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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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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26 views

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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40 views

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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151 views

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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32 views

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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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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173 views

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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30 views

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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153 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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197 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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309 views

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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94 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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260 views

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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266 views

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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39 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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187 views

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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634 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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361 views

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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890 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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478 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. ...
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182 views

Credible Interval Format?

I'm reading a datasheet that has listed a "credible interval". By my novice understanding, it is similar to a confidence interval. It is formatted like this: (-8.13, -2.69). So would the -8.13 be the ...
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147 views

Confidence/credibility intervals for a bernoulli trial

Say we have $$ X \sim \text{Bernoulli}(p). $$ I am interested in finding a 95% confidence and credibility interval. For the credibility, I am assuming a uniform prior, giving me a posterior ...
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275 views

Confidence Interval vs Credible Interval for the Variance

I understand the conceptual difference between confidence and credible intervals. But I have difficulties applying these concepts to my application. I would like to know the concrete difference ...
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71 views

Numerical estimation of binomial confidence interval

I have two measurements from two different distributions. I know both of these distributions are binomial and I measure $k_1$ successes from $n_1$ trials for distribution 1 and $k_2$ successes from $...
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480 views

The Bayes credible interval / Bayes credible region of the posterior distribution of a multinomial Dirichlet conjugate pairs

I have a posterior distribution of Dirichlet form with new parameters (alpha1, x1), (alpha 2, x2) and (alpha 3, x3) and the posterior mode of each dependent variable as the Bayes estimator. I wish to ...
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Credible intervals from bayesian NN misbehave

I am using Edward, to create a Bayesian neural network for non-linear regression. My training set consists of time series that all end at 0 but after doing inference the predictions (and thus the ...
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440 views

What exactly does it mean that a 90% Credible Interval is computationally stable?

I was reading the rstanarm documentation and came across this about its use of 90% intervals as the default. I was hoping someone might be able to provide some clarification. Default 90% intervals ...
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Bayesian nonparametric estimate of median

I've been working on estimating the population median of a variable with a complex distribution that is not easily characterized as a parameterized probability distribution. So far, the best I have ...
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216 views

Proof that the HPD region is the smallest

In one dimension it can be shown that the highest posterior density (HPD) interval is the shortest; I found a proof in Subjective and Objective Bayesian Statistics (Section 8.4) by S. James Press ...