Questions tagged [hierarchical-bayesian]
Hierarchical Bayesian models specify priors on parameters and hyperpriors on the parameters of the prior distributions
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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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Conjugate inverseGamma posterior and Multivariate normal?
If I have a multivariate normal distribution for the mean with an InverseGamma for the variance.
Lets say $$ p\left(\mu, \sigma^{2}\right) = NIG\left(m_{0}, V_{0}, a_{0}, b_{0}\right) = N\left(\mu|...
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Could laplace method be applied to discrete distribution such as bernoulli, poisson-binomial etc?
I am trying to reproduce the result of the essay:
Variational inferences for partially linear additive models with variable selection(K. Zhao, H. Lian,2014)enter link description here
and encounter a ...
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Recovering samples from a density estimation with an additional prior on the samples. Used for Gibbs sampling
Abstract Idea:
Given a noisy measured density ($d_j$ at position $p_j$) and a density model, sample from the model parameters under the following stochastic model:
Stochastic Model:
Prior for model ...
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What is the correct bayesian formulation for the zero-truncated Poisson lognormal model?
In ecology we use compound distributions to describe species-abundance data. One example is the Poisson Lognormal (PLN) distribution which is a Poisson distribution with rate parameter $\lambda$ that ...
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How to combine priors for empirical bayes
I am trying to estimate $Z_{i} = P(Y=1 | A=a_{i}, B = b_{i}, C =c_{i})$ using something like empirical Bayes as in http://varianceexplained.org/r/empirical_bayes_baseball/
aggregating a massive amount ...
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Question about likelihood function in Bayes Rule and marginalization
The following is an instance of Bayes' Rule:
$$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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Is it appropriate and recommended to use a multi-level model when the data isn't strictly hierarchical?
I am running a between-subjects vignette-study in which participants get 3 highly similar versions of an abstract: one neutral version, one version with treatment X, and one version with the opposite ...
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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),$$
...
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What means that the model is "singular"?
My question might be rather basic, theoretical.
I am running spatial and spatial-temporal bayesian models in INLA. I have areal data and a continuous response variable with spatial and temporal ...
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Hierachical Bayesian modelling using brms: how to insert a prior that reflects cut-offs of Reaction Times distribution?
I am running a hierarchical Bayesian model using brms on reaction times (RTs) of a GoNogo task. The predictors are categorical and include the 3 stimuli/condition that participants observed and the 2 ...
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brms: prior of categorical predictors in Multilevel Bayesian model
I want to run a Bayesian Multilevel model on reaction times using two categorical factors (conditionStimuli = 3 levels; sequenceTrials = 2 levels). Initially, I run the model with default priors:
<...
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Hierarchical model for uniform random variable
I am thinking about the following model:
$$
\theta \sim \mathcal{U}[c- \epsilon, c+\epsilon],\\
x \mid \theta\sim \mathcal{U}[\theta - \epsilon, \theta + \epsilon].
$$
I want to derive the marginal ...
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In Bayesian Statistics, can the data be a random variable drawn from the posterior of a separate model? Will the uncertainty flow through?
In the traditional Bayesian hierarchical approach, you typically have a hierarchy built on your coefficients. That is to say, you might have your coefficient of interest distributed around some ...
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Exchangeability, causal inference, and partial pooling
In Statistical Rethinking, Richard McElreath writes the following concerning the use of partial pooling (i.e. varying/random effects) in Bayesian hierarchical models:
Could we also use partial ...
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Understanding Exchangability in Bayesian Hierarchical Models
I have a group of $k$ experiments (indexed by $j=1...k$) and each experiment $j$ produces a set of $n_j$ datapoints denoted as $y_{ij}$ such that $i\in\{1,...,n_j\}$. Meanwhile, $y_{ij}$ is ...
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Minimum sample size to obtain 5 "successes" with high probability using Bayesian estimation and a pilot dataset
I have a population of patients, each with a given type of disease ($D$). The set of disease types is mutually exclusive and exhaustive. Each patient is tested for the same genetic variant, for which ...
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Understanding Bayesian Hierarchical Model in Practice
I have a Bayesian hierarchical model with datapoints $y_{ij}$ which are samples from distributions with parameters $\theta_j$. For each distribution parameter $\theta_j$, there are $n_j$ datapoints ...
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How do updates of parameters in bayesian hierarchical models work?
I'm starting my way in the bayesian data analysis and I'm struggling to understand one thing. Let's say we define a hierarchical model like this one:
$$
y_i| \theta _{g_{i}} \sigma ^2 ∼ N(\theta_{g_{i}...
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Data-informed grouping of covariates in Bayesian Hierarchical Modeling?
Is there a way to place a prior on the first stage's betas that allows the second stage groups to be determined from the data? I am working with co-exposures where I am not super confident in how they ...
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Why is my proportion estimate 0? (BEST MCMC Bayesian inference)
I am trying to run a simple Bayesian inference on my y1 vector as shown below.
...
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How to improve model accuracy fitted by extreme skewness distribution in brm()
I first fit a bayesian model by brm() in R. Because I’m not very familiar with Bayesian frameworks, I didn't specify priors distribution. My data ranges from -1~12 ...
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One-step prediction in JAGS of a dynamic model with unknown variance
I have the following problem. I have a linear dynamic model as follows: $$\theta_{0}\sim N(0,10)$$ $$v,w\sim \text{InverseGamma}(0.1,0.1)$$ $$\theta_{t}\sim N(\theta_{t-1},w), \hspace{0.3cm} y_{t}\sim ...
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Interaction term to hierarchical model
Assume we have a response variable of interest Y and a predictor of interest X1, that might be associated with another predictor X2, and we do a linear regression with interactions: Y = B0 + B1X1 + ...
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Conjugate Hyperpriors
I heard it was possible to have a Bayesian model with likelihood, prior and hyperprior that has a posterior of closed form, by choosing a conjugate prior and conjugate hyperprior. But I struggle to ...
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Bayesian hierarchical model inference problem image segmentation
it might be really confusing question. I am working on my thesis and I am stuck at a problem.
It's a problem in image segmentation and finding parameters of border lines of continuous region in an ...
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Using Hierarchical Gaussian Linear Regression to Inference of a Breakpoint Position
Edit: this question is not a duplicate of imposing a perpendicularity constraint in Gaussian linear regression. Here, the question is about hierarchical reference of the breakpoint position. The other ...
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Modeling with Negative Probabilities?
Background:
As I understand the role of probability in Quantum Mechanics, the idea is that no observable event can have negative probability, but that it can make sense for unobserved quantities to ...
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Posterior-understand troubles in Bayesian Normal hierarchical model
I´m studying the Bayesian Data Analysis book, third edition (link at the end), and has a little trouble dealing with some algebra... here is the context (See pages 113-117)
$y_{ij}|\theta_{j}\sim N(\...
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A posterior predictive check using R
Question:
On page 120,
the data from the SAT coaching experiments were checked against the
model that assumed identical effects in all eight schools: the expected order statistics
of the effect sizes ...
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Predicted treatment effect for causally related conditions
Let's say there are two medical conditions $A$ and $B$ that are causally-related, i.e. that they share a common etiological process $C$. For example, $A$ and $B$ could be related autoimmune conditions ...
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Analyzing a likert-type item data with repeated measures with logistic ordinal regression
I'm analyzing some Likert-type item for my thesis. After a quick research, I figured out, that instead of using a least-squares regression as conventionally, a logit or probit ordinal regression model ...
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