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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6
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4answers
113 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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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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25 views

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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32 views

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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3answers
357 views

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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1answer
32 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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1answer
25 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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1answer
108 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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2answers
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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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31 views

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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25 views

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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1answer
112 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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123 views

Ranking/Sorting Star Ratings - Bayesian Credible Interval

I recently started analyzing episode polling data from a website that uses a 1-10 rating system. I've been reading about ranking star rating systems using Bayesian Credible Interviews as explained by ...
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1answer
192 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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1answer
72 views

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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1answer
79 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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82 views

calculating credible intervals for Bernoulli trials with a beta-distribution

I'm interested in finding credible intervals for Bernoulli trials with a variable probability of success. So, the process (as I understand it) is the probability of success, p, is modeled with a ...
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1answer
227 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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1answer
67 views

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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24 views

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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1answer
213 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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1answer
38 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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31 views

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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3answers
159 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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1answer
533 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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1answer
276 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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2answers
2k views

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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1answer
743 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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1answer
78 views

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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0answers
24 views

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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2answers
94 views

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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59 views

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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1answer
370 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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3answers
154 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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1answer
131 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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260 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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1answer
61 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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1answer
359 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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58 views

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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1answer
365 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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34 views

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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1answer
177 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 ...
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Are credible intervals confidence regions? [duplicate]

I understand that I'm asking a very pedantic question, but as far as I'm aware a confidence region is a multi-dimensional generalization of a confidence interval and therefore as a credible interval ...
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1answer
169 views

Is there a Bayesian analogue to a simultaneous confidence band?

A simultaneous confidence band denotes the probability $$p \big(\hat{f}(x) - w(\hat{f}(x)) \le f(x) \le \hat{f}(x) + w(\hat{f}(x)) \ \ \forall x \big)=1-\alpha$$ where $f$ a function of $x$, $\hat{...
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1answer
31 views

Terminology in interval estimation

(1) Confidence intervals (2) Posterior probability intervals (3) Fiducial intervals Is there some conventional term that encompasses these three sorts of intervals, but does not include prediction ...
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1answer
576 views

Why is the Bayesian credible interval in this polynomial regression biased whereas the confidence interval is correct?

Consider the plot below in which I simulated data as follows. We look at a binary outcome $y_{obs}$ for which the true probability to be 1 is indicated by the black line. The functional relationship ...
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1answer
3k views

How to construct the highest posterior density (HPD) interval

Please, anybody could explain the steps to compute the highest posterior density (HPD) interval, when the posterior distribution is known? For instance, when the posterior distribution is Beta ...