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Questions tagged [hierarchical-bayesian]

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

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156 views

throwing away all Gibbs samples after approximation

This is more of a theory question, consider: $$P(w_1|D)=\int P(w_1|S)P(S|D)d(S)$$ which we approximate via Gibbs sampling $S$ (assume the initial state of the Gibbs sampler is denoted by $M_0$), ...
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1answer
351 views

Hierarchical bayes

I am programming in R using hierarchical bayes for a choice-based conjoint task and wondering how I code the "none of the above" option in the design matrix? The <...
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49 views

What kind of book provides an introduction to free-energy minimization?

I have a basic understanding of free-energy minimization methods from doing some reading in neuroscience on prediction error minimization--primarily from Rafal Bogacz's beautiful tutorial on the free-...
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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 ...
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61 views

Sampling a pymc hierarchical posterior with small population sample size - Spread variance adjustment correction question

I've created a pymc poisson hierarchical model to forecast sports scores. If I use a smaller sample size of the season, say the first month, (10 games per team) versus using the entire season (100 ...
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72 views

How to set the index valued M in Hierarchical model to compute Bayesian model probability?

I'm incorporating a Bayesian Model Averageing(BMA) approach in my research and strapped in trapped in the estimated of Pr(theta|D). Professor John K. Kruschke(2014)'s book in chapter 10 offers an ...
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1answer
578 views

Bayesian 1 sample t-test (paired / repeated measures)

I'm a neuroscientist trying to move away from frequentist to bayesian statistics, please bear with me... I'm after a hypothesis test on some of my data: e.g., let's say I have reaction times for two ...
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1answer
212 views

How can I identify market regimes with a Hidden Markov Model?

I am trying to identify market regimes (2 states: bull or bear) with percent changes in equity returns. Can you help me in the mathematicl modeling of this? So far, I thought that for each day, there ...
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910 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
68 views

Shrinkage in hierarhical models based not on observations

When we have a hierarchical model, such as: $$log(y_{i,t})=\beta_0 + \beta_i*log(x_{i,t})+\epsilon_{i,t}$$ Where $\beta_i$ ~ $N(B,\Sigma)$, and the sampling model is normal (normal disturbances.) ...
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83 views

Hierarchical Bayesian model or ensemble of predictors?

My model has 3 independent parameters $\{\rho, \alpha, \beta\}$ (polar coordinates), and a set of observables $\{Q_i\}$ and $\{T_{ij}\}$ where $i=1,2,...,642$ and $j=1,2,...,Q_i$ (if $Q_i=0$, there is ...
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168 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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319 views

Implementing a hierarchical bayesian model with latent independent and dependent variables for spatial analysis (in stan)

I am moderately familiar with frequentist hierarchical modeling, structural equation modeling, and hierarchical structural equation modeling. I am also moderately familiar with bayesian graphical ...
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2answers
2k 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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213 views

Translating user-defined joint-distribution from PyMC to PyMC3

I'm attempting to set up a simple beta binomial hierarchical model with an uninformative prior in PyMC3. I've read that the uninformative prior for this model should have alpha and beta hyper-...
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1answer
397 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 $\sim \mathcal{N}(0, l_k)$ ...
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278 views

How to run a count time-series multi-level Bayesian regression in R?

I have an upcoming project that involves the following: A client will provide measurements of traffic counts on a daily basis over the period of a calendar year for about 60 out of 300 locations. ...
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1answer
160 views

pyMC produces values outside range of uniform distribution while sampling from Bayesian hierarchical model [closed]

I have a hierarchical Bayesian model consisting of a Uniform prior distribution, between a minimum and maximum value (hyperparameters) at the top level of the hierarchy. I sample a "mean" from the ...
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1answer
211 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 ...
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147 views

what is the use/meaning of taking the partial derivatives of a joint distribution?

This is probably too broad, but is worth asking: Assuming an unknown distribution (from which you would like to sample), is there any benefit in looking at the gradients of the joint with respect to ...
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68 views

How to construct prior for two variables based on known distribution of their product?

Building a hierarchical Bayes model, and I am interested in Bayesian inference of two parameters $a > 0$ and $b > 0$. Right now I am using uninformative priors on both $a$ and $b$. But I ...
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31 views

How do I model of Y| X, M when data has only X and Y not M, an upper bound on Y?

Suppose I have a set of Bernoulli random variables $y_j$, corresponding to molecules in an excited or unexcited state. I have a random variable $Y$ which is the concentration of the molecules in a ...
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125 views

Seeking help in Bayesian Mixed Effects Model

I am implementing a Bayesian Mixed Effects model in my research problem. The model is written as, $y_i = X_i(\alpha + \beta_i) + \epsilon_i$, where $i = 1, 2, \ldots, m$ is the index of response, $j = ...
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1answer
534 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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134 views

How the De Finetti's Representation Theorem works in this case?

For the special case of infinite sequence of $\{0,1\}$ valued random variables the theorem is stated as $$ Pr(x_1, \ldots, x_n) = \int_0^1 p^{(\sum_{i=1}^n x_i)}[1-p]^{(n - \sum_{i=1}^n x_i)} dQ(p)\,. ...
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78 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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166 views

Multiple linear regression as a Hierarchical model in Bayesian framework, cant solve

In lecture notes on introductory graduate course on Bayesian statistics, there is a short discussion of how Multiple linear regression may be treated in the paradigm of "borrowing strength" aka "...
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1answer
307 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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1answer
214 views

How did you learn Bayesian statistics and what would you recommend as a reliable source? [duplicate]

I'm running a Bayesian model and I'm stuck on some aspect of the model that I have difficulty to understand. Since my knowledge about Bayesian inference is limited, I would like to have some good ...
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1answer
584 views

Extending a Hierarchical Beta-Binomial Model to account for higher-level groups

I searched all over but was unable to find an answer to this question. Please forgive me if I missed something obvious. In order to analyze an experiment, I recently implemented a Hierarchical Beta-...
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268 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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130 views

How does this Sampler work for the Concentration parameter of Dirichlet Process?

I am puzzled by how this Gibbs sampler on section 6 of Escobar & West (1995) works. To put it in simple words, the aim is to sample $\alpha$. The defined terms are: $$\eta\sim \texttt{Beta}(a,b)$$ ...
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1answer
137 views

Sampling Concentration Parameter of DP according to Escobar and West

I am reading Escobar&West paper and in particular am interested in their Gibbs sampler for the concentration parameter of Dirichlet Process (eq 13, eq 14). Given this, $$p(\alpha,\eta|k)\propto p(...
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1answer
102 views

Escobar and West Sampler for Dirichlet Process Parameters

I am reading Escobar&West paper and in particular am interested in their Gibbs sampler for the concentration parameter of Dirichlet Process. The issue I have is at the end of their section 6, ...
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112 views

Bayesian mixed ANOVA

I am analysing temperature series measured by different sensors under different conditions controlled by three experimental factors. I am interested to learn how much of the variability in the ...
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60 views

Sampling concentration parameter of DP via Slice sampling?

Is there a published work which shows how sampling the Dirichlet Process's concentration parameter can be done via Slice sampling?
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1answer
797 views

Bayesian estimation of Dynamic Linear Models with RStan

I'm reading the Dynamic Linear Models with R book, where most of chapter 4 is devoted to bayesian estimation of parameters. They code most of it manually though, and it seems it can get quite tricky ...
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1answer
96 views

resampling hyperparameters in a Hierarchical Dirichlet Process

The sampling scheme for the hyper-parameters of hierarchical dirichlet process (HDP) is explained in the appendix of the original paper by Teh et al. I agree that the auxiliary variable $s_j$ is a ...
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1answer
433 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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274 views

Can we use MLE estimates as hyperparameters of bayesian linear regression?

Given a linear regression \begin{align} y_i = \mathbf{x}_i^T \mathbf{b} \qquad i = 1,..,N \end{align} or in matricial form: \begin{align} \mathbf{y} = \mathbf{X}^T \mathbf{b} \end{align} MLE ...
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606 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 ...
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3answers
202 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 ...
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1answer
78 views

Power of Uniform Distribution?

In the Bayesian analysis, $\mathtt{rjags}$ in particular, it is very frequent to see the code: sigma ~ dunif(0, 100) sigma.1 <- pow(sigma, -2) But, what does ...
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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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390 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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1answer
40 views

If I have a nested multi-level model, how can I find the conditional expectation easily of the middle variable?

Suppose I have the following model: $$ y_i | x_i, V_1 \stackrel{ind}\sim N(x_i, V_2) $$ $$ x_i| V_1 \stackrel{iid}\sim N(0, V_1) $$ $$ V_1 \sim Unif(-V_2, \infty) $$ where the data is $y = (y_1, \...
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1answer
906 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-...
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268 views

How to specify the Bayesian version of a clustered-robust standard error OLS in BUGS/JAGS or Stan?

I am trying to reproduce a simple OLS model fitted with clustered-robust standard errors within the Bayesian framework (be it with BUGS/JaGS or with Stan). In R, my frequentist model is the following:...
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1answer
147 views

Relationship between 0-1 Loss and error Type I and II in Neyman Pearson

In the context of hypothesis test $$H_0:\theta\in \Theta_0$$ $$H_1:\theta\notin \Theta_0$$. Find the relationship between the 0-1 loss defined by $$L(\theta,\delta)= \begin{cases} 1-\delta & \...
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1answer
198 views

Marginal likelihood for simple hierarchical model

Suppose that $X$ is a $k$ dimensional normal variate with diagonal covariance matrix. $$ X \sim N(\mu, \Sigma), $$ where $\Sigma=\textrm{diag}(\sigma_i^2)$. The problem I am trying to solve it to find ...