Questions tagged [prior]

In Bayesian statistics a prior distribution formalizes information or knowledge (often subjective), available before a sample is seen, in the form of a probability distribution. A distribution with large spread is used when little is known about the parameter(s), while a more narrow prior distribution represents a greater degree of information.

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Is it practical to derive the prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?

Is it practical to derive the optimal prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"? Suppose you assume a probability distribution. You ...
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When to specify multivariate versus univariate priors on parameters?

Suppose a linear regression model: $$y \sim Normal(\beta X, \sigma)$$ For our purposes, assume $y$ is a univariate outcome and $X$ is a design matrix containing an intercept and one additional ...
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How to improve the predictions of a model when we have too few predictor variables?

I tried to use a linear model to explain a variable "age" with two variables "x1" and "x2". I can clearly see a decreasing slope inside my scatterplot for age vs x1, or ...
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Using a Generalized Beta Distribution of the Second Kind as a Prior in Stan Linear Regression

So I'm considering a simple linear regression model with $p = 1$ predictor $$y = \beta x + \epsilon$$ where $\epsilon \sim N(0,\sigma^2)$. I want to use a generalised beta distribution of the second ...
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Priors that do not become irrelevant with large sample sizes

This may be a weird question. My colleagues and I are working on a medical estimation problem, where relevant prior knowledge regarding plausible values of some physiological parameters exists. In ...
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Bayesian replication, but with new variables

Suppose I have data I've collected containing predictor variables $X_1, X_2$, and $X_3$. I build a main effects statistical model predicting $Y$ from these predictors and estimate the relevant ...
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How to use priors on the parameter number with an information criterion (AIC, BIC, …)?

Example The example is made up because I hope that it’s more accessible than my actual problem. I want to determine the number of planets of a star. I have: data for some astronomical observable of ...
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How do Sparsity Priors help for Identifiability?

Let's say we have a Factor Analysis model with a latent variable $\mathbf{z}_t \in \mathbb{R}^k$: $$x_t = A z_t + \epsilon_t, \qquad \epsilon_t \sim \mathcal{N}(0, \Sigma)$$ Let $A \in \mathbb{R}^{g\...
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What is the statistical model for a multi-label problem?

In a setting with a binary $y$ like dog/cat, a reasonable statistical model is to posit that the probability parameter $p$ of a $\text{Binomial}(1, 0)$ distribution is some function $f$ of features $X$...
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How should I deduce the conjugate prior and corresponding posterior for a geometric distribution

The given pmf is for a geometric distribution and is $f(x_i|\theta) = (1-\theta)^{x_i - 1}\theta; ~x_i = 1, 2 ,\cdots, $ and the 1-parameter exponential family I have obtained is; $$f(x|\theta) = \exp ...
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Prior on a dirichlet distribution [duplicate]

I would like to know if there is a "conjugate" prior that we can place on the Dirichlet distribution parameters. Thanks in advance,
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How to use Gibb's sampling when the conditional probability doesn't depend on the observations [closed]

I have a model that looks like this $$ x(k) = \sum_{m}^{M} e^{i (U_m k + \beta_m)} + n(k)$$ Where $U_m$ has a Gaussian distribution with parameters $\mu$ and $\sigma^2$. $$ U_m \sim \mathcal{N}(\mu, \...
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Posterior distribution when the domain of the likelihood depends on the parameter

I am trying to calculate a posterior density given distribution and a prior. And I am a bit confused about how I should act as the domain of the distribution depends on the parameter. I am talking ...
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How to update a prior probability distribution of hurricane occurrence based on absence of hurricanes to date?

For a forecasting tournament, I am trying to forecast the number of Atlantic basin hurricanes in the 2022 hurricane season. I have reason to believe that my prior distribution looks as follows: At ...
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Is there an implicit independence assumption in Bayesian inference between X and parameters?

I often see things like $$ p(w|X,y) \propto p(y|X,w) p(w)$$ where $w\in\mathbb R^p$ denotes some parameters, $y\in \mathbb R^n$ denotes some observed outcome values, and $X\in \mathbb R^{n\times d}$ ...
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For multivariate normal posterior with improper prior, why posterior is proper only if $n\geq d$

This is related to Gelman's BDA chapter 3 section 5's noninformative prior density for $\mu$. Let $\Sigma$ be fixed positive definite symmetric matrix of size $d$ by $d$. Let $y_1,\dots, y_n$ be iid ...
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Parameterizing priors with related data-sets

I have a Bayesian model I'm using to compare two data-sets X_1 and X_2. I have a prior distribution I would like to use. This distribution could be parameterized by the mean and variance of some data-...
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Bayesian Logistic Regression to Optimize Model Weights

I am new to Bayesian Inference and I want to understand how Bayesian Logistic Regression optimizes the weights of a regression model. To elaborate on a specific example I came across weights, coef1 ...
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Reparametrizing a Uniform Prior Distribution to Multivariate Standard Normal

Problem Description I have a posterior distribution $$ p(\theta\mid y) \propto p(y \mid \theta) p(\theta) $$ with a uniform prior $p(\theta)= \mathcal{U}([a, b]^n)$, which is bounded. However, for my ...
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Does it make sense to do a prior sensitivity analysis if using flat priors (Mplus default)?

If it does make sense to do a sensitivity analysis how should one determine which priors to use? If flat "non-informative" priors are chosen to begin with because of a lack of information ...
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Find a likelihood to calculate a posterior probability

I am having trouble understanding a basic Bayesian inference exercise: Suppose we are interested in inferring the proportion $\theta$ of individuals in a given population suffering from a certain ...
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Conjugate Prior for Multivariate Normal Variances and Correlations

Is there a way to separately specify conjugate priors for the variance and correlations of a multivariate normal? The inverse Wishart is conjugate if you want to specify the covariance, but covariance ...
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Which form of Jeffrey's prior can be used for a three-parameter distribution?

Let X be a random variable which follows a distribution, say S with parameters a, b and c. Knowing that or Assuming that a, b and c are independent of one another, which one is reasonable to do? a) Is ...
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Is a data size in a binomial distribution random variable?

Supposing that there is a binomial distribution ${\rm Bin}(m|N, \mu)$, I think usually $N$ and $\mu$ are parameters and not random variables (or events), thus the notation here ${\rm Bin}(m|N, \mu)$ ...
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How can I find the posterior distribution for gammadistributed data and prior?

I'm working on a project where I believe Bayesian statistics should be useful. However, my knowledge about bayesian statistics are very scarce. Suppose I got data following a Gammadistribution with a ...
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Bayesian SEM: Selecting (Weakly) Informative Prior

I am looking for recommendations regarding how to select informative (or minimally weakly informative) priors for a Bayesian SEM model? I have no specific predictions with regards to my model / ...
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Mixtures vs Multi-level models?

I'm confused on how mixture models and multi-level models are different (if at all.) Are there general rules for when to use one and not the other, pros/cons, etc?
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References for the conjugate prior to the beta distribution?

The Wikipedia article about "Conjugate Prior" has a table containing information about Likelihood Distributions with their Conjugate Priors. In the "Continuous Likelihood" table, ...
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Nested sampling: What does "uniform sampling over the prior" mean?

I'm reading up on Nested Sampling in the book "Data Analysis - A Bayesian Tutorial" (Sivia and Skilling, 2006), and I do not understand the following: What I understand: Given a prior $\pi(\...
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How to find the marginal prior distribution?

Suppose that $\beta$ has the following prior $$ \beta|\zeta \sim f(\beta,\zeta) $$ Then I know that the marginal prior distribution of $\beta$ is given by $$ \int f(\beta,\zeta) d\zeta $$ However, ...
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Is it possible to estimate the parameters of a superposition of Poisson processes through Bayesian inference from a binarized sequence?

My question is complementary to a previous problem : Bayesian inference on binarized Poisson distribution. I retake the previous notations. Problem description : I am counting the number of balls ...
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Creating a better prior based on past observations

Based on this post, In plain english, update a prior in bayesian inference means that you start with some guesses about the probability of an event occuring (prior probability), then you observe what ...
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Bayesian Prior definition [closed]

The prior of an inference problem where we try to infer $x$ from observations $y$ is defined as $P(X)$. Often (e.g.) I see another definition where the prior is defined as $P(X|Q)$, what exactly is $Q$...
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In Bayesian hierarchical models, what is the difference between an Empirical Bayesian approach to parametrising priors vs using flat hyperpriors

Say I have a simple hierarchical model, where: $y_{g,i} = \beta_g x_{g,i} + e_{g,i}$ where $g$ represents the group, $i$ represents the individual within the group, and $e$ is the error. So the ...
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Inverse or reciprocal distribution of a discrete random variable

https://en.m.wikipedia.org/wiki/Inverse_distribution Above wikipedia article only talks about continuous random variables. If Y=1/X, where X is strictly positive, and density function for X is given ...
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Non-Dirichlet Prior for $Cat(\theta)$ parameter that can tractably be integrated out (for Latent Dirichlet Analysis)?

In LDA Topic Models, it is standard to 'integrate out' the $\theta$ parameter, which contains a document's Categorical probabilities of drawing each topic. QUESTION If one uses the standard Dirichlet ...
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Bayesian statistics: what is the variable we are integrating in?

This is a screenshot from Bayesian Data Analysis by Gelman. I am a little bit confused by Equation 1.4 (first and second lines), having read Equation 1.3. In Equation 1.3, the variable of integration ...
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Incorporating prior evidence of predictor having no effect in bayesian linear regression model

Say we start with a linear regression model of the form $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon, \quad \epsilon \sim N(0, \sigma^2)$$ with the conjugate prior $$ \begin{align*} &\sigma^...
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Prior selection in Gaussian - an application to height measurement

Say I have just purchased ACME's Tree Height Measuring Device (THMD). ACME states that the error $\epsilon$ in tree height measurement from this device can be modelled as a normal distribution with ...
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The PDF of the Data (Marginal Likelihood) Given the Prior of a Gamma Distribution with Prior on the $ \beta $ Paraneter

Given a model where $ x_i | \beta \sim \mathcal{Gamma} ( \alpha, \beta ) $ where $ \beta \sim \mathcal{Gamma} ( \alpha0, \beta0 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what'...
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The PDF of the Data Given (Marginal Likelihood) the Likelihood and the Prior of a Normal Distribution with Prior on the Mean

Given a model where $ x_i | \mu \sim \mathcal{N} ( \mu, \sigma^2 ) $ where $ \mu \sim \mathcal{N} ( \mu_0, \sigma_0^2 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what's $ p (...
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Is there any strong argument about objective/non-informative improper prior?

Decades ago improper objective priors - e.g. $\pi(\sigma) \propto \sigma^{-1}, \sigma > 0,$ for a scale parameter - were considered problematic because some authors thought they were leading to the ...
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Beta distribution equivalence with two redondant parameters [duplicate]

context In Factor graphs on discrete variables, the parameters are contained in factors associated each with a subset of the random variables in the system. Each factor provides a different positive ...
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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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Bayesian statistics

Assuming I have that $Y_i\mid \mu$ is an iid ~ $N(\mu,\sigma^2)$, for $i \in (1,\dotsc,n)$ with $\sigma_i$ known and improper prior $\pi(\mu)=1$ for all $\mu$. i. How can I derive a formula for the ...
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Bayes: How to use results from a single study to shape data-driven priors

One way to construct informative priors for a subsequent Bayesian analysis is to carry out a meta-analysis of previous studies. Here, substantial research has been done. However, what to do when there ...
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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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Non-informative prior of a geometric distribution [duplicate]

If we are given a standard geometric distribution $(1-p)^{x-1} p$, with $0<p<1$ what would be a suitable non-informative prior for this?
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Informative priors for standard deviation (or variance)

Suppose I want to perform Bayesian estimation of the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution. Is there a standard way to specify an informative prior over $\sigma$, ...

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