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

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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 ...
4
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44 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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17 views

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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24 views

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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10 views

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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36 views

Setting up posterior and likelihood of Bayesian for more than one model

If I have a data-set and I would like to fit a model and determine its two or three free parameters, while I know that I can fit twice or three times the model to my data and obtains the free ...
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51 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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82 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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29 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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29 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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29 views

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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14 views

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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65 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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16 views

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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18 views

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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37 views

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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33 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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20 views

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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142 views

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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34 views

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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39 views

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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16 views

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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101 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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64 views

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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26 views

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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35 views

MCMC: The posteriors are too narrow

I have this very simple MCMC there I have ...
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38 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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22 views

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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43 views

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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24 views

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 ...
2
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1answer
27 views

Iteratively solving for prior probabilites.

I'm using Bayes theorem to classify data into two groups, where the conditional probability is known but the prior is not. So I assume that the ratio of prior probabilities is 1 and calculate the ...
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48 views

Bayesian credible intervals: “superiority” even if 1 is included?

In a recent medical publication comparing a cardiac device to anticoagulation ("blood thinners") using a Bayesian statistical model to evaluate the efficacy of preventing strokes and cardiovascular ...
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45 views

Is summing posterior probabilities valid for classification problems?

A classification for two mutually exclusive problem can be formulated by having a decision hinge on whether $P_0(x) > P_1(x)$ or $P_0(x) < P_1(x)$ where $P_0(x)$ and $P_1(x)$ are posterior ...
2
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30 views

marginal conditional distribution from MCMC output [duplicate]

I have a MCMC sampler that targets $$\mathbb{P}(U_1,U_2,...U_n \mid G(U) \leq 0)$$ where $U=(U_1,U_2,...U_n)^T$. I realize now I am more interested in estimating the conditional density $$p_k = p(u_k ...
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13 views

How can i get prior information using my few data set from the whole data? [duplicate]

I have a data set (x1...x500, y1....y500 ) I want to know about bayesian regression I want to know the prior information , few data set(400) from the whole data (500) using MCMCregress( packages in ...
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1answer
53 views

Posterior covariance from GPML toolbox

I am currently using the GPML toolbox to perform regression. Generally, after learning the hyperparameters we can extract the posterior mean and variance by using the function in the toolbox as ...
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A problem with deriving posterior of particle filter [duplicate]

At the normal particle filter, there is an equation to deriving posterior $p(x_{0:k}|z_{1:k})$. In the article ,equation 45 , it says that: ...
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1answer
42 views

Question about posterior mean calibration

I'm reading the article "Prior distributions for variance parameters in hierarchical models" by Andrew Gelman(link). This is an extract that I don't understand very well: Posterior inferences can ...
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57 views

Why is the most probable assignment for all variables in MRFs called MAP assignment?

I am new to graphical model, especially Markov Random Fields. I have a question about MAP assignment. Let say we have the graph structure and all the potential functions. MAP estimation is finding ...
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22 views

Ranking estimation with partial data

Consider a problem where we ask a number of people to select and rank their top three choices out of a number of options. The set of options is the same for everyone, and they all have to rank their ...
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40 views

What does posterior “over” parameters $\alpha$ exactly mean? [closed]

From my understanding the posterior "over" parameters $\alpha$ is $$p(D|\alpha)$$ and not $$p(\alpha|D),$$ is it correct?
2
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1answer
42 views

Derivation of Bayesian Posterior

Simple question here about deriving posteriors. Suppose I have some likelihood in mind for my data, $p(y|\theta)$, and I also have a particular conjugate prior in mind $p(\theta)$. Now, I have ...
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39 views

posterior predictive of standard Normal

I want to derive the predictive posterior distribution of $y$ for the case where $$p(\tilde{y}|\theta) \sim N(\theta,1)$$ and $$p(\theta|y) \sim N(\bar{y},\frac{1}{n}) \sim N(6,\frac{1}{9})$$. By ...
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64 views

How to sample the degrees of freedom of a Wishart distribution?

SHORT VERSION: Given K precission matrices drawn from a single Wishart distribution, I try to infer the degrees of freedom of this Wishart. How can I do it? Is there some place where this derivation ...
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21 views

Overestimating variance with MCMC

I'm working with a very specific type of proposal distribution in MCMC algorithm. To validate it I use a simple multivariave Gaussian with $\mu=0$ and $\Sigma$ an identity matrix. The proposal ...
4
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2answers
84 views

What are the parameters of a Wishart-Wishart posterior?

When infering the precision matrix $\boldsymbol{\Lambda}$ of a normal distribution used to generate $N$ D-dimensional vectors $\mathbf{x_1},..,\mathbf{x_N}$ \begin{align} \mathbf{x_i} &\sim ...
4
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Why do we lose conjugacy when assuming unknown $\mu$ and unknown $\sigma^2$ in a normal distribution?

Model: The following model corresponds to samples drawn from a Gaussian distribution with unknown mean and unknown variance: \begin{align} x | \mu, \sigma^2 &\sim \mathcal{N}(\mu, \sigma^2 )\\ ...
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204 views

How to calculate the Bayesian posterior analytically and by simulation?

I am working with this model: Prior: $P(\lambda)$~ N(0, 1), only the positive part likelihood: $P(x) = 1 - e^{-\lambda x}$ or $P(\vec{x}|\lambda)=\prod(1-e^{-\lambda x})$ Posterior: ...
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Incorporating Risk Aversion in Bayesian Expected Loss functions

In Berger's Statistical Decision Theory and Bayesian Analysis, he presents the following expected loss function for decision theory: $\rho(\pi^*,a)=\int_\Theta L(\theta,a)d\pi^*(\theta)$ Where ...