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Questions tagged [posterior]

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

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bayesian decision making - comparing expected loss

The problem is like this: Suppose that I am considering which country should I invest on, country A and country B, based on their GDP growth rate $\alpha$. There are two possible choices for each ...
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1answer
26 views

Bayesian posterior pmf for weighted dice with uniform prior

We want to find posterior probability mass function for dice tossing with uniform prior. We are interested in rolling of weighted dice. The outcome is 1,2,...,6. We assume that prior probability ...
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Posterior distribution of mixture models

In the context of mixture models in bayesian inference, one can assume that the general form of the joint posterior for a mixture model of $k$ components is $$ \begin{equation} p( \boldsymbol{\...
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118 views

Joint posterior distribution of $(\mu,\sigma^2)$ in the Normal model

Find the joint posterior of $(\mu, \sigma^2)$ given Normal data. I've found the joint prior of $\mu$ and $\sigma^2$ (where $\displaystyle\sigma^2\sim\chi^{-2}(v_o,v_os_o^2)$ and $\mu|\sigma^2\sim N(\...
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The distribution of a posterior predictive p-value under certain assumptions

I am wondering if anyone can check my understanding of the following passage concerning posterior predictive p-values in the textbook "Bayesian Data Analysis 3rd Edition" on page 151: In the ...
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Likelihood raised to a power; how to set the power?

Suppose ${\bf{\theta}} = (\theta_1 , \ldots, \theta_d)$ and you have a posterior as below: $$\pi(\theta | D ) \propto L(\theta |D ) \pi(\theta)$$ Suppose we are in active learning setting and need ...
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Viterbi Algorithm vs Maximum of Variational Posterior for HMM

I have a HMM with observed values $x$ and latent values $z$, upon which I've performed variation inference to get an approximate posterior distribution $q(z|x)$. If I want to calculate a "most likely ...
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Setting Average posterior probability value in Stata traj plugin

In complement to Aikake Information Criterion, I want to use posterior probability to select the best model for group-based trajectory modeling. In this paper: https://drc.bmj.com/content/bmjdrc/4/1/...
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Simulating the Posterior Density of a Transformed Parameters

I am reviewing an example (p. 180-181, Example 11.3 and 11.4) from All of Statistics by Larry Wasserman. The example intends to illustrate that the posterior can be found analytically and can be ...
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Posterior sampling without using pm.Potential in pyMC3

I'm going through the Price Is Right example in chapter 5 of Probabilistic Programming & Bayesian Methods for Hackers and I have problems understanding the solution. I have tried to change the ...
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1answer
46 views

Bayesian consistency in compact uncountable parameter space

Let $p(y_i \mid \theta)$ be the likelihood we are using of a single data point, $p(\theta)$ be the prior, and $f(y_i)$ the true distribution of the data. Also, let $\theta_0$ be the parameter that ...
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1answer
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Posterior of random variable with normal prior and normally-distributed observation

Suppose $X$ is normally distributed with mean $10$ and standard deviation $1$. I take a normally-distributed noisy measurement of $X$ with standard deviation $0.1$, and the measurement is $5$. I am ...
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Estimating posterior probability from a random grid

I am simulating the evolution of galaxies, and want to find the distributions of input parameters that best reproduce an observation of a particular galaxy. I have a measurement $y \pm \sigma$ of a ...
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84 views

Failing to implement Bayesian Chi2 goodness of fit test

I am trying to implement one of the methods described in Valen Johnson's A Bayesian Chi-Squared Test for Goodness of Fit. It presents a couple of variants depending on whether the random variable of ...
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Posterior predictive for the beta distribution? [duplicate]

I'm looking to do Bayesian inference using the beta distribution. Are there any sources for a derivation of the posterior predictive? It's an exponential family, so it should have a conjugate prior, ...
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Basic question on proportionality in Bayesian Inference for Normal distribution

I have a nagging question regarding the Normal distribution and maintaining proportionality in Bayesian Inference. Say for example that: $\pi(\theta|Y) \propto L(Y|\theta)\pi(\theta)$ $Y | \theta \...
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What is the posterior kernel lengthscale of a Gaussian process?

If I have access to multiple samples from a Gaussian process with known covariance kernel but unknown parameters (i.e. unknown lengthscale), it is straightforward to estimate the lengthscale using ...
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Bayesian Linear Regression - Creating a distribution for a new prediction

I'm using MCMC to fit a linear regression model with the end goal of making predictions for new observations. See reproducible example below: ...
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How does one compute the posterior odds ratio between two models

Apologies if this question has been asked before but I couldn't find anything that matched my problem. I have some astronomy based data sets and wish to find which of three models fits the data the ...
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Confusion about the use of the MLE & the posterior in parameter estimation for logistic regression

In classification one usually computes $$ C = \operatorname*{argmax}_k p(C=k\mid X) $$ where $p(C=k\mid X)$ is the posterior distribution. In a simple logistic regression setting with $C \in \{0, 1\}...
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what do you mean by writing out the posterior distribution of µ up to a normalizing constant? [duplicate]

I think i am stuck at the basic what do you mean by writing out the posterior distribution of µ up to a normalizing constant ? How to compute the value of C here?
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Likelihood of one datapoint given $k$ models

Introduction I'm currently facing a problem where I'm constraining a set of (physical) parameters $\theta_k$ with $k\in [1,2,...,K]$ via several independent datasets. One of those datasets, however, ...
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How to get posterior probability from Bayes factor

I ran into this question in my class and am not sure how to solve it: A positive test result gives you a Bayes factor of 71 in favor of being sick. If your prior probability of Being sick was 0.05, ...
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Show posterior mean can be written as a weighted average of the prior mean and MLE

Suppose $Y_1, \dots Y_n$ are exponentially distributed: $Y_i | \lambda \sim Exp(\lambda)$. Find the conjugate prior for $\lambda$, and the corresponding posterior distribution. Show that the posterior ...
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Posterior and Predictive Density [closed]

Let X1 be a claim from an auto insurance policy. Suppose X follows an exponential distribution with rate lambda, where lambda follows a gamma distribution with mean 2 and variance 2. What is the ...
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What Is Meant by “Maximising” Posterior Probability?

My textbook says the following: The optimal coding decision (optimal in the sense of having the smallest probability of being wrong) is to find which value of $\mathbf{s}$ is most probable, ...
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Gaussian mixture model - does an improper uniform prior give a proper posterior?

We draw $n$ i.i.d. points $x_1 , x_2 , ..., x_n$ from the following Gaussian mixture: $$p(x|\mu_1,\mu_2) = \frac{1}{2} \text{N} (x|\mu_1,1) + \frac{1}{2} \text{N} (x|\mu_2,1).$$ The prior is the ...
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Finding the pdf of the posterior distribution and the corresponding Bayes Estimate of $\theta$

Suppose that we observe i.i.d random variables $X_1, X_2, \ldots , X_n$ having pmf $$f_{X}(x\mid\theta) =\theta(1−\theta)^{x−1}I_{\{1,2,3,\ldots\}}(x)$$ where $\theta\in(0,1)$. Consider the ...
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Why Does the $\propto$ Symbol Replace the $=$ Symbol When Using Bayes' Rule to Convert Posterior Density to Unnormalised Posterior Density?

My textbook says the following: In order to make probability statements about $\theta$ given $y$, we must begin with a model providing a joint probability distribution for $\theta$ and $y$. The ...
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549 views

Can anyone help explain this basic example of posterior

I am having trouble understanding the authors reasoning here. It is from "The Bayesian Choice" I am confused about why the posterior is initially written without depending on the data, and why we ...
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Normal - Inv chi squared posterior calculation

Given that for a known mean $\mu$ and unknown variance $\sigma^2$ the normal distribution is $$X_i|\sigma^2 \sim \mathcal{N}(\mu, \sigma^2) = \frac{1}{\displaystyle\sigma\sqrt{2\pi}}\exp\left[-\...
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How to obtain the posterior distribution of a given problem?

Problem: Compute the conditional distribution of a random variable $X$ given $Y$. If a random variable $X$ is Bernoulli distributed with probability $q$ for $X = 0$. The conditional ...
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Clarifying a proof of a particular paper on Steins Estimator

I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases. The doubt is a s follows: Given $$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
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Inference with Generalized Additive Models: Bootstrapping and Posterior Simulation

It has been argued in the statistical literature (and by Simon Wood, the creator of the well-known and widely-employed 'mgcv' R package that bootstrapping does not perform very well for Generalized ...
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When is the posterior distribution equal to the prior?

So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I ...
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Can a proper prior and exponentiated likelihood lead to an improper posterior?

(This question is inspired by this comment from Xi'an.) It is well known that if the prior distribution $\pi(\theta)$ is proper and the likelihood $L(\theta | x)$ is well-defined, then the posterior ...
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Slice sampling of a model with continuous and discrete parameters

I have a model with 5 continuous and 1 discrete parameter. I am using PyMC2 to implement slice sampling. I have a custom likelihood function that returns the log likelihood value that gets passed to ...
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1answer
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Proper prior leading to improper posterior

Preface I must say I am aware of previous discussions (e.g. this one) and also of this excellent, didactic proof using Fubini's theorem as presented by Jared Niemi [I'm not saying Jared Niemi is the ...
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Diffuse priors Bayes Factor

In textbooks I always read that it is necessary to have a proper prior on the parameter that we want to test with Bayes factor, otherwise we would always posteriori favor the model with less ...
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1answer
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Bayesian p-value in wrong direction using step function in JAGS / BUGS

I have estimated a Weibull regression model in JAGS using rjags and R2JAGS. The estimated posterior predictive p-values using the step() function confuse me. They make sense (comparing them to lower ...
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Predictive Posterior Distribution of Normal Distribution with Unknown Mean and Variance

Suppose that $x_{i}|\mu,\sigma^{2} \sim \mathcal{N}(\mu,\sigma^{2})$ for $i = 1,...n$. Assume that the assigned prior distributions are $\mu$ ~ $\mathcal{N}$($\mu_{0}$, $\sigma^{2}_{0}$) and $\tau \...
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Joint Posterior Distribution

I have 4 groups, each has a probability of developing gout (Bernoulli distribution), with a total of 400 individuals. I am confused how to derive and present the joint posterior distribution for each ...
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81 views

marginal posterior distribution

Given $n$ normally distributed observations $f(X|\sigma_X)=\mathcal{N}(\mu_X, \sigma_X^2)$ and assuming a uniform prior on $\log(\sigma_X)$ and known $\mu_x$, I'm trying to find marginal posterior ...
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1answer
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Posterior Predictive Distribution as Expectation of Likelihood

Say we have a posterior predictive density: $$p(\tilde{y}|\mathbf{y}) = \int p(\tilde{y}|\theta)p(\theta|\mathbf{y})d\theta$$ In Hoff's Bayesian Statistical Methods text, he suggests that to obtain ...
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Calculating expected loss of posterior distribution

I'm working with 2 posterior distributions from AB tests. For the sake of simplicity let's assume: $$ A\sim Beta(10, 20) $$ $$ B\sim Beta(5, 25) $$ I want to calculate the posterior expected loss of ...
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Find joint maximum of a sampled density

I used a sampling method to fit a model with three parameters to data, by supplying the likelihood function and priors. (I'm using JAGS but I think this applies to any method). I obtain triplets of ...
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MAP estimation as regularisation of MLE

Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this: It is closely related to the method of maximum likelihood (ML) estimation, but employs ...
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Posterior distribution of weight vector tends to Gaussian distribution as data size increases: is it true?

I'm working on Pattern Recognition and Machine Learning(Bishop), Chapter 6, which is about Gaussian Processes. Author says in page 315 : The usual justification for a Gaussian approximation to a ...
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clarifying exponential-gamma conjugate prior

I'm referring to page 22 of this white paper. On page 22, it says the following: given that $s_i \sim \text{Exp}(\theta), i = 1,..,c$ $\theta \sim\text{Gamma}(k, \Theta)$, Then the posterior ...
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Distribution of posterior probabilities of samples from MCMC seems to be made up of several chi square components

I am running an MCMC sampler with a model that uses Cash's C statistic for the likelihood (along with gaussian priors), which is supposed to resemble a chi square distribution in the limit of large ...