Questions tagged [posterior]

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

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Find the posterior of Bernoulli likelihood and reparametrized prior

Supposed that we have $Y_{i} $ ~ $Bern(a)$, and an improper prior p of $Beta(0,0)$. Let $g(a)=\phi = \log(\frac{a}{1-a})$. We would like to find the posterior distribution of $\phi|y$. My trial: For ...
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Posterior predictive distribution of Gibbs Sampling compared to original data

I have implemented a Gibbs sampler to simulate from the joint posterior: $ln(y_1),...,ln(y_n)|\mu,\sigma^2 ∼_{iid} N(\mu,\sigma^2)$ Where both $\mu$ and $\sigma^2$ was unknown. I have simulated values ...
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Bound on the expectation of a function of random variable having a strictly log-concave probability density

let $\theta \in \mathbb{R}^d$ be a random variable having a strictly log-concave probability density function, i.e \begin{equation} p(\theta) = e^{-\phi(\theta)} \end{equation} where $\phi(\theta)$ is ...
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How is highest posterior density interval estimated in this code snippet?

I found the following (Julia) implementation for estimating the highest posterior density interval from a posterior sample (link). Below, I turn it into pseudocode for simplicity. ...
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Deriving Bayesian Credible Intervals for AUC using R brms

I am trying to estimate the posterior distribution for the AUC of a predictive biomarker using R brms. However, whenever I calculate the AUC using the posterior distribution of the model parameters, ...
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bayesian parameter estimation of hawkes process

I have encountered a problem of modeling event data with hawkes process. I define the intensity function as: $$ \lambda(t) = \lambda_{0} + \alpha\cdot\beta\cdot\Sigma_{i}^{n}exp(-(t-t_{i})) $$ where $...
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Calculate posterior distribution given two normal distributed measurements

I have two measurements $m_x$ and $m_y$ from two sensors $X$ and $Y$. $m_x$ can be approximated by $m_x \sim \mathcal{N}(m_x|m, 2)$ and $m_y$ can be approximated by $m_y \sim \mathcal{N}(m_y|3m, 5)$. ...
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Posterior distribution is impossible depending on which prior hyperparameters are used?

Suppose we randomly select one of two coins and flip it. In that situation we have random variables $\alpha$ and $\delta$, where $\alpha$ tells us which coin we select, and $\delta$ tells us whether ...
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How to determine uncertainty of data from a bayesian posterior distribution

I am a bit confused as to how we determine the uncertainty of a set of data from Bayesian Analysis. In my specific case, I am asked the following: Assume $f(x,x_0)$ as the correct model for the ...
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Bayesian update with the shifted and scaled data

Suppose I have data $y$ (N observations) which follows a normal distribution: $y \sim N(\alpha+\beta*\mu,\sigma^2)$ while $\alpha$ and $\beta$ are known parameters. I want to update $\mu$ and the ...
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Updating prior in MCMC with new estimates for parameters

I'm new to doing Bayesian analysis and I wanted to learn by using baseball data. I took a group of players and found their hits and at bats for various years and I want to be able to get estimates for ...
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Posterior distribution when it's only known that data belongs to some known interval

Suppose we know the prior distribution. When we observe a data point, say $x$, it allows us to form a posterior distribution. I was wondering what if we only know that the data point $x$ belongs to a ...
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Variational Inference Mean-Field Gaussian

I am new to variational inference and got very confused about some basic ideas. We want to use the mean-field gaussian family to approximate a complicated high-dimensional distribution. I want to ...
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Posterior Predictive Distribution of Latent Function Values in Bayesian Linear Regression

In the book Gaussian Processes for Machine for Machine Learning, the authors review Bayesian linear regression. For the setup, we have $f = x^Tw$ and $y = f +\epsilon$ where $\epsilon \sim N(0, \sigma^...
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Highest Posterior Density Interval (HPDI) using kernel

I'm trying to compute the 95% HPDI of a posterior from 10000 draws from a distribution. I've been instructed to use density() in ...
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Kl Divergence between factorized Gaussian and standard normal

Given two distributions, one a parameterized gaussian and the other a standard normal gaussian: $q(x) \sim \mathcal{N}(\mu,\sigma)$ $p(x) \sim \mathcal{N}(0,I)$ We want to compute the KL Divergence $...
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Samples from Conditional Posterior Distribution in Pymc3

Let us consider the following Hierarchical Bayesian model: $w \sim\ Beta(20, 20)$ $K = 6$ $a = w * (K - 2) + 1$ $b = (1 - w) * (K - 2) + 1$ $theta \sim\ Beta(a, b)$ $y \sim\ Bern(theta)$ The above ...
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Is the mean posterior fit the same as the fit of the mean parameters?

Say we are fitting a parametric model $y(x, \theta)$ to some data (e.g. logistic regression). Given a prior distribution over the model parameters $\theta$ and observed data $x$, we arrive at a ...
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Bayesian Test in hypothesis testing with Bernoulli random variables

Let $X_1,\ldots , X_n$ be iid Bernoulli$(p)$ $(0<p<1)$. Our sample is $x_1=\ldots x_n=1$. Suppose that the prior distribution of $p$ is Uniform$(0,1)$. Consider a Bayesian test for $H_0: p\geq 0....
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Posterior distribution of $\mathcal{N}(\theta, \theta^2)$ with normal prior

Are there any analytical/closed-form results on the posterior distribution of $x \sim \mathcal{N}(\theta, \theta^2)$ using a standard normal prior?
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Obtaining approximate posterior probabilities with Bayesian cross-validation

(Apologies to anyone that may have been following this question: I have decided to rewrite it to make it more succinct. As a result, comments below now appear out of context.) Given a set of models $\{...
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Sampling-based posterior predictive of binomial likelihood and neg-binomial prior

Suppose the following distribution for $ssc$ and its parameter $mcc$: $$ssc \sim Binomial(trials= mcc, success= P)\\ mcc \sim NegativeBinomial(\lambda) $$ Where $P$ is a known parameter. Is there an ...
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Derivation of posterior distribution under Dirichlet prior distribution:

suppose that $\mathbf{y}=(y_1, y_2, \cdots, y_n)$ is a vector of $n$ observed sample points drawn from a mixture of $g$ components, and $\mathbf{z}=(z_1, z_2, \cdots, z_n)$ is a vector of latent ...
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Partially specified Bayesian prior?

In bayesian linear regression for example, we may specify a model as: $$y_i \sim N(\beta_0 + \beta_1 x_i, \epsilon^2) \\\\ \beta_0 \sim N(0, \tau_0^2) \\\\ \beta_1 \sim N(0, \tau_1^2) \\\\ \epsilon \...
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Bayesian estimator under transformation of the parameters

Suppose we have $x=(x_1,...,x_n)|\mu,\sigma^2\sim f(x_i|\mu,\sigma^2)$ $iid$, also let $\mu\sim p(\mu)$ and $\sigma^2\sim \pi(\sigma^2)$ be prior distributions. Here $f,p,\pi$ are generic ...
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Posterior standard deviation, from normal sample with discrete prior

Suppose the sample $(7,2,6,12,10,9)$ is well approximated by a normal distribution with mean $\mu$ and standard deviation $6$. Use the discrete prior distribution $(8,9,10,11)$ of possible values for $...
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Extension of normal-gamma to gaussian process prior

I am trying to solve a problem where the solution involves both the mean and the variance of a multivariate normal distribution, modelled through a Gaussian Process prior. The standard Gaussian ...
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Posterior for Discrete prior from normal sample with known variance

Given a sample $(1,2,5,2.5,3,7)$, and known $\sigma^2 = 4$ with possible values for prior mean of $(0,0.5,1,1.5)$ all equally likely, find the posterior distribution and the posterior probability that ...
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Posterior distributions --- what's the correct way to see it?

When running models from a bayesian perspective — a regression for example — we get posterior distribution for every parameter/statistic we want, right? I’m wondering whether I should see this this ...
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Find posterior distribution given a prior normal and a normal rv [duplicate]

Suppose $X$ has $N(\theta, \phi)$ distribution for a known $\phi$. Suppose that the prior distribution for $\theta$ is $N(\theta_0, \phi_0)$ (both parameters known). The question asks me to find the ...
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Samples from Marginal Posterior Distribution in Pymc3

Let us consider the following Hierarchical Bayesian model: $mu \sim\ Beta(1, 1)$ $k \sim\ Exponential(1)$ $a = k*mu$ $b = (1-mu) * k$ $theta \sim\ Beta(a, b)$ $y \sim\ Bern(theta)$ The above example ...
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(multilevel) posterior mean vs. ordinary mean

Suppose I would have a large dataset from pupils nested in 1000 schools, a typical multilevel dataset, with say 50 pupils from each school. I'm looking for some advice on the difference between ...
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Sampling for Approximate Bayesian Computation without Simulation

I am trying to use ABC for a physical black box phenomenon. Both the input space and output space are 3D, and there is a proper distance function for the performance space (CIEL*A*B* ΔE). It is not ...
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Gaussian closed form solution of marginal likelihood

i have tried some time now to understand a specific step in the derivation of what I think is a marginalization integral. I am still learning about these statistical things and I think I miss ...
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Find posterior distribution for the variance in a regression with normally-distributed X

Consider a univariate regression $$ y_t = \alpha + \beta x_t + \epsilon_t,$$ where $x_t \sim N(\mu_x, \sigma^2_x)$ and $ \epsilon_t \sim N(0,\sigma^2_\epsilon)$. The prior distribution is of the form $...
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Find a decision rule using Rao-Blackwell Theorem

Suppose that an observation $x \in (-1,1)$ comes from a sample model with a parameter $\theta$, with density function: $$ f(x\mid\theta) = \begin{cases} \theta\ &\text{if}& -1 < x < 0\\...
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Find posterior distribution and Bayes estimator [duplicate]

Suppose that an observation $x \in (-1,1)$ comes from a sample model with a parameter $\theta$, with density function: $$ f(x\mid\theta) = \begin{cases} \theta\ &\text{if }\ -1 < x < 0\\ ...
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How to derive a regularized machine learning objective function with the maximum a posteriori for random features?

My question is at the end of the post. I tried to give as much information as I can to clarify my understanding and to point out as precisely as possible where I am stuck. Independent variables or ...
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Why Gibbs Sampling for mixture models?

I am studying MCMC and in the book I'm reading there is this example on Gibbs algorithm for inferring the posterior of a gaussian mixture. I understand how the algorithm works and the fact that its ...
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Marginal posterior distribution of error variances

I have been working on Bayesian statistics recently and have came across the term called Marginal distribution of error variances. Though I understand what is a marginal distribution and that an error ...
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Finding the posterior under Jeffreys prior

Let $U \sim \text{Unif}(0,1)$, $\alpha$ the rate parameter of interest and $a>0$ a fixed known constant, define $$ X = a U^{-1/\alpha}$$ Find the Jeffreys prior, the posterior for $\alpha$ and ...
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Location-scale parameter with non-informative (improper) prior : at what condition is the posterior proper?

Consider the setup: Let $(X_i | \mu = m, \sigma = s)$ be a continuous random variable with pdf$$f_{X_i | \mu, \sigma}(x | m, s) = f_{X_i | \mu , \sigma}\big( \frac{x-m}{s} | 0,1 \big) \ s^{-1}, x \in \...
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Posterior predictive distribution, linear regression

The following quotation is from Machine Learning: a Probabilistic Perspective Chapter 7, page 234 In machine learning, we often care more about predictive accuracy than about interpreting the ...
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Linear regression, input centered and mean of the output

Why "If the inputs are centered, the mean of the output is equally likely to be positive or negative"?
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In Bayesian Statistics, can the data be a random variable drawn from the posterior of a separate model? Will the uncertainty flow through?

In the traditional Bayesian hierarchical approach, you typically have a hierarchy built on your coefficients. That is to say, you might have your coefficient of interest distributed around some ...
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Exponential Posteriori with a Uniform Prior

I'm studyng for a final exam and found this problem from another generation, but I don't know how I should continue... I will be gratefull for any help, thanks you. Let be $X|\theta\sim U(0,\theta)$ ...
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2 votes
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Multi-armed bandit algorithm for finding the best performing bandit in the least amount of trials

I'm wondering if there's an algorithm that minimizes the expected posterior loss for the best performing bandit where regret is calculated as the number of trials to achieve a threshold for posterior ...
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How to average several posteriors distributions from a Monte Carlo Simulation

Say you produce several posteriors distributions from different runs of the same model under different seeds. That is to say you have something like the following: ...
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Bayesian statistics probability while knowing posterior predictive distribution

The data for this exercise consists of the monthly number of van drivers killed in trac accidents in Great Britain from January 1969 to December 1984 (a total of 192 observations). Seatbelt ...
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How to obtain the distributions of linear model parameters?

For a linear model $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + …… + \sigma^2$$ Assuming $\beta_0, \beta_1,……$ are random varibles, how to obtain the distributions of these parameters? It seems that ...
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