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

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Summing multiple posterior distributions

I have obtained separate posterior distributions of regression coefficients of several variables, and would like to know what the most probable sum of these coefficients is. This is because the sum of ...
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Latent Dirichlet Allocation yields different posterior distribution than simple Bayesian model

Method A: out of the box LDA I am using a package to run LDA on a sample of size m with n words in the vocabulary. The end ...
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Marginal Posterior Distribution of Random Effects in Bayesian Logistic Regression

Suppose I'm fitting a logistic regression, and I would like to include individual variability into the estimation process via a random effect. So I have something like: $$ y \sim ...
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36 views

Is this posterior probability integral right?

From Wiki: where , k is binomially distributed, and I'm not sure about u. I'm thinking that the second line should be: I mean, if we let X represent the toss of a die, then $P(X = 1, 2, ...
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Is this denominator of a posterior distribution the marginal distribution of Y?

From Wikipedia: , where Is the denominator (above pics are from Wiki) the marginal distribution of Y? Intuitively, it seems that way so that when we cross-multiply, LHS and RHS are mirrors. ...
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How do frequentists guess a distribution?

With competing hypotheses such as testing if a coin is fair, frequentists and Bayesians have their own approaches. What about for coming up with a distribution? In An Essay towards solving a Problem ...
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What exactly does it mean to and why must one update prior?

I'm still trying to understand prior and posterior distributions in Bayesian inference. In this question, one flips a coin. Priors: unfair is 0.1, and being fair is 0.9 Coin is flipped 10x and ...
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how to model and implement bayesian change point with longitudinal data

I am working on Bayesian change point of Poisson data with different identities that is longitudinal. I have understood to degree the hierarchical structure of the posterior with hyper priors for data ...
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Metropolis Hastings Algorithm - Prior vs Proposal vs Numerator of Bayes Theorem

I've been using this technique in 'black-box' form for a little while as a physics student. I have been struggling to understand what's happening under the hood for some time and I think I almost ...
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Covariance of noise in Posterior PDF for Bayesian General Linear Model

I'm reading about Bayesian estimation in Steven M. Kay's Estimation Theory vol. 1. I understand the basic philosophy behind the Bayesian approach, but I think there's a fundamental insight I haven't ...
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How do I compute the posterior predictive distribution of a logit model?

So I used stan to take samples from a logit model. I want to compute the posterior predictive distribution of this model, but I am having trouble figureing out the logit link function and how it ...
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Reference prior for a three-parameter model and likelihood factorization

Let a (regular) statistical model with three parameters $\phi_1$, $\lambda_2$, $\mu$, and three observations $x_1$, $x_2$, $y$. Assume the likelihood has form $$ L(\mu,\phi_1,\lambda_2 \mid y, x_1, ...
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Bayesian neural networks: very multimodal posterior?

Question: How do Bayesian treatments of neural networks address the fact that the posterior has an exponentially large number of modes? Background: There seems to be a lot of interest in Bayesian ...
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Bayesian approach for comparing the predictability of different datasets for another

Suppose I have three datasets A, B and C with not necessarily the same amount of data. Now, I want to know whether dataset A or dataset B is better in predicting C. I want to use a Bayesian approach ...
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23 views

How to use the output of a RStan model to predict test set data? [duplicate]

I'm uber-new to Bayesian statistics and am tasked with implementing a housing price estimator using RStan (the Stan package for R). I'm quite consfused as to how to design the modeling activity in ...
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74 views

Find a posterior distribution [closed]

I came across this task that I have no idea how to solve, because I'm not very good at statistics, so I was wondering if someone could help me understand it. 7 scientists with very different ...
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37 views

Using empirical priors in PyMC

I'm using PyMC to sample the posterior distribution and I've run into a roadblock with using priors from samples, not models. My situation is as follows: I have some empirical data for a parameter ...
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Estimating posterior probability using kernel frequency estimation

given a dataset containing a numeric random variable $X$ and a class label $Y=\{+,-\}$ , the posterior probability $P(y|x)$ should be estimated using kernel frequency estimation. I can't seem to ...
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Computing marginal posterior in a multivariate setting

I am computing posteriors using individual level data and would like to know if my formulation in the end is right. Let the sequence of choices made by individual $i$ be $y = y_1, y_2 ......y_j$ in ...
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48 views

A simple question about MAP and MLE

I recently got this simple question from a friend. But I am quite confused about it. Suppose we toss a coin $N$ times, and got heads $m$ times. Assume the binomial distribution with $p$ which is the ...
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33 views

Prior predictive density given by $f(y) = {f(y\mid \lambda) g(\lambda)}\big/{g(\lambda | y)}$?

(I guess stats.SE is the right place for this) I'm reading Albert's book "Bayesian computation with R". To get theprior predictive density, he extensively uses this formula $$f(y) = \frac{f(y\mid ...
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55 views

Derivation of Normal-Wishart posterior

I am working on the derivation of a Normal-Wishart posterior but I'm stuck at one of the parameters (the posterior of the scale matrix, see at the bottom). Just for context and completeness, here is ...
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Combining independent MCMC samplers from different models

I am interested in sampling from the joint posterior $p(\theta_k,k \mid y)$ where $\theta_k$ belongs to the parameter space of model with index $k$. One way of doing this is with the reversible jump ...
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Why can I use the posterior probability of a classifier as a new classifier?

I have read that, when doing discriminant analysis, you can use the posterior probability you obtain using your classifier as a new fine-tuned classifier. Can anyone talk me through the rationale of ...
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How to choose between two models on the basis of the normalised posterior distributions?

Suppose you are given two normalised posterior densities $\pi_1(\theta|y)$ and $\pi_2(\theta|y)$, based on the data $y$, and arising from model 1 and model 2, respectively. You are asked to find ...
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60 views

Introductory examples in computational statistics class

I'm looking for an example of Bayesian inference for a class with the properties: The problem is easy to state, and the model & prior are both pretty reasonable, and R can't really calculate the ...
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125 views

Interpreting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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46 views

Posterior Conditional on Beta in Bayesian Linear Regression with Factor Analysis

This should be an easy question if you're familiar with the terms involved. I am performing some research using a hierarchical Bayesian regression model that incorporates factor analysis into the Beta ...
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62 views

Posterior predictive for Gamma distribution with unknown scale and shape

I have a question that needs clarification. The posterior predictive distribution can be described as the distribution that a new i.i.d. data point $\tilde{x}$ would have, given a set of $N$ existing ...
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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 ...
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Can Platt Scaling to calibrate probabilities be used for classifiers other than SVM?

I am using Gaussian Mixture Models as classifiers and I compute posterior probabilities from them for a 2 class problem. However, the probabilities are pushed towards 0 and 1 due to very skewed ...
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187 views

Evaluate posterior predictive distribution in Bayesian linear regression

I'm confused on how to evaluate the posterior predictive distribution for Bayesian linear regression, past the basic case described here on page 3, and copied below. $$ p(\tilde y \mid y) = \int ...
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Binary decision, evaluating Bayesian probit regression?

Laplacian logistic regression. I have a training set of data and an evaluation set. The response is binary. I have to verify the models by calculating posterior predictive on the evaluation set. Last ...
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Predicting with Posterior PMF(discrete non-pram dist) ? Predictive posterior distribution?

Sorry in advance if my question is abit awkward, I'm somewhat confused because most of the tutorials on the Internet mention that you should use Posterior-predictive dist to predict new data. The ...
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Sampling Twice and Posteriors

I have a random variable with some unknown distribution with support over $[0, 1]$. Every turn, I sample a $p_t$ from this distribution. However, I am unable to observe $p_t$ directly. Instead I ...
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36 views

How can we estimate the predictive interval in Lasso regression

Dear Community members, I am using lasso to solve an inverse problem (a Fredholm) which I can reframe as \begin{equation}\min_{\mathbf x ~~{\rm with}~~x_n\geq 0} \ell_{\rm Lasso}(\mathbf x, ...
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Posterior predicted distribution, practical question

I'm new here to this place but I have already learned so much here. Yet I still remain with a large question involving my thesis in econometrics and medical scoence. For a starter, I have read ...
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What to do when your likelihood function has a double product with small values near zero - log transform doesn't work?

I currently have a likelihood function defined as the following: $$ L=\prod_{i=1}^{N}\left[\prod_{s=1}^{S_i}L_{is}(y\space|\space \rho_A)\times\phi + \prod_{s=1}^{S_i}L_{is}(y\space|\space ...
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Finding the posterior distribution

Suppose that an observation $X$ is drawn from the following distribution $f(x|\theta) = \begin{cases} \frac{1}{\theta} & \text{ if } 0 < x < \theta \\ 0 & \text{ if } otherwise ...
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How to include a pattern for'unknown' for an SVM classifier?

I am doing a classification of heart beat with SVM. There are five kinds of beats in my training data. I plan to add a new kind of data named 'unknown' beat. If there is no unknown beat, one ...
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Verifying posterior distribution is proper when integration is not feasible

I am wondering how we can verify the posterior distribution is proper if the expression we have for it is difficult (or impossible) to integrate. For example, I have a model that is based on a ...
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104 views

Calculating marginal distribution via integration

Suppose that we have an IID random sample $\mathbf{x} = (x_1, \dots, x_n)$ from a given distribution with the following PDF: $$\theta (1 - e^{-x})^{\theta -1}e^{-x}, \, x > 0, \, \theta > 0$$ ...
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Posterior predictive test quantities

I've been trying to figure out problem 6.2 from Gelman's book, second edition, page 192 on Bayesian data analysis. Can anyone help? a) Set up predictive test quantities to check the following ...
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Combining several posteriors

Is there an accepted method of combining the posterior distributions from a model fit to several participants to obtain a posterior for the entire group of participants? The reason I am asking is ...
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MCMC: The posteriors are too narrow

I have this very simple MCMC there I have ...
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50 views

How to get the error and error variance after hierarchical bayes using bayesm

I estimated individual part-worth based on multiple paired comparisons by using Bayesian logistic regression in R (bayesm, model: rhierBinLogit by Rossi). rhierBinLogit implements an MCMC algorithm ...
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How to calculate the posterior probabilty of Gaussian Mixture Component

If the mean vector and the Covariance matrix of a Gaussian Mixture model are known, how could I calculate the posterior probability of each of the Gaussian Component in the mixture.
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Why is it called “mode” in MAP estimation?

When estimating parameters with MAP, why is it written that we are estimating the "mode"? I thought it would be the mean of the posterior distribution?
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Why does Empirical Bayes work in my simple case?

I have a problem where I am trying to classify data into two groups using a single parameter. The distribution of this parameter is Gaussian for two groups, so what I'm dealing with is two overlapping ...