# 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 ...
1 vote
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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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### What is the "effective sample size" of the prior in Bayesian statistics?

In Bayesian statistic, what is the mathematical definition of "effective sample size" of the prior? Could you provide what the "effective sample size" is for the well known classes ...
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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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### 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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### 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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### Default Priors for Intercept and Standard Deviations in R package brms

The only resource I found explaining the default priors in brms is its manual (newest version, updated 03/14/2021) for function set_prior(). For the intercept, the manual does not specify how the ...
1 vote
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### How to calculate the confidence interval with weighted data?

I've done some search for similar questions, but they're not the same as what I'm trying to get. Assume that there's a server that handles requests $r$ and returns a set of items $I_{r}$ of random ...
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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 ...
1 vote
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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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### Lasso (or Ridge) vs Bayesian MAP

This is the first time I have posted here. I am looking for some feedback or perspective on this question. To make it simple, let's just talk about linear models. We know the MLE solution for the $l_1$...
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### Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?

I’d like to share and submit for (dis)approval and discussion yet another, simple but original (to the best of my knowledge) Bayesian solution to the classical problem of comparing k samples or groups,...
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### Bayes-Poincaré solution to the Behrens-Fisher problem 2: calculations for Jeffreys’ priors [closed]

In a previous post Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?, the classical Bayesian and likelihoodist solutions to 2-sample tests for comparison and the ...
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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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### 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 ...
1 vote
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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 $X\in \mathbb R^{n\times d}$ ...
1 vote
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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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### pymc3: Updating the standard error prior

I am estimating a Bayesian multiple regression using continuous data on both the dependent variable and the regressors. My goal is to iteratively estimate the coefficient distributions as more data ...
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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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### A transformation from uniform random variable to Gaussian mixture

I am attempting to describe a prior_transform for a multivariate Gaussian mixture in order to estimate the evidence integral of that prior convolved with another likelihood distribution. This is ...
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### What prior distributions could/should be used for the variance in a hierarchical bayesisan model when the mean variance is of interest?

In his widely cited paper Prior distributions for variance parameters in hierarchical models (916 citation so far on Google Scholar) Gelman proposes that good non-informative prior distributions for ...
1 vote
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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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### How to derive the noninformative prior for location parameters and scale parameter?

I am reading this paper, in it: I have a lot of confusion reading it, I will list it one by one: Let $X$ be distributed as $f(x-\theta)$, which is a location invariant density. Q1: The sentence <...
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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, ...
1 vote
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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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### How do I get around this "Argument 'coef' may not be specified when using boundaries."

I have a model, the brms code is given below. It is a system of equations (I am estimating demand for two categories of goods). Economic theory tells me that the intercepts have to be restricted to ...
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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, ...
1 vote
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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 ...
1 vote
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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 $Q$...
1 vote
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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 ...
1 vote
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### location/scale invariant priors

I'm trying to understand what's the motivation behind these priors, and why they are used. I understand that for location parameters of some distribution, you want it to be invariant of movement. e.g....
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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, ...
1 vote
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### Priors and nested random effects in MCMCglmm?

I am trying to construct a zero inflation Poisson GLMM using MCMCglmm(). I am new to Bayesian Statistics and this function and I am struggling to understand a couple of things. For my data I am ...
### 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 ...