Questions tagged [hierarchical-bayesian]

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

Filter by
Sorted by
Tagged with
5
votes
1answer
924 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 ...
5
votes
1answer
231 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 ...
5
votes
1answer
297 views

Are predictive distributions supposed to be distributions of future data?

In frequentist analysis, we define a 95% prediction interval as an interval that will contain the next observation 95% of the time under repeated sampling of the entire experiment and prediction. If ...
5
votes
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:...
5
votes
1answer
389 views

Probability distribution to represent group mean of multiple beta distributions

Say I have two coins from a particular mint in the US. I flip coin one 20 times and receive 4 heads, giving me a beta distribution for the bias of coin one of $Beta$($\alpha$=5, $\beta$=17). I then ...
5
votes
0answers
649 views

Gibbs sampling deriving complete conditionals with mixture priors

My question is about the derivation of the complete conditionals for Gibbs sampling in a hierarchical model where some of the parameters are mixtures of point-masses and Normal distributions. The ...
5
votes
0answers
276 views

Hierarchical model: question on frequentist estimation

I am interested in understanding the differences between Bayesian and Frequentist estimation in the context of hierarchical models. Consider $n$ subjects, where for subject $i$ there are $k_i$ ...
4
votes
2answers
3k views

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: ...
4
votes
2answers
92 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 ...
4
votes
2answers
458 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 ...
4
votes
2answers
325 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 \;|\; \...
4
votes
4answers
2k views

“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 ...
4
votes
1answer
3k 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
votes
2answers
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 ...
4
votes
2answers
5k 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: ...
4
votes
1answer
433 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 ...
4
votes
2answers
345 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 ...
4
votes
3answers
215 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 ...
4
votes
1answer
1k 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 ...
4
votes
1answer
117 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
votes
1answer
176 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(\...
4
votes
2answers
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 ...
4
votes
1answer
50 views

Matt's trick (reparametrization) makes my models slower, not faster [closed]

I am currently programming a hierarchical model in Stan. Following the advice from section 22.7 from the Stan manual, I reparametrized my model so it samples the individual differences from a $N(0,1)$ ...
4
votes
1answer
248 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 ...
4
votes
2answers
538 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,...
4
votes
1answer
274 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 ...
4
votes
1answer
1k views

Choice of a model for Bayesian Change Point Detection

I am getting my hands dirty with Probabilistic Programming using Bayesian approach to change-point detection. I read a number of tutorials provided with PyMC and reading the book by Cameron Davidson-...
4
votes
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 ...
4
votes
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 ...
4
votes
2answers
851 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} = \...
4
votes
1answer
138 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 ...
4
votes
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 ...
4
votes
0answers
73 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, ...
4
votes
1answer
277 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 ...
4
votes
0answers
291 views

How to fit newer cohorts using Pareto/NBD or Beta/Geo for CLTV

I am trying to fit the popular Pareto/NBD or Beta/Geometric models for non-contractual, continuous customer data. On top of that I then fit the Gamma/Gamma model for monetary value (using the very ...
4
votes
1answer
185 views

How to predict using Spatial temporal hierarchical bayesian models

I am using the R package CARBayesST to fit a Spatial-temporal Bayesian models. I want to use piece-wise ST model proposed by Lee and Lawson, 2017. The package does not have a built-in predict ...
4
votes
0answers
63 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 ...
4
votes
0answers
35 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 ...
4
votes
0answers
247 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 ...
4
votes
0answers
445 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-...
4
votes
0answers
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 ...
4
votes
0answers
272 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 ...
4
votes
0answers
360 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 ...
4
votes
0answers
645 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)...
3
votes
1answer
130 views

Formulating posterior predictive distribution from hierarchical model

I have been reading a couple related papers using Bayesian inference in hierarchical models1,2,3 but am struggling to bridge the gap in one aspect of the papers. I think the struggle is in relation ...
3
votes
1answer
55 views

Are analytically tractable posterior distributions exclusively the result of a conjugate relationship in Bayesian hierarchical models?

I have been building a few of my own MCMC algorithms for hierarchical Bayesian models. If the posterior distribution of say $\alpha$ is analytically tractable, I sample $\alpha$ using an R function ...
3
votes
1answer
519 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 ...
3
votes
1answer
36 views

Is random effects model in meta-analysis a Bayesian approach?

Unlike fixed effect model meta analysis, assumption in the random effects is that the effect size of an individual study deviates from the true effect, not only because of the sampling error but also ...
3
votes
2answers
270 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)\,...
3
votes
2answers
534 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 ...

1
2
3 4 5
11