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

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FDR for Posterior Probability?

I searched this community for any explanation for Posterior Probability FDR and found nothing that answers the question. I googled for it and found different solutions for particular cases. If there ...
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44 views

Find the posterior distribution of $\pi$

An observation $x$ is taken from a negative binomial distribution $X \sim \text{Negative-Binomial}(k,\pi)$. The parameter, $\pi$, is allocated a beta prior $\pi \sim (\alpha,\beta)$. My attempt: ...
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19 views

What is the difference and relationship between posterior distribution function and likelihood function in MCMC?

I am learning MCMC in class, and I encounter one question about the relationship between posterior probability and likelihood function. In our lecture, the professor asked us to take samples from ...
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Wishart distribution posterior samples do not seem to be supporting the observed covariance matrix [closed]

I am having a variance covariance matrix $S= \begin{bmatrix} 16 & 10\\ 10 & 25 \end{bmatrix} $ based on 60 observations of a bivariate normal distribution with some known ...
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1answer
50 views

Beta approaching Binomial

If we have a Beta likelihood and a binomial prior, we get a beta posterior. Can someone please explain why this approaches a binomial as $n\rightarrow\infty$. I plotted it and this appears to be the ...
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23 views

Posterior Pred. Distribution for Bayesian Hierarchical Regression Model for Existing Group Parameters

For a hierarchical regression model, I understand that there are two posterior predictive distributions potentially of interest: (1): The distribution of future observations $\tilde{y}$ ...
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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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64 views

Posteriors and Sample Sizes

Suppose we have a two dimensional parameter $\theta=(\mu,\sigma^2)$, and a prior distribution $p(\theta)$. Let our sample come from a normal distribution with mean $\mu$ and variance $\sigma^2$. The ...
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22 views

P.d.f for Gamma posterior with Exponential data

I am trying to perform a simple exercise: Sample $N$ points from $\text{Exponential}(\lambda=0.1)$ Assume a $\text{Gamma}(\alpha,\beta)$ prior for the parameter $\lambda$ above Build a p.d.f for the ...
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40 views

Bayesian inference when observed variable contains uncertainty

I have a very simple graphical model to describe the relationship between two categorical variables $c \in \{0,1\}$ and $l \in \{A,B,C\}$: $$c \rightarrow l$$ I know all the conditional ...
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1answer
21 views

Marginal Posterior distribution with Normal observations

According to chapter 3 of Gelman's Data Bayesian Analysis[DBA], when we have $y_i\sim N(\mu,\sigma^2)$, and $p(\mu,\sigma^2)\propto (\sigma^2)^{-1}$ Then, $p(\mu,\sigma^2|\mathbf{y})\propto ...
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73 views

Multilinear loss in Exponential-Uniform model

Let a prior $\pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta))$ and $f(x\mid\theta)=\theta e^{-\theta x}$. Taking the multilinear loss ...
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How to find the posterior distribution and posterior mode of Beta given an exponential prior distribution and binomial data?

This is the question I'm working on: I have already completed part (a). For part (b) i, I modeled y1 given Beta and n1 as Binomial(n1, p1 = d1/(d1+Beta)). For part (b) ii, I showed that the ...
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1answer
35 views

Bayes risk of Normal-Normal model

Consider $x\sim N(\theta,1)$ and $\theta\sim N(0,n)$. Show that the Bayes risk is equal to $\frac{n}{n+1}$. I know that $$r(\theta,\delta)=\int_\chi\int_\Theta ...
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26 views

How to treat fixed observations in probabilistic programming?

Suppose I have a collection of light bulbs. I record their ages, then plug them in, record the power input and then measure the heat output. Then I increase the power input and measure the heat output ...
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Does a check failing to compare observed and predicted data qualify as a posterior predictive check?

I consider a Gaussian mixture distribution and I want to implement posterior predictive checks for choosing the model with the correct number of mixture components. I know the true number of ...
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34 views

maximum a posteriori corresponds to a minimization problem?

I encountered some question related to maximum a posteriori. These questions say finding a parameter which maximizes a posteriori corresponds to a minimization problem. the questions are always binary ...
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15 views

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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84 views

Posterior distribution of normal with gamma prior on the precision

Find the posterior distribution when $$x|\sigma\sim \mathcal N(0,\sigma^2),\:\:\: 1/\sigma^2\sim \mathsf{Gamma}(1,2)$$ I'm stuck in this exercise, I know that $$\pi(x|\sigma)\approx ...
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26 views

Finding the posterior of this conjugate prior

According to Wikipedia the first posterior hyperparameter of a normal likelihood function and a normal prior with known variance is $$ \left.\left(\frac{\mu_0}{\sigma_0^2} + \frac{\sum_{i=1}^n ...
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3answers
602 views

Where is wrong with my formulation of estimating the probability of a biased coin?

I represent a biased coin with a discrete distribution $p(\theta)$, where $p(\theta=h)=\pi$ is the probability of heads, and $p(\theta=t)=1-\pi$ the probability of tails in one toss. I have a ...
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52 views

Bayesian statistics: Can a posterior probability be exactly 1?

I have a question regarding bayesian statistics. Is it possible to end up with a posterior probability of 1, that a slope is positive? My likelihood data shows a greatly significant relationship, ...
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38 views

Multiplication of two normals Gaussian Processes

I am working with Bayesian statistics for gaussian processes and I want to derive the posterior distribution. In general, I am clear about how to derive a posterior using Bayes rule. However in this ...
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42 views

Practical situation in which the posterior mean is prefered to the MAP

Sometimes experts for which we design models are interested in having a point estimate and in practical situations, they always say me "give us the most probable parameter value". And whether the ...
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2answers
111 views

Conveying Bayesian Statistical Analysis to Non-Bayesian

I am trying to convey the results of a Bayesian statistical analysis to an audience uneducated with Bayesian statistics but familiar with the interpretation of p-values (verbal, non-publication ...
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30 views

Posterior from a Poisson likelihood and prior

I have the following Poisson mass function: $$p(y| \theta) = \frac{\theta^y e^{\theta}}{y!} $$ Which has a corresponding likelihood for n independent realizations of y as follows: ...
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37 views

Relation Between Bayesian Estimation and Maximum a posteriori estimation

Is maximum a posteriori estimation some kind of Bayesian Estimation? If yes, can you point out other Bayesian estimators? Edit: So I've come to know the following (don't know if they are correct): ...
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36 views

Multivariate posterior probability

This is a 2-dimensional pattern recognition system that I am working on. It is known that the distribution between the two classes are $1/2$ and $1/2$ respectively for class $\omega_1$ and class ...
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21 views

Is there any way to convert from a posterior probability to p-value, or the opposite?

I have results of a study from associations of a variant with a phenotype in the form of posterior probabilities but I was wondering if there is any way to convert these to p-values, even making ...
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1k views

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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1answer
66 views

Bayesian GARCH(1,1) Forecasting

I am using the following bayesGARCH here package in R. I am interested in forecasting $h_t$, the model setup is given bellow. $r_t$ = $\varepsilon_t(\frac{v-2}{v}\omega_th_t)^{1/2}$ $\quad$ with ...
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47 views

Updating Bayesian prior & likelihood for A/B test

I'm fairly new to bayesian. I'm trying to edit a bayesian python code for $A/B$ test analysis. I'm using uninformative priors as a beta distribution, so my $\alpha$ & $\beta$ parameters are $1$ ...
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26 views

Bayesian approach and ML approach for distributions

Suppose that we have two sets of discrete random variables X ~ f(θ), Y~g(θ) where X and Y are independent, and the parameter θ is the same in both cases. We are interested in predicting Y on the basis ...
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35 views

get probabilities of inference in bayesian network in R

I have a question about how continuous variables can be used for building models and prediction in a bayesian network. With some help, I was able to get it to work for continuous variables as follows ...
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113 views

Forecasting Bayesian GARCH(1,1) volatilities

As a beginner in Bayesian statistics, I was wondering how one can make a GARCH(1,1) volatility point forecast using a Bayesian approach in the following model: ...
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62 views

Linear regression with t-distribution prior for beta coefficients

Having: $$y\sim N_n(X\beta, \sigma^2 I_n)$$ with prior distributions: $$\beta\sim t_\nu(\beta_0, B_0)$$ and $$\sigma^2 \sim IG(\alpha_0/ 2, \delta_0/2)$$ What would be the conditional posterior of ...
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Posterior distribution dependent on two variables make inferences about one

If i have some model for X that depends on THETA1, THETA2 and has a posterior P(THETA1,THETA2 | x1,...,xn). How would I make inferences just about THETA1? What I am thinking so far is just to ...
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28 views

exponential likelihood uniform prior

Say I have a sample $x_1,...,x_n$ from an exponential distribution where $x_i$ is i.i.d exponential with parameter $\lambda$. 1) Suppose the prior for $\lambda$ was a uniform $0$ to $\beta$, what ...
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Posterior distribution in uncertainty traps

I was reading a bit about uncertainty traps in which there is a model that describes firms having investing opportunities in a competitive environment. A self-contained explanation can be found in ...
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153 views

Why is a Frequentist confidence interval usually referred to as “exact” in comparison to the Bayesian posterior probability interval?

I have noticed that a lot of literature usually refers to the frequentist confidence interval as "exact" in comparison to the Bayesian probability/credibility interval, which is calculated off the ...
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48 views

how to compute this posterior joint density function?

If I want to compute the probability $p(y,z|\theta,\lambda)$,then how to? I know the answer is $p(y|z,\theta)p(z|\lambda)$, but I do not know how to? Please help me, thanks a lot
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286 views

Binomial uniform prior bayesian statistics

Suppose to have a binomial distribution where the prior of the parameter is uniform. How can I get the posterior distribution of the parameter?
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33 views

Given a marginal posterior distribution, when should I report an upper limit on the parameter?

I've been utilizing MCMC in the analysis of $\gamma$-ray spectra. For many of my model-parameters, the marginal posterior distributions, $p( \theta_i |X)$ where ($\theta_i\in(0,1]$), are normal ...
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266 views

Why use mean of posterior distribution instead of probability?

I'm reading the Think Bayes (pdf link) by Allen B. Downey, and on this example I don't understand well the purpose of Mean in the chapter 3.2 The locomotive ...
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2answers
61 views

Dirichlet conjugate update derivation

I am attempting to derive the update equations for the conjugate to the Dirichlet distribution, as outlined here: http://mathoverflow.net/questions/20399/conjugate-prior-of-the-dirichlet-distribution ...
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1answer
37 views

Using MCMC to sample from a posterior, are our posterior beliefs on parameters independent?

I've been given a classification problem in which MCMC (slice-sampling) is used to sample from a hierarchical posterior. After getting $n$ samples, the Monte Carlo method can be used to give an ...
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When can't frequentist sampling distribution be interpreted as Bayesian posterior in regression settings?

My actual questions are in the last two paragraphs, but to motivate them: If I am attempting to estimate the mean of a random variable that follows a Normal distribution with a known variance, I've ...
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Prior pdf decay in Recursive Bayesian Estimation

I'm doing Recursive Bayesian Estimation numerically. I have a state vector, x, that I'm trying to estimate by regularly taking noisy measurements, z. I use Posterior = Likelihood x Prior / ...
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52 views

Calculating Bayesian posterior distribution for an exponential distribution

I want to calculate the Bayesian posterior distribution of an exponential distribution where $\lambda$ is distributed according to gamma distribution. I know that I need to calculate ...
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56 views

Is MCMC needed for a Dirichlet prior with multinomial likelihood?

Basically, I have a multinomial distribution with probabilities $\theta_1, \theta_2 ... \theta_m$, and a Dirichlet prior (arbitrarily set as $\alpha_i = \alpha_{i+1} ... = 1$ (i.e. every $\alpha = ...