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

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What are the hyperparameters? [duplicate]

I find the meaning of hyperparameters not always clear. The hyperparameters are defined as "the parameters of the prior". Suppose that one has prior information about a certain parameter $\theta$. ...
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44 views

Interpretation in histogram of empirical posterior distribution

I'm having trouble to understand the following histograms I know that the posterior distribution in this case is just the empirical cumulative $$P(\rho\leq c)=\frac{1}{n}\sum_{i=1}^n \mathbb{...
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75 views

Posterior of Dirichlet distribution parameters

I want to obtain posterior distribution for parameters of a Dirichlet distribution $x = (p_1,p_2,p_3) \sim Dir(p_1,p_2,p_3; a_1,a_2,a_3)$ with uniform $P(a_1,a_2,a_3)$ and observed data $X=\{x_1,x_2,.....
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27 views

How should I compare posterior samples of the same parameter from two Bayesian models?

I have run 2 Bayesian regression models and would like to compare the posterior samples of a parameter that is common to both models. For example, if model A is $y=\alpha + \beta_1x_1$ and model ...
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12 views

Posterior pointwise uncertainty of multivariate normal-Wishart (variational GMM)

Given a variational mixture of Gaussians (as per, e.g., Chapter 10 of Bishop, 2006), we can compute the posterior predictive pdf: $$ \left\langle p(x|\alpha,\beta,\nu,\mu,V) \right\rangle $$ where $\...
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inferential structure determination

I'm trying to get my head around Bayesian inference and the difference between the posterior and likelihood. Going off the back of these answers, I'm under the impression that the posterior is $P(data|...
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30 views

Is it possible to estimate the posterior distribution of the difference of two means using published t statistics?

I am interested in using summary statistics from published papers to obtain posterior distributions for the difference of two means. In the setting of the classic t-test, we can imagine measurements ...
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84 views

Distribution of posterior mean from different datasets

This question has originated from this question. Suppose we have the following simple setup, for $i = 1, \dots, n$ $$y_i \mid \mu \sim N( \mu, 1) \text{ and } \mu \sim N(0,1). $$ Then due to the ...
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Sampling from posterior

Assume that the likelihood is available in closed form $p(Y|X)$. And also the prior is available in close form $p(X)$ and it is easy to sample from. Then the posterior $p(X|Y) = c p(Y|X)p(X)$ ...
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log ratio of Poisson means versus logit of binomial probability

Assume that two observed counts C1 and C2 are independent realizations of two Poisson processes: ...
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49 views

Why can I combine gamma and factorial coefficients for a beta conjugate posterior

A commenter reminds me to be clearer. In a Bayesian context, the product of a binomial likelihood and a beta prior probability is $$\left( {\begin{array}{*{20}{c}}n\\x\end{array}} \right)p_{}^x{(1 -...
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Inverse gamma posterior

I am working on bayesian analysis and I have a normal likelihood function and an inverse gamma (IG) prior for the parameter $\lambda$. I have the following posterior: $$ \propto \frac{1}{\lambda^{\...
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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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94 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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39 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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57 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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29 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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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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34 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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45 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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27 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 \sigma^{-...
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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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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 L(\theta,\delta(x))\pi(\theta|x)d\...
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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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35 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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18 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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89 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 f(x|\sigma)\...
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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 x_i}{\...
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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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58 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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39 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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43 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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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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31 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: $$\frac{e^{-n\...
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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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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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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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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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81 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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48 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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27 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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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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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: $$\sigma^2_{t+1}=\alpha_0+\alpha_1\...
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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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29 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 ...