Questions tagged [hierarchical-bayesian]

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

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Why Reversible jump mcmc has only one step increase/ decrease?

I was applying reversible jump MCMC for joint estimation of model order and parameter estimation. I've a conceptual question in my mind. First of all, the algorithm has 3 steps, namely the birth, ...
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Is the term “latent effect” in statistics has the same meaning as “latent variable”?

Is the term “latent effect” in statistics has the same meaning as “latent variable”? If not, then what would be its substantive interpretation? UPDATE: There is a whole class of models called “Latent ...
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Estimating Direct Effect with Conditional Models

I was recently considering the following: Suppose we have an experimental set-up where we have collected observations over thousands of locations (S) before and after treatment (T). Further, we have ...
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Is it wrong to use sufficent statistc estimated from the data as a prior in Bayesian data analysis?

First I want to state that I got unexpected feedback from a reviewer in regard to my question and I am simply interested in others' views in this regard (I have already sent in my rebuttal). Suppose ...
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How to test for differences between groups in EEG trial data?

Experiment Setup Each participant must perform volutary biceps flexions (each flexion is referred to as a trial $t_i$) while staring at a screen. EEG and EMG signals are recorded throughout the ...
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Best way of modelling date of month seasonality/cyclical patterns in daily timeseries data, an bayesian approach is proposed!

I have several datasets which exhibits clear seasonality/cyclical patterns w.r.t date of month. The days after the 25th seem to be clearly correlated between months suggesting that in this case ...
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Gradient Descent for Multi-Level / Mixed / Hierarchical Regression Model

How would gradient descent work in a multilevel regression setting? This is fairly clear to me in a standard linear regression formulation, but haven't been able to wrap my head around parameter ...
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Comparing two randomly loaded dice

Say I have two six-sided dice, A and B, which are loaded in different ways, and I'd like to compare their probability distributions. So far I've constructed the priors for the probabilities $\vec\pi = ...
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In what ways do conjugate priors compose?

A lot of conjugate priors are known for a lot of likelihood distributions (mostly the exponential family). But most Bayesian models in practice don't just consist of one distribution. Usually, you ...
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Bayesian (Stan) advantage for estimating small variance components in a multilevel model

Imagine you have some data with one continuous predictor x, a categorical predictor A, and some dependent variable y. You have some amount (enough) data for several subjects S across all conditions. ...
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Understanding rare definition of the likelihood function and corresponding posterior from research paper

Reading the paper https://storage.googleapis.com/pub-tools-public-publication-data/pdf/b20467a5c27b86c08cceed56fc72ceadb875184a.pdf i came across a rare definition of the likelihood function that in ...
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How would you compare competing Bayesian hierarchical models that are estimating the same time-series?

Suppose we have two models $\mathcal{M}_1$ and $\mathcal{M}_2$ that attempt to estimate a time-series $\boldsymbol{\theta}$. Let's assume that I actually know the true values for $\boldsymbol{\theta}$....
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How to interpret the standard deviation of the slope random effect in a multilevel model

How do you interpret the standard deviation of the slope random effect in multilevel models? Suppose I want to show how Urbanization percentage changed across time in Western and Eastern Europe, so it ...
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Hiearchical Beta-Binomial model via rjags: How to draw posterior sample/do inference on posterior exactly?

I have the following code for bugs model which I want to use with rjags: ...
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Estimating a model with predictors themselves being random variables

Consider the following scenario: we have iid realizations from $Poi\left(\lambda\right)$, where $\lambda = \alpha + \beta t$. ($t$ is a predictor variable.) So far this is a standard GLM. However, ...
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What is the prior corresponding to a weighted sum of pdfs?

Recently, I have read a paper (Bayesian Bridge Regression , H. Mallick, 2018) which states that since $$ \frac{\lambda^{1/\alpha}}{2\Gamma(1+1/\alpha)}e^{-\lambda|\beta|^\alpha}=\int_{u>|\beta|^\...
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Recommend a book on Bayesian Hierarchical Modeling

I would like to learn about Bayesian Hierarchical Modeling and its applications to time series analysis. I would appreciate a good book / publication / course which will both teach some fundamental ...
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Loss function definiton for relabelling

Taken from the appendix to the paper (Yongning Wang & Ruey S. Tsay) of this (2019) paper Clustering Multiple Time Series with Structural Breaks. Appendix to be downloaded her Appendix To fix label ...
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Latent variables for spatio-temporal Extreme Value in R [closed]

Latent variables models are often used for spatial extremes modeling see e.g., Davison, Padoan and Ribatet. A typical application use block maxima such as annual maxima of temperature, assumed to ...
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Categorical model divergences/high parameter density near zero in Stan

I'm working on a hierarchical categorical/multinomial logit model in Stan. I thought I'd expand my question to stack exchange to see if anyone has any suggestions on the statistical model, since it's ...
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How to calculate Bayesian marginal credible interval?

I have a Bayesian model that I've fit using Stan, and I'm trying to figure out the best way to calculate the correct credible interval that I am interested in. The model is a hierarchical GLM with a ...
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Bayes graph model parameter dependencies problem with prior density

Suppose I have a graph model with underlying density (denote $f(\theta_1|\theta_0):=f(\theta_1)$) $$f(\theta|x)\propto \prod_{i=1}^k f(x_i|\theta_{1:i})f(\theta_i|\theta_{1:i-1}). $$ Suppose markov ...
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In Bayesian hierarchical models, what is the difference between an Empirical Bayesian approach to parametrising priors vs using flat hyperpriors

Say I have a simple hierarchical model, where: $y_{g,i} = \beta_g x_{g,i} + e_{g,i}$ where $g$ represents the group, $i$ represents the individual within the group, and $e$ is the error. So the ...
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Question regarding the Bayes estimator of a parameter $\theta$ which has a continuous distribution

Let $c > 0$ and \begin{equation} L(\theta,a)=\left\{ \begin{array}{@{}ll@{}} c|\theta-a|, & \text{if}\ \theta < a \\ |\theta-a|, & \text{if}\ \theta \ge a \quad. \end{array}\...
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Bayesian Model(Write out likelihood and prior)

I am working with a dataset regarding transmission rate for a disease spreading among cattle at different farms during a 5-month period. The goal is to estimate the transmission parameter $\alpha$ ...
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Is stationarity of variables neccessary condition for Bayesian VAR?

I am trying to run a BVAR on 5 variables. Four out of five are non-stationary. So shall I do the first difference of the non-stationarity variables or take them in level for running the BVAR? And what ...
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Could laplace method be applied to discrete distribution such as bernoulli, poisson-binomial etc?

I am trying to reproduce the result of the essay: Variational inferences for partially linear additive models with variable selection(K. Zhao, H. Lian,2014)enter link description here and encounter a ...
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Recovering samples from a density estimation with an additional prior on the samples. Used for Gibbs sampling

Abstract Idea: Given a noisy measured density ($d_j$ at position $p_j$) and a density model, sample from the model parameters under the following stochastic model: Stochastic Model: Prior for model ...
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What is the correct bayesian formulation for the zero-truncated Poisson lognormal model?

In ecology we use compound distributions to describe species-abundance data. One example is the Poisson Lognormal (PLN) distribution which is a Poisson distribution with rate parameter $\lambda$ that ...
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How to combine priors for empirical bayes

I am trying to estimate $Z_{i} = P(Y=1 | A=a_{i}, B = b_{i}, C =c_{i})$ using something like empirical Bayes as in http://varianceexplained.org/r/empirical_bayes_baseball/ aggregating a massive amount ...
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Question about likelihood function in Bayes Rule and marginalization

The following is an instance of Bayes' Rule: $$P(\alpha, \beta, \gamma, \delta, \epsilon|\mathbf{X}) = \frac{P(\mathbf{X}|\alpha, \beta, \gamma, \delta, \epsilon)P(\alpha, \beta, \gamma, \delta, \...
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Estimating the likelihood of a Dirichlet process

I am not sure if what I'm trying to achieve makes sense or is even possible, but I'd like to do MLE on a Dirichlet process mixture model. My reasoning is the following: If we can write out the ...
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Hierachical model causing too much shrinkage?

I use population-wide, 12-year data for examining regional differences in rehabilitation use (zero-inflated, lognormally-distributed outcome variable). I analysed the outcome variable with 2 models: <...
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Parallel with Weighted Least Squared in Bayesian Regression

I have a dataset with a column of ratios $Y = z_1 / z_2$, which will be my depending variable, and a set of columns that explain $Y$. Here $z_1$ means "imports" and $z_2$ means "exports&...
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Bayesian estimation under transformation on the paramater

Consider the classical model Normal-Normal-Inserse-Gamma model: $$ x=(x_1,...,x_n)|\mu,\sigma^2\sim N(\mu,\sigma^2)\,\,(iid),\,\,\mu\sim N(m_0,\tau),\sigma^2\sim IG(a,b), $$ where $m_0,\tau,a,b$ are ...
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Samples from Conditional Posterior Distribution in Pymc3

Let us consider the following Hierarchical Bayesian model: $w \sim\ Beta(20, 20)$ $K = 6$ $a = w * (K - 2) + 1$ $b = (1 - w) * (K - 2) + 1$ $theta \sim\ Beta(a, b)$ $y \sim\ Bern(theta)$ The above ...
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Is it possible for a hierarchical model to forecast a sub-region with complete lack of data as "borrowing strength"?

I am looking at sales time-series data (weekly from Jan to Dec 2021) which has a natural hierarchical structure by geography. For example, storeA1, storeA2, .... storeA99 located in Neighborhood A, ...
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Using Pairwise Differences Between Two Conditions as Data (Bayesian)?

Suppose I have measurements for the expression-level of a "gene" from two groups of arbitrary (possibly different) sizes. Maybe one group is a control and the other treated. $x$ = <4.5, 5....
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Hyperprior in Gibbs Sampling

Following up from this question, I have managed to derive the following posterior distributions $$ \lambda_z | \boldsymbol{y}, \Theta^{(-\lambda_z)} \sim Gamma(a + \sum_{i=1}^{n_z} y_{ij}, \quad a + ...
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Can we define inverse gamma priors in stan_glmer()?

Although I tried to read the manual, I don't quite see how I can incorporate the following model in stan_glmer(). $Y_{ij}|\mu_j,\sigma_y \sim N(\mu_j,\sigma_y^2)$ $\...
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Poisson-Gamma Hierarchical Model

I am fairly new to Gibbs Sampling and I am trying to build a Gibbs Sampler for a Poisson-Gamma hierarchical model. In this model, there are $m$ restaurants in a city, with $n_z$ number of observations ...
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inverse gamma (0.001,0.001) prior on the variance in the Bayesian hierarchical model

This 8 schools data is from Gelman 2006 paper: http://www.stat.columbia.edu/~gelman/research/published/taumain.pdf. In Figure 1 (c), the prior density of inverse gamma (0.001,0.001) was overlain on ...
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marginal likelihood in relevance vector machines RVM

I need to do a fit of 1d response versus 1d input, both real quantities. I wanted to implement Linear Regression, but because I wanted uncertainty in the resulting fit-params, I studied about Bayesian ...
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Choosing priors for the parameters of Gamma distribution

Suppose that $X_1, X_2, \cdots, X_n$ is a sample drawn from a Gamma distribution with parameter $\alpha$ and $\beta$. Then, the likelihood function can be written as follows: \begin{equation} L(\...
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Why are prior distributions sometimes conditional probabilities?

I came across the following Bayesian equation in textbook of evolutionary biology: $f(t, r, \theta|X) \propto f(X|t, r, \theta)f(t|\theta)f(r|t,\theta)f(\theta)$ $f(X|t, r, \theta)$ is the likelihood ...
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Modeling "Pay as Much as You Want" with a Bayesian Model

I have data of sales of a certain product which is sold "Pay as Much as You Want". The daily data is in the form of number of sales per day and the total revenue per day: Day Sales Revenue ...
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Extension of normal-gamma to gaussian process prior

I am trying to solve a problem where the solution involves both the mean and the variance of a multivariate normal distribution, modelled through a Gaussian Process prior. The standard Gaussian ...
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Samples from Marginal Posterior Distribution in Pymc3

Let us consider the following Hierarchical Bayesian model: $mu \sim\ Beta(1, 1)$ $k \sim\ Exponential(1)$ $a = k*mu$ $b = (1-mu) * k$ $theta \sim\ Beta(a, b)$ $y \sim\ Bern(theta)$ The above example ...
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How can I find the acceptance probability for a joint Metropolis-Hastings proposal?

Suppose that I want to generate a proposal $(x^*,y^*,z^*)$ with the following: $$z^*\sim p(z|\alpha,\beta)$$ $$x^*\sim p(x|z^*,\boldsymbol{\gamma}_x)$$ $$y^*\sim p(y|z^*,\boldsymbol{\gamma}_y),$$ ...
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What means that the model is "singular"?

My question might be rather basic, theoretical. I am running spatial and spatial-temporal bayesian models in INLA. I have areal data and a continuous response variable with spatial and temporal ...

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