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

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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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11 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 ...
4
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1answer
49 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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3 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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1answer
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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30 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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19 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 ...
3
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2answers
114 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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2answers
33 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 ...
4
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2answers
34 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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9 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 ...
4
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2answers
96 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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1answer
41 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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25 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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34 views

MCMC: The posteriors are too narrow

I have this very simple MCMC there I have ...
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1answer
28 views
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34 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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16 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.
3
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1answer
42 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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21 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
25 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 ...
2
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0answers
47 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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1answer
44 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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1answer
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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12 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
48 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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13 views

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
39 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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2answers
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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1answer
39 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
38 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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38 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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50 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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19 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
votes
2answers
72 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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1answer
80 views

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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2answers
201 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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24 views

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 ...
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1answer
66 views

Posterior probabilities with decision trees or decision forests

Is there a way to get posterior probabilities $P(C | \vec{x})$ (probability that a data item $\vec{x}$ belong to one of the given classes) in a multiclass classification problem using decision trees ...
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1answer
38 views

Clarification on a paper regarding estimating N from a Binomial Distribution

I was wondering if someone could clarify the following for me. In the paper "Inference for the binomial $N$ parameter" by Adrian Raftery, his first example outlines the posterior of $N$ given $x$ as ...
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1answer
74 views

Skewed posterior distribution on constrained parameter space for Bayesian inference of MCMC. Advice on what to do?

I am running a fully Bayesian MCMC procedure to estimate some time series models, and my model has a lot of parameter estimates. In particular, one of these parameters, $\phi$, is $\in [-1,1]$. The ...
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27 views

find posterior distribution parameters by simulation

I've calculated the posterior distribution parameters of a variable X analytically and by simulation. But doesn't mach. X ~ Normal(mu,s=6).And the prior distribution of X is a Normal(mu=100,s=20). As ...
6
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1answer
160 views

Unrealistically high significance when marginalizing over large number of parameters

The setup I've tried to reduce the problem to a self-contained subset for this question, but it still ended up being pretty long. Sorry about that! I have a set of observations of a set of objects ...
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1answer
55 views

General questions on MCMC

This is a continuation of the following question. The previous link was related to rejection sampling. This is related to MCMC. General questions on rejection sampling 1a. As far as I understand, ...
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7 views

How do I calculate posterior probabilties from HMM?

The title may be misleading but I don't know if a better one exists. Consider the following figure: Basically, I want to replicate that. Its a bit easier for me because my HMM only has 12 states. ...
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82 views

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): ...
6
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1answer
250 views

Posterior predictive check following ABC inference for multiple parameters

I am relatively new to Bayesian statistics so please be gentle. I have just performed Approximate Bayesian Computation (ABC) for the inference of a multi-parameter model. Now I am looking to perform ...
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21 views

Meta analysis, joint posterior distribution of study effect

Meta analysis (with common study variation $\sigma$) often assumes that: $$ X_{i,j} | \theta_i \overset{ind}{\sim} N(\theta_i,\sigma)\\ \theta_i \overset{i.i.d.}{\sim} N(\mu,\tau) $$ where ...