determine which covariates are relevant at the population level to include. Suppose I have $G$ groups with $N$ observations ...
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9 votes
2 answers
501 views

Comparison between Bayes estimators

Consider the quadratic loss $L(\theta,\delta)=(\theta-\delta)^2$, with prior given $\pi(\theta)$ where $\pi(\theta)\sim U(0,1/2)$. Let $f(x|\theta)=\theta x^{\theta-1}\mathbb{I}_{[0,1]}(x), \...
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9 votes
2 answers
2k views

Why is the mixtures of conjugate priors important?

I have questions about the mixture of conjugate priors. I learned and saw the mixture of conjugate priors a couple of times when I am learning bayesian. I am wondering why this theorem is such ...
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9 votes
1 answer
907 views

Hierarchical Bayesian modeling of incidence rates

Kevin Murphy's book discusses a classical Hierarchical Bayesian problem (originally discussed in Johnson and Albert, 1999, p24): Suppose that we are trying to ...
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9 votes
0 answers
2k views

Horseshoe priors and random slope/intercept regressions

I'm interested in using the horseshoe prior (or the related hierarchical-shrinkage family of priors) for regression coefficients of a traditional multilevel regression (e.g., random slopes/intercepts)....
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8 votes
4 answers
7k views

Covariance matrix proposal distribution

In a MCMC implementation of hierarchical models, with normal random effects and a Wishart prior for their covariance matrix, Gibbs sampling is typically used. However, if we change the distribution ...
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8 votes
1 answer
806 views

What level to use when comparing subjects in a hierarchical Bayesian analysis?

Say that I have an experiment where I test the reaction time of a number of subjects where each subject makes many reaction time trials. In a Bayesian framework the reaction times ($y$) could be ...
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8 votes
1 answer
471 views

Is rstan or my grid approximation incorrect: deciding between conflicting quantile estimates in Bayesian inference

I have a model to achieve Bayesian estimates the population size $N$ and probability of detection $\theta$ in a binomial distribution solely based on the observed number of observed objects $y$: $$ p(...
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8 votes
3 answers
2k views

What is the relationship between graphical models and hierarchical Bayesian models?

I've searched a good bunch of literature but have failed to find an exact distinction between the two. My impression is that in the Machine Learning literature you'll find allusions to hierarchical ...
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8 votes
1 answer
2k views

Hierarchical Bayesian analysis on difference of proportions

Why Hierarchical? : I've tried researching this problem, and from what I understand, this is a "hierarchical" problem, because you are making observations about observations from a population, rather ...
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8 votes
1 answer
364 views

Plotting a "posterior median surface"

As part of reproducing a model I described partially in this question on Stack Overflow, I want to obtain a plot of a posterior distribution. The (spatial) model describes the selling price of some ...
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8 votes
0 answers
3k views

Hyper-prior for negative binomial in hierarchical model using JAGS/BUGS

Below I'm using a negative binomial because it is more flexible than a simple poisson model. The data are counts $y$ of events for 16 individuals $x$. There are 14 counts (i.e. counting periods) for ...
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7 votes
2 answers
3k views

Use of Bayesian hierarchical model

What is the purpose of Bayesian hierarchical model? When should I use such models? I've found many questions here and references on the web but they are all too technical. My doubts are about the ...
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7 votes
1 answer
3k views

Define Priors for Dirichlet Distribution parameters in JAGS

I'm defining a Multinomial-Dirichlet model in JAGS and want to assign some hyperpriors to the parameters of the Dirichlet distribution. In the WinBugs manual I read that "the parameters of Dirichlet ...
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7 votes
3 answers
671 views

Serial correlation AR(1) model for residuals: how to generalize to irregular times

I am working on a Bayesian serial correlation model for binary and ordinal logistic models (proportional odds model). I am modeling the serial correlation structure on the random effects of the model ...
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7 votes
1 answer
360 views

Can Kalman Filtering be done hierarchically - estimated from multiple time series with the same parameters?

I have a large number of of noisy time series recordings (trials), for which I wish to estimate the state transition model underlying them using the Kalman filter. The process generating the time ...
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7 votes
2 answers
2k views

What level to use when making inferences on the group mean in a hierarchical Bayesian analysis?

(This question is a bit related to a previous question of mine, but that question was about between subject comparison while this question is specifically about making inferences the group mean.) ...
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7 votes
0 answers
153 views

Why does Quadratic (Normal/Laplace) Approximation fail on multilevel models?

In Statistical Rethinking, 2nd Edition, section 13.1, Richard McElreath says: Why doesn’t simple quadratic approximation, using for example quap, work with multilevel models? When a prior is itself a ...
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7 votes
1 answer
379 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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7 votes
0 answers
258 views

Why does increasing number of observations in linear mixed model cause Bayesian modelling approach to fail?

I have a fairly good understanding of the theory behind Bayesian modeling and I have started to attempt some practical modeling using jags in R. I have been ...
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7 votes
0 answers
415 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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  • 350
7 votes
1 answer
898 views

Generate Posterior predictive distribution at every step in the MCMC chain for a hierarchical regression model

I'm trying to fit a Bayesian Hierarchical regression model with a random correlated coefficients using R ,I'm using data having 160 groups (schools) to fit a model of math score as a function of one ...
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6 votes
2 answers
4k views

Normalizing constant irrelevant in Bayes theorem? [duplicate]

I've been reviewing Bayesian literature in an attempt to utilize Bayesian inference for hypothesis testing when I have very well established priors, but there's one thing I cannot get my head around: ...
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6 votes
1 answer
4k views

Downsides of inverse Wishart prior in hierarchical models

I am working with a Bayesian hierarchical model that has a number of parameters for each experimental unit (6 parameters). I really do not know all that much about them a-priori, but it is quite ...
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6 votes
1 answer
303 views

Calculate mean and variance of a distribution of a distribution

I am using hierarchical distributions, of the following form: $\theta\sim N(\mu,\sigma)$ $\mu\sim N(a,b)$ I can calculate the mean, and variance using Mathematica, and find: $\mathrm{E}(\theta)=a$ ...
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6 votes
1 answer
5k views

What is posterior predictive check, and how I can do that in R?

I am using Bayesian hierarchical modeling to predict an ordered categorical variable from a metric variable. For example, I want to regress Happiness (in 1-5 ratings) on Money (a metric variable): ...
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  • 327
6 votes
2 answers
2k views

Bayes-factor for testing a null-hypothesis?

I heard somewhere, that I can directly test (or gather support for) a null-hypothesis using the Bayes-Factor. In my specific experiment, I hypothesize that an experimental manipulation does not have ...
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  • 915
6 votes
1 answer
365 views

How can we convert values proportional to probabilities to Bernoulli probabilities?

According to Wikipedia, the parameter in a Bernoulli distribution should be $0<p<1$. I am reading this famous paper proposing Hierarchical Dirichlet Process, and on page 1580, A.6 and the ...
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  • 1,052
6 votes
1 answer
404 views

Bayesian Aproach: Infering the N and $\theta$ values from a binomial distribution

I am doing a homework about infering the N value of a binomial distribution for my Bayesian Statistics Course and I have seen a paper in Biometrika magazine published in 1988 for doing so. The ...
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6 votes
2 answers
2k views

Update rule for beta distribution with fixed K/confidence/sample size

Normally you have a beta distribution with shape parameters $a$ and $b$. The mean of this distribution is $a / (a + b)$ and the sample size, or the confidence (or K) is $a + b$. Now, if you do some ...
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