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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Hierarchical clustering with a prior

I would like to perform a clustering (in the best case scenario a hierarchical clustering) of N entities and the distance among those entities is a known input. I also have an a priori on the ...
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Noninformative prior distribution: uniform or normal? [closed]

The uniform distribution, with the support that has a finite measure, guarantees that the entropy is maximum(as stated in this answer), but in our daily life, normal distribution seems more ...
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Can a range of priors being used for a linear regression be applied to a logistic regression?

I have trial level data from a study in which participants responded to a series of stimuli. I have a predictor of interest. For the sake of this example, let's call it the size of the stimulus. There ...
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Theoretical Justification for Zellner's g Prior

What is the theoretical justification for Zellner's g prior for linear regression? I cannot see how it is possible to justify from a purely Bayesian perspective, in which probabilities are epistemic, ...
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Prior and posterior distributions involving a prior Beta distribution [duplicate]

Question: A poll is conducted to help ascertain whether the Labour party candidate or Tory candidate will win in a forthcoming election for Coventry Mayor ( there are no other candidates, and the ...
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What does it mean to say that “the prior over $f$ induces a prior over probabilistic classifications $\pi$”?

I am currently studying the textbook Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams. Chapter 1 Introduction says the following: We now turn to the ...
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Gaussian processes: The uncertainty is reduced close to the observations?

I am currently studying the textbook Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams. Chapter 1 Introduction says the following: In this section we ...
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Why does a function being smoother make it more likely?

I am currently studying the textbook Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams. Chapter 1 Introduction says the following: Given this training ...
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Numbers of draws on a modified Bernouilli process

Here is the setup: Bob runs an experiment: he flips a coin N times (between 0 and +$\infty$). The coin has a probability p of landing on heads. Bob starts with zero points. For each head, Bob scores a ...
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Posterior distribution of two normal samples

I have a task to find the posterior distribution $θ|(x,y)$ of two random samples $x=(x_1,...,x_n)\sim N(θ,σ_1^2)$ and $y=(y_1,...,y_n)\sim N(θ,σ_2^2)$. The prior I've got is $θ\sim N(μ_0,σ_0^2)$. I've ...
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What options does a person have for showing evidence in favour of the null hypothesis?

I have a linear mixed-effects model with a theoretically important null result. Of course a reviewer asked for a Bayesian approach to "show evidence" for it. However I am struggling with ...
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How do I implement a default prior of cauchy(0,1) in rstanarm?

What I intend to do is use a default prior on my coefficients, and then to compute Bayes Factors for those coefficients. Rouder and Morey (2012) say: "When using the Cauchy prior, s describes the ...
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R- Non-informative vs Informative Prior for Bayesian Logistic Regression

I'm kinda new to Bayesian Statistics and I'd like to try to fit Bayesian Logistic Regression but I don't have prior knowledge about my dataset. So, I guess I have to use non-informative prior for ...
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Showing that a posterior is Normal given improper prior

I am having difficulty showing the following problem and I suspect it has something to do with my lack of understanding of the question. The question is this: Suppose we have an improper prior ...
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How to implement a default prior in a stan_lmer() model?

I have found Rouder and Morey (2012) suggesting a default prior of cauchy(0,1). I would like to implement this in a linear mixed effects model I’m computing using stan_lmer(). However I have both ...
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How to prepare a dichotomous predictor for the same prior as continuous predictors?

I would like to use a standard weakly informative prior in my model (i.e., normal(0, 1)). I believe that I would scale this to the mean and sd of my dependent variable. For example, if my DV is ...
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Can the posterior mean always be expressed as a weighted sum of the maximum likelihood estimate and the prior mean?

See this question. Is this always true? Can the posterior mean always be expressed as a weighted sum of the maximum likelihood estimate and the prior mean (after choosing some appropriate prior)?
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How to check the robustness of a null result of an lmer() model using a Bayesian analysis?

I have an lmer() model that has a theoretically important null result. I would like to use a Bayesian analysis to check the robustness of this null result. What is the best way to do this? I had ...
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Under what circumstances can an improper prior be used in bayesian analysis?

I am attempting to gain some intuition about the use of priors in bayesian analysis. I have read in some instances that an improper prior can be used when no information is known. However here is my ...
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Should prior distribution reflect stationarity assumptions?

In the paper Dynamic Hierarchical Factor Models they present a four-level dynamic factor model and estimate it using a Gibbs sampler. One interesting feature of the model is that the error terms are ...
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Question on working out a pdf from a posterior distribution

Need help with part (a) of this exercise. The exact step I'm concerned about is calculating the pdf from the relationship given in the exercise. I will appreciate any explanations on how should I ...
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Bayesian decision boundary for bivariate uniform distributions

I would like an analytical derivation of the Bayesian decision boundary between 2 bivariate uniform distributions. Let me explain with an example. Suppose the two distributions are $$U_1 = 0.25 \text{ ...
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How to parametrize a posterior to use it as a prior in Bayesian statistics?

In my problem, I have two sets of parameters, $\theta_1$ and $\theta_2$, and two datasets $d_1,d_2$ that constrain them with a known likelihood function. There is a certain 'hierarchy' in the model: ...
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Fitting logistic regression with positive priors

I'm trying to fit a Logistic regression with positive priors for the coefficients. I tried bayesglm package but the only priors permitted are t-student, normal and Cauchy. For rstanarm, the prior ...
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Update beta distributed prior with data that is a probability

In my experience with Bayesian statistics, beta distributions are typically used to estimate the posterior for parameter, $p$, of a binomial distribution that has been used to generate some data. But ...
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Linear combination of conjugate prior

Let's say we want to find the posterior distribution for $\Theta$, where the likelihood model $X|\Theta$ ~ $Binom(8000, \Theta)$. Suppose instead of one distribution for the prior, we use a linear ...
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Are there conjugate priors used in nonparametric density estimation methods?

I am still quite new to Bayesian inference and non-parametric methods. I was wondering if conjugate priors feature in non-parametrix density estimation. I understand that in non-parametric density ...
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Likelihood-free inference: what is a tractable prior distribution of parameters?

I am reading the normalizing flows review article by Deepmind and came across a sentence that I don't understand in section 6.2.4 Likelihood-Free Inference regarding use of normalizing flows to ...
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Choosing a prior for the intercept in a logistic regression with increased -INF probability?

I am trying to fit a simple logistic regression of the kind: n ~ binomial(N, theta) theta = inv_logit( a + x * b ) where x is either 0 or 1 depending if a ...
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Parametrization of Dirichlet distribution

Hej! Consider I have a Dirichlet distribution with 4 variables, where the mean (u) values of these are known. $(u1+u2+u3+u4=1)$ Now, I want to obtain the parameters of the Dirichlet distribution ($\...
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Deriving posterior distribution for variance of normal distribution

I have a task to derive posterior distribution for parameter $\sigma^2$, given that the data vector $y^t = (y_1,...,y_t)$ is from $N(0,\sigma^2)$. The uninformative prior for $\sigma^2$ is $h(\sigma^2)...
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Posterior Distribution of a Normal Sample using Jeffreys Prior with a Known Parameter

Suppose I have a sample of $x_1, x_2, ... x_n$, where $X \sim N(\mu, \sigma^2)$, for some known $\sigma^2$, and that $\mu$ is defined only in $\mu \in [0, b]$, for some finite constant $b$. It then ...
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Behaviour of the marginal in the limit for an infinite sequence of hierarchical priors

Consider the following model: $$y \sim \text{Exponential}(\lambda_0) \\ \lambda_i | \lambda_{i+1} \sim \text{Exponential}(\lambda_i+1) \\ \text{for } i=1,2,\dots,d\\ \lambda_{d+1} = k $$ With an ...
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When to stop the chain of priors in Bayesian hierarchical models?

From Wkipedia's article on hyperprior: In Bayesian statistics, a hyperprior is a prior distribution on a hyperparameter, that is, on a parameter of a prior distribution. There will be some ...
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Reparameterising likelihood

In the comments of this question it is mentioned that, when calculating the log posterior of a Normal distribution with a uniform prior on $(\mu, \log\sigma)$, we can write down the same likelihood ...
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How to use maximum likelihood to estimate the prior probabilities?

Suppose we have 2 classes k = 1, 2 and the class conditioned densities are given by Gaussian distributions with a shared covariance matrix. Suppose we are given a training data set {(xi; yi)} where i =...
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Why do we reparameterize before assigning a hyperprior distribution?

I am studying hierarchical models, and trying to understand a point in the book where they try to decide on a non-informative hyperprior distribution. The hyperparameters is $\alpha$ and $\beta$ for a ...
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74 views

What are some common prior/likelihood choices for Bayesian logistic regression?

I'm not really clear on the Bayesian approach to logistic regression. From everything I've read, the prior and likelihood can be can be whatever you want them to be. Well, I've a couple things; namely,...
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Choosing reasonable priors for Poisson GLMM

I am using the package brms in R to fit a generalized linear mixed model using a Poisson distribution with log link. The model takes count data that ranges from 0 ...
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Why prior distribution is not conditioned on X?

I would like to know why in the below formula the prior distribution of theta is not conditioned on X (observations): $$P(\theta|X, y)=\frac{P(y|X, \theta)P(\theta)}{P(y|X)}$$ In my understanding, the ...
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How does Prior Variance Affect Discrepancy between MLE and Posterior Expectation

Suppose that $\theta\in R$ is a parameter of interest, $p(\theta)$ is our prior belief regarding $\theta$, and $\hat \theta$ is the MLE for theta derived from the data $x$. It is my understanding that ...
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108 views

Find the prior distribution for the natural parameter of an exponential family

Show that for the binomial likelihood $y$ ~$Bin(n, \theta)$, $p(\theta) \propto \theta^{-1} (1-\theta)^{-1}$ is the uniform prior distribution for the natural parameter of the exponential family. I am ...
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Estimate with known sample mean, sample size, prior mean, prior standard deviation?

I want to estimate the actual "eval" of a chess move (in this case, expected win rate - expected loss rate, ranging from -100 to +100). I have empirically calculated that on average, random ...
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112 views

posterior distribution of a Poisson mixture model

This is a Poisson-gamma model with mixture prior, thus mixture posterior. I am having some trouble finding the posterior weightings. I have the prior weightings $p_1=1/3$; $p_2=2/3$. The 2 component ...
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How do I put different priors on different levels of a categorical variable in brms?

This is a just a coding query from a bayesian novice. I have a model of this type: ...
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69 views

Regarding the use of non informative priors

I am a beginner to Bayesian analysis and I am trying to fit a logistic regression model using Bayesian approach. For the prior distribution of the $\beta$ regression coefficients , I used a non ...
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Bayesian Inference question: What was the likelihood that my observation originated from one distribution versus another?

While analyzing one of my datasets, I noticed that a subset of my data has some interesting distinguishing features. The light line represents the distributions of the blue/red/green feature for all ...
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How to describe an “incomplete” prior?

I would like to know how to describe sources of uncertainty neglected when I approximate a prior distribution $p(x)$ by a marginal distribution. Specifically, let's say that I have a marginal ...
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385 views

What is a non-informative choice of parameters for a Dirichlet distribution?

Dirichlet distribution is a conjugate prior for multinomial distribution. I want to impose a non-informative prior over sampling weights $\pi$ for a draw $x=(x_1,…,x_N)$ from a multinomial ...
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On the choice of prior in Bayesian Bootstrap

Let $d=(d_1,…,d_K)$ be a vector of all the possible values that the data $x=(x_1,…,x_N)$ could possibly take. Then, each $x_i$ is modeled as being drawn from the $K$ possible values where the ...

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