Questions tagged [credible-interval]

A credible interval is an interval in Bayesian statistics that includes the true value of a parameter with $(1−\alpha)\%$ probability. Credible intervals treat the interval as fixed and the parameter as random.

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How do I generate a confidence region for a set of sample from a bivariate posterior?

I have a set of samples generated from a posterior function as shown below: I want to generate a bivariate High Posterior Density (HPD) credible region for the samples as in the below example ($\...
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How to explain the difference between confidence and credible interval?

The key difference between Bayesian statistical inference and frequentist statistical methods concerns the nature of the unknown parameters that you are trying to estimate. In the frequentist ...
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Calculating credible intervals of group means with sum contrasts?

I have fit a Bayesian mixed-effects logistic regression with sum contrasts +/- 0.5. All predictors are binary. But now I don't know how to calculate the credible intervals for the reference level of ...
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Using a hierarchical model (on pymc3) to compute credible intervals for dependent proportions

I have a dataset composed of 2 conditions, each condition consists of 3 samples, each sample subdivided into different categories. I would like to compute 95% credible intervals for the proportions of ...
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Point estimate and 95% credible interval

The text of the problem as follows: The data follows a normal distribution with $\mu$ and $\sigma^2$ unknown. We wish to perform inference on the mean selling price $\mu$. And our sample data are (...
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How to determine the size of highest density region in high dimensions

I want to calculate the "size" of the highest density region (HDR) that contains p% of the total probability for multivariate samples of a Bayesian posterior obtained via MCMC. In 1D this "size" is ...
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Use a Bayesian approach to construct a 95% confidence/credible interval for the mean and variance using conjugate priors

I am looking for information on how to solve this problem above in R. It does not have to be solved using any specific dataset but I would appreciate an example. I am unsure of conjugate priors. ...
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For a univariate Normal distribution, how to show the interval obtained by transforming the HPD interval by $\exp^x$ is not HPD?

Suppose the HPD interval for the Normal distribution we have is [l,u]. I have tried to show that, when we transform the Normal pdf to obtain the pdf of z=$\exp^x$ and set the value at $\exp^l$ to be ...
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Finding a credible region with a normal posterior

I have a posterior distribution that is normal and I want to find a credible region for which $\mu$ lies. However, I am having trouble integrating the following posterior: $\int \pi(\mu|x) = \int_{0....
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Gaussian Process Confidence vs Credible Intervals

Since Gaussian Process returns a distribution and not a point estimate, why this example (and actually in every example with GP) talk about Confidence Intervals on the analogues for Bayesian ...
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BCI coverage and constant theta

I have perused the exchange on the topic and found this answer to me most salient to my problem. Could someone provide a more detailed answer and perhaps a little code snippet the would allow me to ...
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How to best characterize uncertainty for an incidence rate? [duplicate]

Here is the scenario I am trying to model. I have a population of people who are susceptible to developing a disease. I observe each person for a different amount of time, summing to a total of 3000 ...
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Highest Posterior Density for Poisson with exponential prior

Let $Y$ be random variable with $Poiss(\theta)$ distribution. The parameter $\theta$ is a realization of a random variable $\Theta$ with a priori distribution $Exp(\lambda)$. The task is to find HPD ...
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Is confidence interval determined before observing data?

My professor is comparing the frequentist confidence interval and the bayesian credible interval. He claims that a confidence interval is determined prior to observing the data, while the credible ...
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Am I supposed to change Credible intervals to probabilities when reporting them?

I am using brms with family = bernoulli(). The coefficients, if I understand correctly, are in log odds. Here is a piece of the output of the population-level effects: ...
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Interpretations of negative confidence interval

Let's say I measured the weights of 50 chickens from my family farm, which keeps 1000 chickens. The sample mean is 5 kg, SEM is ± 3 kg, and the 95% confidence interval is 5 ± 3 * 1.96 = -0.88 kg to 10....
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One- vs two-sided credible interval for Poisson process with all zero counts

In a related question, I asked about a confidence interval for the estimate of the mean of 50 observations of a Poisson random variable, for which all 50 observations had a count of zero. In the ...
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Posterior variance vs variance of the posterior mean

This question is about the frequentist properties of Bayesian methods. Suppose we have data ${\bf y}$ generated from a distribution with a single parameter $\theta$, equipped with a prior $\pi(\...
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Estimating a population proportion and credible interval from a biased stratified sample

I am attempting to calculate a point estimate and the upper 95th percentile (one sided credible interval) for a defect rate (i.e. a population proportion) using the results of a biased stratified ...
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Do Bayesian credible intervals treat the estimated parameter as a random variable?

I read the following paragraph on Wikipedia recently: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat ...
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Likelihood, posterior, prior interpretation and credibility/confidence_level with bayesian/frequentist approaches

This question was originally posted on physics exchange but one advised me to transfer it here. I try to understand the following article : testing general relativity from curvature and energy ...
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Why is a frequentist confidence interval equivalent to a credible interval with flat priors?

It's a commonly quoted result that frequentist confidence intervals are equivalent to a bayesian credible interval assuming a flat prior. Ignoring for now questions about invariance under ...
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When does a confidence interval “make sense” but the corresponding credible interval does not?

It is often the case that a confidence interval with 95% coverage is very similar to a credible interval that contains 95% of the posterior density. This happens when the prior is uniform or near ...
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How should I treat categorical variables in Bayesian modelling

I have got a dataframe that contains three categorical predictors and one numerical response. I would like to compare their differences using posterior uncertainty intervals of MCMC draws. The reason ...
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meaning of posterior distribution and credible internal

In Bayesian method, we can get a posterior distribution of a parameter. Now I want to do some simulation to know if the posterior distribution is the same as the true distribution. For example, mean ...
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207 views

Interpretation of confidence interval in Bayesian terms

Motivation: I was standing in front of a class to introduce into the concept of confidence interval using the example of differences in means (purely frequentist setting) and I was torturing the ...
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credible interval in bayes are higher than the nominal level

When I fitted the nonlinear regression using the openbugs, and calculated the 95% credible interval of the coefficient through the high density interval, I found that the total number of the credible ...
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Monte Carlo integration for Bayesian parameter estimation

I want to determine the credible interval of a quantity $\theta_1$. I want to make this estimate using observed data by assuming a certain model which depends on $\theta_1$ as well as about n=15 ...
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Computing the credible region from a pre-computed list of function evaluations

I have a nonlinear function of four variables $f(x_1,x_2,x_3,x_4)$. This function outputs three things: $[y_1, y_2, y_3]$. (In reality, the function $f$ is a simulation which solves the equations of ...
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253 views

How do I calculate the confidence interval from a Bayesian analysis

I got the posterior distribution of a parameter from Bayesian analysis. I want to express it as confidence interval. If I plot the empirical cumulative distribution function of the parameter and ...
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A question on Bayesian credible interval vs frequentist confidence interval

The difference of Bayesian credible interval (BCI) and the frequentist confidence interval (FCI) is well explained with a nice example in this answer. Here is my own summary of the situation in the ...
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Find bayesian credible Interval

I am taking this example from here. They have given the steps but i cannot understand them so a little dumbing down of answer is necessary or you can just explain the answer given there. its on page 5 ...
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98 views

My credible interval never contains the true value when it is at the edge of the data

I am estimating an unknown variable which is between 0 and 1. I start off with a uniform prior on [0, 1]. Then I update it given some evidence (flip the coin a few times). Now I have a posterior. ...
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The basic logic of constructing a confidence interval

Consider a model with a parameter of interest, $\theta$, and its point estimator, $\hat\theta$. For simplicity, assume $\hat\theta\sim N(\theta,\sigma^2/n)$ (in numerous instances this could be ...
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What formula for a Confidence Interval of the difference in proportions when sample sizes are small

Suppose that we are interested in comparing two approximately normal sampling distributions described by random variables $ \displaystyle \frac{Y_1}{n_1} = N(p_1,p_1q_1) $ and $ \displaystyle \frac{...
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Highest posterior interval and monotone changes of variables

Suppose $X$ is distributed with a unimodal pdf $f(x)$ and let $Y = g(X)$ for some strictly monotone function $g$. Hence $g$ is invertible. Is there an analytically tractable relationship between the ...
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Predict credible interval of Poisson-distributed response based on Lambda credible interval

I am approaching Bayesian inference. Could you review my steps and give me a hand with my model predictions? I am using a N-mixture model to predict how many individuals N of a rare species inhabit a ...
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40 views

credible intervals for functions of hyperparameters

If I have a statistic $\nu(x, y)$ which is a function of hyperparameters (say just two for ease of explanation) $x$ and $y$ of a distribution $F(t|x , y)$ and associated prior $g(x,y)$ which is the ...
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Constructing credible interval for store opening times

Overview: Suppose I have a log which records the time at which customers visit a store. For the sake of this example, say I have 10 stores in the dataset. Example data for Store 1: Customer 828: 9:...
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Joint credible regions from MCMC draws

Lets say I have $n$ posterior samples of $\theta_1$ and $\theta_2$. I suppose that any region $R$ which contains exactly $(1-\alpha)n$ of the points will be an approximate $(1-\alpha)\times100$ ...
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693 views

Highest Posterior Density (HPD) region of the marginals vs. of the joint distribution

In a Bayesian context, to analyse the posterior distribution, one can define the Highest Posterior Density (HPD) region or interval as $$\{\theta; \pi(\theta \mid x) \geq k\} $$ in both unidimensional ...
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If a credible interval has a flat prior, is a 95% confidence interval equal to a 95% credible interval?

I'm very new to Bayesian statistics, and this may be a silly question. Nevertheless: Consider a credible interval with a prior that specifies a uniform distribution. For example, from 0 to 1, where 0 ...
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Computing Highest Density Region given multivariate normal distribution with dimension $d$ > 3

Background: Suppose I have a unimodal symmetric distribution of dimension $d$ > 3 such as the multivariate normal distribution ~ N(0, H), where H is a known $d$-dimensional covariance matrix. ...
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Bayesian credible interval calculation: posterior distribution or marginal distribution

Let's say we want to obtain the 95% credible interval for theta, where theta|x ~ Beta(a,b), which one of the following is correct and why? [E(theta|x) - 2*se(theta|x), E(theta|x) + 2*se(theta|x)] [E(...
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970 views

How to calculate a 95% Credible Interval for a Bernouli sampling in R?

How does one calculate a 95% Credible Interval for Bernouli sampling? I have looked into the package emdbook and the function tcredint() but i am unsure if this the way to go. If anyone has ...
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Estimate the mean and variance 95% HPD credible region using Bayesian inference

I have the following data: 31.0, 30.5, 20.6, 27.2, 26.5, 28.1, 25.8, 29.6, 30.0, 25.8, 25.1, 27.9, 23.0, 29.4, 28.7, 25.0, 31.1, 24.8, 24.8, 27.0, 22.3, 29.5, 31.5, 26.2, 24.6, 23.2, 25.7, 24.2, 28.8,...
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Is it valid to average the credible intervals from many simulations to obtain an average credible interval?

I am currently doing a Bayesian analysis where as output, I obtain a a point estimate. Each point estimate has an associated credible interval. Now, I am hoping to do the analysis 1000 times, then ...
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What does Bayesian Comparison of Groups and Posterior Interval say about my Hypothesis?

I am comparing the score of two groups: A and B. The score is normally distributed and a two sample t-test yields a p-value >0.05. Therefore I have to reject the Hypothesis that there is significant ...
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Finding credible intervals for hyperparameters in Bayesian inference

I'm trying to use Bayesian inference to fit and interpret a linear model of the following form: $$ y=X\beta + \epsilon \hspace{1cm} \text{where } \epsilon_i \sim \mathcal{N}(0,\sigma^2)$$ The ...
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558 views

Reduce credible intervals in Causal Impact model

I'm hitting an issue with a causal impact model that I'm building. I'm trying to create a counter factual for daily sales at one store (nseasons = 7). I've included sales for 5 other stores nearby. ...