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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31 views

Posterior of one observation transform into posterior of several observations

Suppose $\mu$ has prior distribution $\mathcal{N}(M, A)$ and $x |\mu \sim \mathcal{N}(\mu, 1)$ After one observation, the posterior is $$\mu|x \sim \mathcal{N}(M + B(x-M), B), \tag{1}$$ where $B \...
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What happens if I change the range of a flat prior for Bayesian inference?

I am working through an example on doing Bayesian inference on binomial distribution using a flat prior, and trying to understand the impact of choosing a prior. I know that if we work with a flat ...
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Can you use a prior on the population distribution when using BTYD?

In Counting Your Customers: Who-Are They and What Will They Do Next?, David C. Schmittlein, Donald G. Morrison, Richard Colombo, 1987. which defines the BTYD model, ...
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Failing to simulate data for a negative binomial probability distribution [migrated]

I am hoping someone can help me. In a beginners workshop I attended, in the process of fitting a multiple regression model, the instructor initially established a prior predictive check using a ...
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31 views

Prior knowledge in context of nullhypothesis testing

I am currently working through the book Doing Bayesian Data Analysis - John Kruschke and have trouble to reason about a text paragraph [Page 315] Suppose that we are not flipping a coin, but we are ...
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Setting priors for a beta distribution in R-INLA

I'm running a regression in R-INLA, where the response variable is proportion of grid squares suffering deforestation (by year, over a 20-year period). Setting the response for the last 5 years to NA ...
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25 views

How can I choose how confident my beta distributed bayesian prior should be?

I am new to Bayesian statistics, and would appreciate help understanding the Prior. I want to combine a small national dataset with a prior from very large international studies, to give a posterior ...
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59 views

Is the prior in Bayes formula a probability or it can also represent a probability distribution?

Given the Bayes formula: $$ p(\theta|D) = \dfrac{p(D|\theta)p(\theta)}{p(D)} $$ If there is a distribution (let's say $g$) over the parameter $\theta$, how should one rewrite the Bayes formula? $D$ is ...
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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 ...
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Is my Stan model correct? The Jeffreys prior for a heteroscedastic mixed-effects model

I am using rstan to derive MCMC samples from a heteroscedastic mixed-effects model with different residual variances $\sigma_j$ for each experimental condition $j$. One assumption is the Jeffreys ...
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Reproduce results of bayesglm with stan_glm

As indicated in the title, I am trying to reproduce the results of the bayesglm function with the stan_glm. In principle, the ...
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35 views

Prior for covariance matrix?

Given a set of data $\{(x_i\pm e_{x,i},\,y_i\pm e_{y,i})\}_i$ (with uncorrelated uncertainties), I want to model it as a multivariate Gaussian function with an unknown mean $\boldsymbol{\mu} $ and a ...
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What are the bayesian prior distributions to use for a binomial model with unknown $n$ and $p$

I a experimenting with a new MCMC software and before I delve into more complicated models I wanted to run some simple simulations. This is a very very simple simulation, so not meant to be very ...
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56 views

bayesian question: why prior = mu * sigma?

I'm doing a course of Fundamental of Bayesian Analysis in Datacamp and these codes were presented. What is the rationale of prior being mu * sigma ? code: ...
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Metrics for assessing the quality of prior distributions

Clarification: My purpose is to compare different methods for selecting/creating priors (or perhaps I should refer to them as predictive distributions for a quantity of interest/parameter). I do not ...
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What are Large Scale and Complicated Priors?

We use priors in Bayesian networks to include prior knowledge in our models. In this context, what are these two terms: -complicated prior -large scale prior I have seen priors like Laplace, zero-mean ...
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How to get prior distribution based on confidence interval [closed]

If we have mean value and 95% confidence interval of a parameter. For example, sensitivity = 0.5, CI = [0.2,0.78](as you can see, it is asymmetric) How to decide the prior distribution? How to ...
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Decision Theory: Why is it called a “least favorable prior”?

I'm currently reading the chapter on Statistical Decision Theory in Larry Wasserman's "All of Statistics". Reading the section 13.4 about Minimax Rules he introduces the so called Least ...
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Postetior from Jeffrey prior of Normal distribtion

Context I am given a sample from normal distribution $v_i \sim N(\gamma \cdot u_i, \sigma^2)$, $i =1,..., n$. I need to obtain the posterior distribution using Jeffreys prior for $\gamma$. My solution ...
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When does this prior dominate likelihood?

This is a simple Bayesian inference problem, where we are trying to infer some weight parameter $w$. Our posterior distribution is $$ P\propto \exp\left(-\frac{1}{\sigma^2} w^Tw\right) \exp\left(-f(w)\...
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Distributions with negative support in JAGS

I am creating a Bayesian regression model where I want to include a prior for a variable that can only have a negative coefficient. What distribution can I use that only has a negative support and is ...
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Prior/Posterior of the Proxy

I am trying to understand a sentence from Caldara and Herbst (2019), who develop a baysian proxy SVAR model. The paragraph is: "In case of weak identification, the prior plays an important role ...
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How are these priors generated? [closed]

I am trying going through an exercise, I don't understand how the information provided in the text below transitions into the parameters displayed in the beta priors. How are these informative priors ...
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How to create a distribution for a feature that is conditional on more than one other variable

I have a variable, r. It has a distribution P(r). I have found two other variables, A and B that are correlated with r. I want to build a distribution P(r) that is conditional on these two variables. ...
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Prior probability of Normal distribution [closed]

I was solving one problem and got to the point where I needed to find the prior probability of the normally distributed variable, with the known mean and variance. I'm a little confused because I've ...
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Prior for Variance Covariance Matrix [closed]

Why in Bayesian Hierarchical Modelling the prior corresponding to a Variance Covariance Matrix is taken to be Inverse Wishart Distribution not Wishart Distribution?
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Bayesian Multinomial in rating

I have a dataset that has two columns:2)The rating of a product from 0:5 and 2) the numbers of votes.Can i make a Bayesian analysis with rating following a multinomial distribution (categories 1-5) ...
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What is the posterior distribution $p(\textbf{f} | \textbf{y})$ for a Gaussian Process regression?

What is the posterior distribution $p(\textbf{f} | \textbf{y})$ for a Gaussian Process regression? Suppose that $p(y_n |x_n, f) = N(f(x_n), \sigma^2)$ with prior on $\textbf{f} = [f(x_1), \ldots f(x_n)...
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51 views

Prior distribution on Bayesian T Test?

I have two subgroups of structure (bone structure) and I want to test if there is any difference of size (area) between them, and if there is, how important this difference is. The first set is a ...
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108 views

What is Cromwell's rule and why is it important for Bayesians?

I have just heard of Cromwell's rule, but I'm not sure I understand it very well. What is Cromwell's rule and why is it important for Bayesian statistics?
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121 views

Showing $X\sim \operatorname{Poi}(\lambda)$ is minimax

Assume that $X$ has $\operatorname{Poisson} (\lambda)$ distribution and the loss function is $\ell(\lambda,a)=\frac{(\lambda-a)^2}{\lambda}$. Now, I want to show that $X$ is minimax. A hint that is ...
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How should past experiments inform power analysis?

One crucial step in power analysis is to guess the effect size. Luckily, I do have 1 similar past experiment. So I do have 1 data point instead of completely guessing what the effect size is. However, ...
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21 views

Converting posteriors to likelihoods by removing prior

I have a set of MCMC chains (i.e., unnormalized posteriors) for a parameter I modeled for a sample of objects. I have a model that requires that I condition on the likelihoods of this parameter. My ...
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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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1answer
97 views

Maximum a posteriori estimate with exponential prior

Lets say that I have N observations that are poisson and i.i.d. The prior is an exponential with parameter 2. I know that the exponential distribution is given by $ \lambda e^{(-\lambda x)} $ But how ...
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82 views

Mathematical foundations to justify the “existence” and appropriateness of using prior distributions?

I claim that there is no a priori reason I should use a probability distribution to express uncertainty about models, claims or parameters. Apparently, (some) Bayesians disagree with that, as they ...
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Should you multiply every observation with the prior when calculating the maximum a posterior?

Lets say I have a number of observations and a prior. The observations are poisson distributed, are i.i.d and the prior is exponential with paramater 2. I want to calculate the maximum a posterior ...
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23 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 ...
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Understanding of scale loss with Gaussian prior

I am reading a paper called PHOSA wherein a scale loss is used. I had the following questions regarding this scale loss in equation 8: What part of this loss function is actually incorporating a ...
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49 views

Determining the Likelihood function from a Uniform Prior

I am trying to find the Bayes factor between two models, which I understand is the ratio of the likelihood functions of each model. The second model has a uniform prior described by: $U(A; -a, a) = \...
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1answer
36 views

How is the likelihood different from the posterior? [duplicate]

I come from an applied mathematics background and have never looked at statistics, but I started studying Machine Learning recently. One thing that I am struggling to understand is: what is the ...
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19 views

Determining the mean of a posterior probability

There is a comet whose model show it is moving along a straight line $$ x(t) = mt + b $$ The posterior predictive distribution of a comet's position, which is $$ PPD(x) = p(x|t,d) ∝ \exp\left( -\frac{...
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1answer
83 views

Confusion about prior used in Recursive Bayes Filter

I'm currently using this thesis to understand key concepts about probabilistic inference in computer vision which is being a great source. The frame of the question is the following: Let us assume we ...
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250 views

How does one place an uninformative prior on a Gamma Distribution?

I'd like to choose an uninformative prior for the scale and shape parameters of the Gamma distribution. Any help and suggestions will be appreciated.
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43 views

How to calculate the posterior distribution from the density

I'm stuck on a answer from an old exam. The task is to use a Poisson distribution and a Gamma distribution as prior to calculate the posterior density: $$ p(\lambda|x) \propto L(\lambda)p(\lambda)\...
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How to jointly model $N$ groups where data in each group is i.i.d. Normal and infer the posterior distribution?

I am given the following data of income scores of individuals from $N$ groups: $$(\textbf{x}_1, \textbf{x}_2 \ldots \textbf{x}_N),$$ where $$\textbf{x}_j = (x_j^1, x_j^2 \ldots x_j^{N_j}),\quad j = 1, ...
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35 views

Choosing the Dirichlet prior in a mixture model

Consider the following mixture model with $K < \infty$ components, $$ f\left(x \mid \theta_{1}, \ldots, \theta_{K}, \pi_{1}, \ldots, \pi_{K}\right)=\sum_{k=1}^K \pi_{k} \varphi\left(x \mid \theta_{...
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What is the “prior standard deviation of the modelled predictive means” and how do you calculate this?

In the book Regression and Other Stories (Gelman et al., 2021, p. 208), there is an example where a multi-linear regression model has: $26$ coefficients; standardised predictors with mean $0$ and ...
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Bayesian Statistics: Properly updating the Prior for new analysis

I have three tables of information about $A$ and $B$ (gray cells, black font), their row and column marginal totals (black cells white font), and the grand total (white cell black font). The first two ...
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26 views

How to choose priors for experimental data

My question results from the subjectivity of priors, and if there are bodies of work that help to create a more objective approach towards prior choices. My question specifically is to do in the realm ...

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