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

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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?
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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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299 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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50 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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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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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: ...
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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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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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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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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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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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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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why use Mean of posterior distribution instead of probability?

I'm reading the Think Bayes by Allen B. Downey, and on this example I don't understand well the purpose of Mean in the chapter 3.2 The locomotive problem. ...
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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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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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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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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 = ...
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Simulate mixture of betas

Suppose that we have $X_1, ..., X_n$ iid such that $X_i| \theta \sim Ber(\theta)$ and $\theta \sim g(\theta)$ such that $$g(\theta) = 0.6 Beta(2,1) + 0.4 Beta(1,1) = 1.2 \theta + 0.4$$ Doing the ...
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In a mixEM class object, does the $posterior describe the probability of an observation belonging to a particular grouping?

Disclaimer: Probably a trivial stats question, though the documentation in R seems sparse on useful examples/description of this to a non-professional statistician I am trying to work out how to ...
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Signal Extraction About a Signal Extraction Problem

This is a slightly weird variant of a classical signal extraction problem that somehow eludes me. Consider a random variable $X$ with a prior normal distributionwith mean $x_0$ and variance $1/a$, ...
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What's a good model for predicting the binomial distribution of a target variable that represents the number of successes?

Let's say one of my students is taking a test with 50 questions, and I want to predict the distribution of how many questions they will get correct. I have some information about the student and the ...
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Simple and direct application of Bayes Theorem

Suppose that $\theta \in \Theta=\{0,1\}$ such that $P(\theta = 0) = 0.1$. Let $X$ be a r.v such that, given $\theta=0$, $X \sim N(50,1)$ and given $\theta = 1$, $X \sim N(52,1)$. Show that the ...
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Posterior of the linear regression model with g-Prior

Assume a linear model of the form $$Y=X\beta+\epsilon$$ where $\epsilon$ has a multivariate Normal distribution with mean $0_N$ and covariance matrix $\sigma^2 I_N$. I would like to perform simple of ...
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Bayesian regression with singular $(X'X)$ - Is the posterior well-defined?

SE community, I hope to get some insights into the following problem. Given a simple linear regression model $$Y=X\beta+\epsilon\text{ , where } Y\in\mathbb{R}^T,X\in\mathbb{R}^{T \times N}.$$ Under a ...
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Are the mean of samples taken from Metropolis-Hasting MCMC normally distributed?

I've come across the following theorem while studying MCMC. It seems to suggest that the sample mean taken from the MCMC – the posterior marginal expectation – should be normally distributed, using ...
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210 views

What would be the reason that the posterior distribution looks like the prior using MCMC

I am trying to use MCMC to obtain the posterior probabilities of the free parameters of a model. I have tried first to leave two free parameters for my model and I was able to estimate the posterior ...
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Using Bayesian econometrics to forecast macro data (BVAR model)

I am in the middle of a Bayesian class. I have to make a project where I implement Bayesian statistics. I have chosen to do this on macro data. As far as I can see the optimal model to forecast ...
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Likelihood function and plot Pareto distribution for posterior distribution

I implemented two ways to obtain the likelihood of a Pareto distribution with unknown $\alpha$, to find the posterior distribution for $\alpha$. ...
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Posterior of alpha parameter (Shape) of Pareto Distribution

Im trying to generate the posterior distribution of $\alpha$ parameter of Pareto Distribution. I did all the job correctly on paper, but when i go to implement in R i have some problems.I have a ...
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162 views

What is posterior predictive check, and how I can do that in R?

I am using Bayesian hierarchical modeling to predict an ordered categorical variable from a metric variable. For example, I want to regress Happiness (in 1-5 ratings) on Money (a metric variable): ...
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probability of a hypothesis being true given a positive or a negative result

Link to the paper: https://www.aeaweb.org/articles.php?doi=10.1257/jep.29.3.81 Coffman, Lucas C., and Muriel Niederle. 2015. "Pre-analysis Plans Have Limited Upside, Especially Where Replications Are ...
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Does a proper prior lead always a proper posterior?

Does a proper prior lead always a proper posterior? I cannot check whether the posterior is proper, so I was wondering if this assumption is always satisfied .
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Posterior distribution in Bayesian linear regression - why not include $p(X | \beta, \sigma^2)$?

Given parameter/s $\theta$, data $X$ and prior on the parameter/s $p(\theta)$, Bayes' theorem allows us to estimate the posterior distribution $p(\theta | X)$: $p(\theta | X) = \frac{p(\theta) p(X | ...
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Factoring a joint posterior

Suppose the joint posterior density for parameters $(\theta_1, \theta_2)$ can be expressed as \begin{align} \Pr(\theta_1, \theta_2 \mid y) &\propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1, ...
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Clarification on marginal, posterior and likelihood distributions?

I would like to clarify a basic question about how to use the concepts of marginal, conditional, and likelihood distributions. When we observe data from phenomena $x$. We are looking at realizations ...
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Why aren't there two integrals in 'Model evidence' in 'Bayesian linear regression'?

From the Wikipedia article on Bayesian Linear Regression: Model complexity is already taken into account by the model evidence, because it marginalizes out the parameters by integrating ...
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Maximum A Posteriori Confusion

I'm a bit confused by maximum a posteriori estimation. I think I've got a handle on maximum likelihood estimation, however (so maybe this'll be helpful in figuring out the former). MLE: Say I flip ...
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Finding the Complicated Posterior Probability Distribution of $θ$

Suppose, we are given a likelihood function, $f(x|θ)$ corresponding to a shifted-exponential distribution and the prior distribution on the parameter $θ$ is a standard Cauchy distribution. Now I am ...
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Summing multiple posterior distributions

I have obtained separate posterior distributions of regression coefficients of several variables, and would like to know what the most probable sum of these coefficients is. This is because the sum of ...
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Latent Dirichlet Allocation yields different posterior distribution than simple Bayesian model

Method A: out of the box LDA I am using a package to run LDA on a sample of size m with n words in the vocabulary. The end ...
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Marginal Posterior Distribution of Random Effects in Bayesian Logistic Regression

Suppose I'm fitting a logistic regression, and I would like to include individual variability into the estimation process via a random effect. So I have something like: $$ y \sim ...
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Is this posterior probability integral right?

From Wiki: where , k is binomially distributed, and I'm not sure about u. I'm thinking that the second line should be: I mean, if we let X represent the toss of a die, then $P(X = 1, 2, ...
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Is this denominator of a posterior distribution the marginal distribution of Y?

From Wikipedia: , where Is the denominator (above pics are from Wiki) the marginal distribution of Y? Intuitively, it seems that way so that when we cross-multiply, LHS and RHS are mirrors. ...