# Questions tagged [posterior]

Refers to the probability distribution of parameters conditioned on data in Bayesian statistics.

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### Dealing with dependent data in a Bayesian model

Background: Consider a series of dependent data points, $$y_1,y_2,y_3,\cdots,y_N.$$ In cases where the dependence is well described by an exponentially decaying auto-correlation function, it is ...
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### Including feature-dependent priors on output class, in bayesian logistic regression

When doing logistic regression with data $D_N = \{(x_i, y_i)\}_i^N$ with $x_i \in \mathbf{X}^N$ (each data point has N features) and $y_i \in \mathbf{Y}$ being assigned output classes, in a Bayesian ...
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### Implementing Predictive Posterior Distribution Using Stan

Background I had an example that sought to demonstrate the posterior predictive distribution in the context of a normal measurement model. The data that was used is as follows: ...
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### Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
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### Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
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### Parameter Estimation for Naive Bayes - Maximum a posteriori and Maximum Likelihood

I am wondering if I understand those terms correctly. To summarize my thoughts: In naive Bayes, our decision rule is basically the Maximum a posteriori (MAP) estimate of our hypothesis. We assign an ...
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### 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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### Density estimation/approximation from MCMC samples

I'm looking to accurately describe the density function of a multivariate posterior probability distribution based on samples from MCMC. As far as I know, in most cases this is done either with a ...
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### How do I sample from the posterior distribution with gamma likelihood with unknown alpha and beta?

I realize that this Wikipedia page provides the proportional form of the conjugate prior to the gamma distribution with unknown $\alpha$ and $\beta$ parameters, as well as the posterior values of $p$, ...
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### Posterior distribution under Cauchy prior?

I have a (I hope) simple question! If I had a linear regression, $Y_t = \alpha + \beta X_t + \epsilon_t$ with $\epsilon_t \sim N(0,\sigma^2)$ and I assume a Cauchy prior for $\sigma$, is it ...
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### Derive ditribution for $\mu | Y_1,…Y_h,\rho$ (Bayesian stats)

I am trying to understand the following paper (http://www.ncbi.nlm.nih.gov/pubmed/20156954). Imagine we have H clinical trials with historical data on control group. $Y_1, ... Y_h$ - are estimates ...
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### Derive Marginal Posterior to set up Gibbs-Sampler

I am currently trying to replicate a Hierarchical Model for multivariate returns proposed in the paper Portfolio selection using hierarchical Bayesian analysis and MCMC methods. However, in order to ...
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### Does a proper prior lead always a proper posterior?

Does a proper prior lead always a proper posterior? I cannot check whether the posterior is proper, so I was wondering if this assumption is always satisfied .
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### Finding the Complicated Posterior Probability Distribution of $θ$

Suppose, we are given a likelihood function, $f(x|θ)$ corresponding to a shifted-exponential distribution and the prior distribution on the parameter $θ$ is a standard Cauchy distribution. Now I am ...
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### Is the posterior a sufficient statistic when observations are conditionally independent?

Suppose there are two random variables, $X_1$ and $X_2$, and we're trying to infer $\theta$. If $X_1$ and $X_2$ are conditionally independent, then is $f(\theta|X_1)$ a sufficient statistic for $X_1$?...
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### Implementation of Bayes posterior predictive check

I have a question concerning the implementation of a bayes posterior predictive check. Let us assume i have this model (implementation is in R and jags): ...
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### Marginal Likelihood Latent Variable Model

I am trying to apply the method proposed by Chib in Marginal Likelihood from the Metropolis Hastings Output to calculate the marginal likelihood of a logit model the includes latent variables. ...
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Let $X = ( x_1, ..., x_n )$ be $n$ samples from a normal distribution with unknown mean. What is the best estimator for this mean? I can think of at least 2 unbiased estimators: The empirical mean $... 0answers 272 views ### Determining the posterior distribution for an Autoregressive or order 1 model Question: For this question, note that the notation$y_{1:T} = (y_1, y_2, \cdots, y_T), ie, a vector of random variables. Consider the following AR(1) model: \begin{align*} y_{t+1} = \phi y_t + \... 0answers 117 views ### Practical problems with difficult posteriors I'm looking for difficult Bayesian inference problems to test out different Monte Carlo sampling methods. I've mostly been looking at Hamiltonian Monte Carlo based algorithms and in particular, I've ... 1answer 4k views ### Find posterior distribution for uniform distribution Given X with uniform distribution in the interval [μ,μ+θ]. Suppose θ is given. Find the posterior distribution with prior distribution on your own. From that, find the Bayesian estimator with ... 0answers 33 views ### Posterior distribution of\sigma^2$In chapter 9 of Jim Albert's Bayesian computation with R it's mentioned that, in the context of Normal Linear Regression, the posterior joint density is: $$g(\beta, \sigma^2 | y) =g(\beta|y, \sigma^2)... 2answers 47 views ### Laplace approximation in high-dimensions Obviously computing the inverse Hessian is hard when a probability distribution is fitted on high-dimensional datapoints. One idea to reduce computational cost would be to approximate the distribution ... 0answers 35 views ### Single point estimate of joint posterior distribution This is a very basic Bayesian beginner question about what to do when one does not want a full parameter distribution but only a single set of parameters. Example: Let's say we have a number of points ... 0answers 61 views ### How to determine the size of highest density region in high dimensions I want to calculate the "size" of the highest density region (HDR) that contains p% of the total probability for multivariate samples of a Bayesian posterior obtained via MCMC. In 1D this "size" is ... 0answers 21 views ### Bayesian inference problem with three parameters Assume that I observe a yearly time series for the number of certain events occurring per year. Also, assume that the data points come from a Poisson process with parameter \lambda. The dataset in ... 0answers 50 views ### Posterior distribution of Bernoulli distribution The pdf of X | \theta is given by \theta^x (1- \theta)^{1-x} and its prior distribution is given by p(\theta) \frac {1} {B(\alpha, \beta)} \theta^{\alpha - 1} (1 - \theta)^{\beta - 1} where ... 0answers 46 views ### Threshold crossing: Integrating posterior distribution over a forecast period Suppose I have some time series \{Y_t\}_{t=1}^T and I have come up with some probabilistic model:$$p(Y_{t+h} \mid Y_{1:t}, \theta)$$for any horizon$h$, and some parameters$\theta$. Now ... 1answer 52 views ### Approximating p-values from Bayesian posterior distribution I'm preparing a manuscript for publication in which I have fit a mixed linear model using Bayesian regression. I'm assessing whether group A is bigger than group B. In the paper, I have reported the ... 0answers 36 views ### Posterior with a much larger uncertainty than the prior I have done an MCMC analysis with many variables. One of my nuisance parameters has a Normal prior distribution with mean 0 and standard deviation 1. The posterior distribution for this parameter has ... 0answers 59 views ### Multivariate bayesian inference: learnig about the mean of a variable by observing another variable I want to derive a Bayesian learning procedure where I don't only learn from my own signal, but also from other signals which are correlated to mine. I thought it could simply work with Bayesian ... 0answers 52 views ### Finding the mode of the posterior distribution I have the following hierachical bayesian model -$\mathbf{x}|\mathbf{c},\sigma^2 \sim \mathcal{N}(\mathbf{x}|\mathbf{c},\sigma^2)\mathbf{c}|\mathbf{c}_1,\sigma^2_2 \sim \mathcal{N}(\mathbf{c}|\...
I would like to perform model selection given a range of $k$ models $\mathcal{M}_1, \mathcal{M}_2, ..., \mathcal{M}_k$, each with some prior probability $f(\mathcal{M}_1), \dots, f(\mathcal{M}_k).$ ...