Questions tagged [hierarchical-bayesian]

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

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Normalizing constant irrelevant in Bayes theorem?

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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2answers
284 views

What are some statistical tests for exchangeability of a data set?

The representation theorem of de Finetti is seen by some as motivation for the use of Bayesian and/or hierarchical modeling. In some settings, it may be plausible to assume measurements are ...
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2answers
58 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
498 views

Relation between Bayesian analysis and Bayesian hierarchical analysis?

I have been studying a Bayesian hierarchical model. In that model all I am dealing is with the estimation of parameters. In Bayesian analysis, loosely speaking, we update our prior knowledge (in light ...
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2answers
322 views

Is the posterior of a random variable's mean necessarily the mean of that random variable's posterior?

Let's say I have a model that's like, $$ Y \;|\; \theta_1 \sim P(Y \;|\; \theta_1) $$ $$\theta_1 \;|\; \theta_2 \sim P(\theta_1 \;|\; \theta_2) $$ $$ \theta_2 \;|\; \theta_3 \sim P(\theta_2 \;|\; \...
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4answers
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“Unidentified” hierarchical model in brms/stan - where to go from here?

I am evaluating an intervention in which participants are grouped in teams and each participant fills in a survey before and after the intervention. As such, the data presents a classic multilevel ...
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1answer
2k 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): ...
4
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1answer
275 views

Ergodicity of MCMC in a hierarchical model

Many of the Bayesian hierarchical models that I am studying use a Markov chain as the model. These hierarchical models use different MCMC techniques to sample low-level and high-level parameters. My ...
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1answer
99 views

MCMC advice: Ignoring some parameters in a MCMC scheme?

I am after some general advice regarding my MCMC scheme, which is causing me some grief. Essentially, I have a large (2N + 9 parameters) MCMC scheme which works great. However, the problem is that ...
4
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1answer
142 views

Bayesian regression with independent variable drawn from distribution

I'm trying to set up a bayesian regression of the form $y_i \sim f(\beta_0 + \beta_1 x_i)$ but rather than $x_i$ fixed, they themselves are drawn from a distribution of (known) mean $x_i \sim g(\...
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2answers
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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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1answer
231 views

Bayes Rule with Model Comparison

In Doing Bayesian Data Analysis 2ed, by Kruschke, in chapter 10, we get two equations (10.1, 10.2) for which no hint as to how they are obtained is given... How does one get the second equality in ...
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1answer
800 views

Crossvalidation in hierarchical bayesian models (HBMs)

I am trying to find a way to cross-validate Hierarchical Bayesian Models used for predicting and modelling abundance in Species Distribution Models. For this purpose, I have tried posterior predictive ...
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2answers
504 views

Particle filter (sequential Monte Carlo) for a non-Gaussian hierarchical model

I have the following, which I am attempting to model with a particle filter. \begin{align*} y_{i,t}&\sim\mathrm{Poisson}\left(\lambda_{i,j,t}\right)\\ y_{j,t}&\sim\mathrm{Poisson}\left(\mu_{i,...
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1answer
1k views

Seeking a closed form for a posterior distribution

In the book Bayesian Data Analysis by Gelman et al. (3rd edition, 2014), a hierarchical model (or one-way random-effects ANOVA) is presented in section 5.4 as follows, \begin{equation}\label{eq:...
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1answer
2k views

How to specify a hierarchical bayesian model with sum-to-zero constraints?

I'm working on the first model described in this paper ("Bayesian hierarchical model for the prediction of football [soccer] results"). The gist of the model is: The model includes two sum-to-zero ...
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4answers
1k views

Is there a desription in the literature of a Normal hierarchical model with hyperparameters for both the mean and the standard deviation?

I'm looking for a comprehensive description of and justification for a Normal hierarchical model where both the means of the groups and the standard deviation are modelled. It is common to find ...
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2answers
783 views

Joint prior distributions in WinBUGS

Suppose we have a hierarchical model summarised by the following: $y_{i} \sim N(\mu_{i}, \sigma^{2})$, for $i = 1, \ldots, n$; (For these purposes, assume $\sigma^{2}$ is known) where $\mu_{i} = \...
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1answer
135 views

Variance pooling when sample size is a predictor

Suppose that I am building a hierarchical model of performance and the data is hierarchically structured (e.g., multiple customers rating a single salesperson). I might want to use variance pooling in ...
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1answer
2k views

Replicate simulation study from a paper and calculate the MSE in R

I have implemented a Gibbs Sampler for the Bayesian Elastic Net (BEN) according to this paper on Penalized Regression by Kyung et al. In this paper, they execute a simulation study that has been used ...
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70 views

How to infer a prior belief after observing a behavior

My participant goes through a maze made of 32 T intersections. At each intersection he must choose whether to go either to the left or to the right: one option will lead to another T intersection, ...
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198 views

Likelihood function of a hierarchical model

I have the following model: $$ y\sim\textrm{MvNormal}\left(\mu,\Sigma\right)\\ p=\textrm{logistic}\left(y\right)\\ k\sim\textrm{Binomial}\left(p,n\right) $$ Where $\mu$ and $\Sigma$ are free ...
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124 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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169 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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236 views

Difference between hierarchical Bayes and random parameter/effects models?

From my limited understanding, the difference is mainly that hierarchical Bayes (HB) incorporates parameter distribution priors that will "constrain" the individual parameters to one side of the ...
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402 views

Effect size for contrasts in hierarchical Bayesian “ANOVA”

Kruschke (2014) shows in his book how to compute posterior distributions of effect sizes (standardized mean difference) for the Bayesian analogues of frequentist independent-samples t-tests, and one-...
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345 views

What convergence diagnostics are appropriate for a Bayesian hierarchical logistic regression model?

Using WinBUGS, I fit several Bayesian hierarchical logistic regression models for the mean of a binary response variable conditional on a set of criteria. I am now using CODA in R to determine if my ...
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562 views

Negative number of parameters in hierarchical bayesian model

I'm using Deviance information criterion to assess the fitness in my Bayesian hierarchical model. The functional form of this criterion is as follows: $$DIC=p_{D}+\bar{D}$$ where $p_{D}=\bar{D(\theta)...
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1answer
539 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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2answers
1k 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 ...
3
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1answer
439 views

Over-parameterization in Bayesian Hierarchical Model

Can someone explain the influence of adding parameters to a Bayesian model? I have read from Kruschke that Bayesian analysis 'accounts' for model complexity by way of multiple priors, however I don't ...
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2answers
3k views

How to generate the posterior predictive distribution for hierarchal model in PYMC3

See iPython notebook for full example The below stochastic node y_pred enables me to generate the posterior predictive distribution: ...
3
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2answers
211 views

Is my Bayesian analysis correct?

This is my first time doing a Bayesian analysis, so I'm not sure whether what I did makes perfect sense. I'm trying to tell if two samples come from the same distribution, more specifically, if they ...
3
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3answers
203 views

Verifying and/or changing priors assumptions on Bayesian ANOVA

I am performing a Bayesian analysis of around 1500 data, divided into 2 factors, one that I am interested x1, and the id-variable for the paired/within-subject x2. x1 has 15 levels, and x2 around 100 ...
3
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1answer
764 views

Bayes risk of Normal-Normal model

Consider $x\sim N(\theta,1)$ and $\theta\sim N(0,n)$. Show that the Bayes risk is equal to $\frac{n}{n+1}$. I know that $$r(\theta,\delta)=\int_\chi\int_\Theta L(\theta,\delta(x))\pi(\theta|x)d\...
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1answer
5k views

Multi-level Bayesian hierarchical regression using rjags

I am trying to to implement a Bayesian hierarchical Model in R. I have a few predictor variables (2 metric and one categorical) and am trying to predict quarterly home sales in the US. Each sales ...
3
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2answers
291 views

Flipping random coins from a bag - equivalent to a single coin?

My first and I think naive question here. I am trying to model a certain business, and the simplest model I am willing to test is: 1. there is a bag of differently biased coins. 2. every step, a ...
3
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1answer
891 views

How to choose t-distribution degrees of freedom in “robust” Bayesian linear models

It is well known that in both frequentist and Bayesian linear models, outliers can greatly influence the parameter estimates. Consider the simple example where one outcome variable, $y$, is predicted ...
3
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1answer
160 views

Setting up a MCMC scheme for Multivariate Stochastic Volatility

I want to understand the survey of Lopes and Polson (2010) regarding MV stochastic volatility. Assume the $p$-dimensional vector $y_t$ follows $$y_t\sim N(\Theta,\Sigma_t).$$ In order to model the ...
3
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1answer
981 views

Gibbs sampling with mixed prior using a Metropolis-Hastings step

My questions are about a sampling procedure for fitting a Bayesian hierarchical model where one of the priors is a mixture distribution of discrete and continuous parts. The model is not my own but I ...
3
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1answer
976 views

Metropolis-Hastings acceptance rate confusion

I ran a Bayesian model that have about 2700 parameters. Among these parameters, Adaptive Metropolis algorithm was implemented to estimate ~790 parameters in the I-group and Metropolis algorithm was ...
3
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1answer
226 views

Stick-breaking construction of Dirichlet process

In the stick-breaking construction of Dirichlet (let me base things on Sethuraman's construction - slide 6 of this) do we sample one $\phi$ vector from the base distribution $H$ and use it for ...
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1answer
1k views

How does Hierarchical LDA compare to Hierachical Agglomerative Clustering?

I have a collection of documents and want to detect a hierarchy of named topics from them, what are the pros/cons for using hierarchical latent Dirichlet allocation (h-LDA) over hierarchical ...
3
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1answer
415 views

KL divergence for a hierarchical prior structure e.g. Linear Regression

For a Linear Regression $\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \epsilon$ with $\epsilon \sim \mathcal{N}(0, \sigma^2\mathbb{I})$, suppose the prior set on $\beta_k$ is $\sim \mathcal{N}(0, l_k)$ ...
3
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1answer
467 views

Interesting / strange behavior of one chane on different [unrelated] variables in STAN

I have a quite complex hierarchical model for which I'm estimating parameters and producing posterior predictive using STAN (rstan) for some psychophyiscal data. I'm (sometimes) observing some ...
3
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1answer
400 views

Need help deriving a gibbs sampler for a normal mixture model with two components

Let $\theta_i$ be an indicator that the i-th eruption is a long eruption. (i.e. $\theta_i = 1$ if the i-th eruption is long and $\theta_i = 0$ otherwise.) Assume the following model and derive a Gibbs ...
3
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1answer
178 views

Data Conversion to Standard data format in hierarchical Dirichlet process

I'm trying to test the performance of posterior inference on a set of documents with hierarchical Dirichlet process for topic modeling. How can i convert my data (document) to standard data format ...
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0answers
130 views

What is the posterior kernel lengthscale of a Gaussian process?

If I have access to multiple samples from a Gaussian process with known covariance kernel but unknown parameters (i.e. unknown lengthscale), it is straightforward to estimate the lengthscale using ...
3
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0answers
333 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 ...
3
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1answer
157 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 ...