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Questions tagged [bayesian]

Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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Is this equation true for any joint probability distribution (used in orthogonality principle proof for estimators)?

Given two random variables $x, y$, is it true that $p(x - \hat{x}|y) = p(x|y) - \hat{x}$ where $\hat{x}$ is a known constant I came across this in this kalman filter derivation, Corollary 3.2.1, ...
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Is it possible for the mean posterior distribution to be higher than the prior distribution?

Basically just the question. I would appreciate a lot if there was an example, if such thing happened. Thanks.
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Stan: Ancova with a Poisson distribution

I am trying to code an Ancova with a block effect for count data. Here I will simplify the Ancova to a simple linear regression with a block effect. As I am using count data with low observed values ...
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Interpretation of confidence interval in Bayesian terms

Motivation: I was standing in front of a class to introduce into the concept of confidence interval using the example of differences in means (purely frequentist setting) and I was torturing the ...
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Bayesian experimental design choosing the support of the distribution

I am not an expert on these topics so any help is very much appreciated. I'm not even sure if this question is trivial. If so, please let me know. General Setup: Consider the problem posed in this ...
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Ridge logistic regression and posterior distribution

We know that glm regression with gaussian prior can be assimilated to Bayesian regression. Let say I fit the model with frequentist approach and I have the optimal ridge parameter. If I want the ...
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Bayesian Statistics in CS229

In this note,there is a posterior distribution of parameter $\theta$ in section 3 like this: $$ \begin{align} p(\theta|S)&={p(S|\theta)p(\theta)\over p(S)}\\&={\big(\prod_{i=1}^mp(y_i|x_i,\...
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Bayesian sensitivity analysis

I need to preform a Bayesian sensitivity analysis for a mechanistic model(written in python) which has around 150 parameters. Is there a pre-existing package/program where I put my model and ...
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applied papers on probabilistic generative models and inference engines

I am looking for applications papers where people choose some task on which they will do Bayesian inferencing and graphical modeling, and then build an inference engine to infer latent parameters. And ...
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Understanding the meaning of $U^I_{ijp}=X_{jp}\beta^I_{i}+\epsilon^I_{ijp}$

I am currently looking into the following research paper: https://www.jstor.org/stable/23012006?seq=3#metadata_info_tab_contents However, I can not seem to understand two concepts that the author ...
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Calculating the Bayes estimator given prior $\pi (\theta) = \theta^{-2}e^{-1/\theta}$?

Let $[X_i|\theta] \sim N(0,\theta)$ and suppose $\theta$ has prior distribution: $$\pi(\theta) = \theta^{-2}e^{-1/\theta}$$ I want to find the Bayes estimator of $\theta$ under square error loss, ...
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how to multiply two conditional probabilities in general

I am trying to understand how to multiply two conditional probabilities. $P(X|C) \times P(C| P,S)$ seems to equal to $P(X,C | P,S)$. How to understand this? I understand the product rule, but how ...
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what is the difference between a multilayered autoencoder and a hierarchical latent variable model?

I have been trying to understand how hierarchical latent variable models are different from multilayered autoencoders and in specific the argument below Autoencoder networks resemble in many ways ...
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Get posterior distribution of categorical variable given empirical continuous-categorical priors?

Suppose I have categorical variable $Z \in D$ defined for some finite domain $D$. I also have a continuous variable $X \in \mathbb{R}$ which is observed. From historical data samples I have the ...
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Can sampling from a truncation of a random variable, rather than the original variable be more Blackwell-informative?

Suppose you are interested in finding the mean of a random variable. You have some prior belief of it and before sampling 1 observation, you can decide whether to sample from the original random ...
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Showing $\delta'(X)$ is a Bayes estimator of $\theta^k$ for specified prior

Problem: Define $\delta(X) = \frac{\sqrt{n}}{1+\sqrt{n}}\bar{X}_n + \frac{1}{2(1+\sqrt{n})}$ We assume $\bar{X}_n|\theta \sim Bin(\theta,n)$. It is known that $\delta(X)$ is the Bayes estimator of $\...
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Applying de Finetti's representation theorem to Dirichlet distribution

Let's begin from the de Finetti–Hewitt–Savage theorem: for an exchangeable sequence of random variables we can always write $$ p(x_1, x_2,\cdots) = \int \prod p(x_i | L) P(dL) $$ where $L$ is a latent ...
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How to assign a prior distribution to a loading matrix that has restrictions?

I came across the paper Fast Variational Bayesian Linear State-Space Model. They work with the following model: $$\begin{align} {\bf{x}}_n &= {\bf A} {\bf{x}}_{n-1} + {\text{noise}} \\ {\bf{y}}_n &...
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How to estimate the intriscs probability error of a string of character

So my problem is as follow : I have a given string of characters, and I would like to quantify the uncertainty linked to the probability of each letter types in the string, based on there observed ...
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Is it rational to select a parameter posterior value because it maximizes utility, even if probability is low?

I did Bayesian parameter estimation and I have now an estimate of the posterior distribution for my model parameters (say I have 2000 samples). Now I would like to make the optimal decision under my ...
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Calculating the posterior distribution of linear predictor

I am currently fitting a linear regression model in a bayesian framework in R with the package ngspatial. To investigate the quality of fit, I would like to calculate the bayes R2, as suggested here ...
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Bayesian chi-squared tests

I have a dataset with two groups of participants. Each participant performed a repeated measures task on which three types of errors could be made. I want to measure the difference in distributions of ...
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How can you deal with volatility of a metrics that depends on the count of events?

I am using Herfindahl Index metrics to measure the degree of concentration of posts by email, device_id, IP and other variables to identify potential fraud events. For example, a high degree of ...
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optimization based interpretation of Bayes' theorem

I read about one equivalent interpretation of bayes' theorem as follows: $P(\mathcal{M}|x) = \frac{P(x|\mathcal{M})\cdot\pi(\mathcal{M})}{\int P(x|\mathcal{M})\cdot\pi(\mathcal{M}) d\mathcal{M}}$ is ...
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How do Bayesians verify their methods using Monte Carlo simulation methods?

Background: I have a PhD in social psychology, where theoretical statistics and math were barely covered in my quantitative coursework. Through undergrad and grad school, I was taught (much like many ...
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relation among loss function / MLE / Bayesian estimation

I have read a lot of stuff on the relation between minimizing a loss function / maximizing the likelihood / choose a centrality measure of the posterior (Bayesian estimation); but I cannot see a clear ...
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Non-informative prior for the covariance matrix

I'm currently working on a project around the Bayesian approach to portfolio selection, and I can't manage to wrap my mind around the specification of the non-informative (diffuse) prior. Assuming ...
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Conditional Probability Table in R

I want to perform Bayesian network analysis in R. I have a large network and i am bit confused with defining conditional probability tables! In my network i have a node with in-degree of centrality ...
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Introduction to Variational Bayesian methods?

I am interested in learning about Variational Bayesian methods. I understand the general idea, explained in Wiki, where the aim is to approximate a posterior using a more tractable distribution, in ...
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JAGS: Posterior Predictive Check for a Logistic Regression Model

I want to perform a posterior predictive check on some simple logistic regression models that I fitted in JAGS. I found a function in the R package jagsUI called pp.check (see doc here: (pp.check ...
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help determining ROPE for bayesian multilevel probit model

I am having difficulty determining a justifiable region of practical equivalence (ROPE) for a parameter from a multilevel probit model Below is the posterior distribution for the fixed-effect of ...
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Posterior convergence in expectation vs probability

Let's assume that we are doing approximate Bayesian inference and compute the convergence of our posterior estimate to the true value of the parameter using Wasserstein distance. Why posterior ...
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Gibbs sampling for drawing samples and estimating parameters

I'm learning Bayesian inference by myself and having a difficulty for understanding Gibbs sampling. From what I understood, Gibbs sampling is to draw samples from a given probability distribution $p(...
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Marginalising over Dependent Random Variables

Suppose I have two RVs, $A$, and $B$. Every place I have looked thus far suggests the following for marginalisation, which for me is fine: $f_A(a) = \int_{-\infty}^{\infty} f_{A,B}(a,b)db $. ...
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Generating data from the posterior distribution

Let $$p(D \mid \mu,\sigma^2) \sim \mathcal{N}(\mu,\sigma^2)$$ where $D=(x_1\ldots x_n)$ is my data. I imposed a normal prior on the mean as $$\pi(\mu) \sim \mathcal{N}(\mu_0,\sigma_0^2)$$ Using Bayes, ...
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Bayesian Linear Regression to Gaussian Process

I'm trying to understand how a Gaussian Process with a squared exponential covariance function can be obtained from Bayesian Linear Regression with a Gaussian prior $N(0,\sigma_p^2 I)$ on the ...
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MCMC - how to compute prior(𝜃)

Question How to compute the prior $P(𝜃)$ and $P(𝜃')$ in MCMC when calculating the posteriors? Prior I thought prior keeps updated with the accepted θ'. However, the way it is computed in the ...
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Can I use Poisson regression to model prevalence ratios if I only have information on events?

I often used Poisson regression models to estimate prevalence ratios. However, in these cases my data contained information on the whole population, including events (1) and non events (0). ...
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How can I determine what values of alpha and kappa to use for Bayesian Optimization?

I'm using the pretty great Bayesian Optimization package for python. I have a very noisy function I'd like to optimize for a given hyperparameter. I've read a little on this, and it seems like if ...
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Using information about optimising function in bayesian optimisation

I know that bayesian optimisation is a strategy for optimising black-box functions. But if i have some information about type of function or it's specific behaviour at some interval what methods of ...
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Bayesian A/B test for LogNormal data

I'm currently working on a (manual) calculation for a bayesian A/B test on logNormal data. I'm currently working with simulated data to increase my understanding. It's giving me some problems, so I ...
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How to calculate Kernel Density for Bootstrap Likelihood

I am attempting to write R code to generate bootstrap likelihood as described in section 3 of this paper https://arxiv.org/pdf/1510.07287.pdf. I am confident that I performed the bootstraps correct, ...
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Can you calculate Bayes Factors for effects in a non-significant regression model

I ran a linear regression model and want to calculate Bayes Factors (BF) for any non-significant effects that are generated by the model. However, the regression model itself is not significant (p = 0....
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dirichlet distribution and excessively large numerator

what I am trying to do is calculating posterior probability using dirichlet distribution as my prior. the situation is like this. a web log have three variables A, B, C, and each variable's value is ...
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Is assigning an inverse-Wishart distribution to a diagonal matrix problematic?

I'm reading the paper Bayesian Vector Autoregressions by Thomas Wozniak. He considers the model $$y_t = \mu + A_1 y_{t-1} + \cdots A_k y_{t-k} + u_t$$ where each $y_i$ is a $N$-vector, each $A_j$ is a ...
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Find normalization constant from factorized density

Please consider the following Bayes Network: We can express density $p(\mathbf{x}_1 | \mathbf{x}_0, \mathbf{y}_1)$ in terms of measurement and motion models by ignoring normalization constants as: $...
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Calibrating LASSO prior (how to select the scale hyperparameter)?

I want to use a LASSO prior (Laplace prior) for a location parameter $\mu$ $$\pi(\mu \mid s) = \dfrac{1}{2s}\exp\left(-\frac{\vert \mu \vert}{s}\right).$$ However, I do not know to calibrate this ...
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Simple up/down vote rating but weighted by number of responses

I am trying to analyse the ratings for restaurants from a website. The rating system on the website is pretty simple: people can up-vote or down-vote. The restaurant is then presented to website ...
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How to calculate mean and variance from very small function proportional to the density?

Problem Say I have the following function $g(x)$, which is proportional to the density function $f_\theta(\theta)$ of random variable $\theta$, i.e. $g(\theta) \propto f(\theta)$, such that $$ \...
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Posterior using Numerical integration - why divided by (sum of posterior / number of samples)

Background Reading Markov Chain Monte Carlo (MCMC) and the numerical integration method to get the posterior. Question The implementation code divides the posterior with (post.sum() / len(thetas)). ...