# 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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...