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.

Filter by
Sorted by
Tagged with
1 vote
2 answers
37 views

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
0 answers
16 views

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 ...
14 votes
2 answers
2k views

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 ...
0 votes
0 answers
17 views

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 ...
0 votes
0 answers
11 views

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 ...
7 votes
2 answers
128 views

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 $X$...
16 votes
4 answers
1k views

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 ...
2 votes
1 answer
417 views

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
1 answer
2k views

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 ...
0 votes
0 answers
22 views

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
1 answer
90 views

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 ...
0 votes
1 answer
33 views

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(\...
5 votes
2 answers
100 views

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\...
3 votes
1 answer
196 views

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 $l_1$...
4 votes
0 answers
1k views

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,...
5 votes
0 answers
833 views

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 ...
0 votes
0 answers
25 views

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 ...
1 vote
1 answer
71 views

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, \...
0 votes
0 answers
17 views

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,
0 votes
1 answer
36 views

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 ...
0 votes
0 answers
40 views

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
0 answers
19 views

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
1 answer
28 views

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 ...
0 votes
1 answer
225 views

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 ...
0 votes
0 answers
13 views

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-...
0 votes
0 answers
14 views

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 ...
0 votes
0 answers
13 views

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 ...
0 votes
0 answers
12 views

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 ...
2 votes
2 answers
39 views

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 ...
1 vote
0 answers
18 views

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 ...
2 votes
1 answer
394 views

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 ...
20 votes
2 answers
8k views

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
1 answer
31 views

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 ...
2 votes
2 answers
671 views

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 <...
5 votes
2 answers
211 views

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 ${\rm Bin}(m|N, \mu)$ ...
1 vote
1 answer
44 views

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 ...
2 votes
0 answers
33 views

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 / ...
0 votes
1 answer
20 views

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?
0 votes
0 answers
33 views

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
1 answer
60 views

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 ...
2 votes
1 answer
157 views

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 ...
2 votes
1 answer
55 views

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
0 answers
31 views

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
0 answers
32 views

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
0 answers
50 views

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
1 answer
362 views

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....
0 votes
1 answer
368 views

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
1 answer
502 views

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 ...
0 votes
0 answers
34 views

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 ...
0 votes
0 answers
44 views

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 ...

1
2 3 4 5
19