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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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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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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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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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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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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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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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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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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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how to create a bayesian model to combine different breakdowns/groups of people in a single population?

Let's assume we can represent a population in two different dimensions and we know the behaviour of the population if they are taking a specific action. Example; 1st dimension: breakdown population ...
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179 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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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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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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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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303 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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128 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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123 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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246 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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56 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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306 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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309 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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153 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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153 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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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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532 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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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 ...
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380 views

Do I evaluate coverage of my credible interval correctly and what are reasons for off-nominal coverage?

For a simulation study I derived the posterior distribution of a parameter $\theta$ with data $D$ as $p(\theta|D)$. I can sample from the posterior, but there is no analytic expression. The posterior ...
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160 views

Is the pth percentile of a mixture distribution a weighted average of the pth percentiles of it's components?

Suppose Y is a mixture distribution defined as follows: If are beta distributed random variables (making Y a mixture of three beta distributions); and are the 95th percentiles of the densities of Y ...
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If I do the same experiment many times then does a 95% credible interval mean 95% of the time the true value lies within that range? [duplicate]

Its common misinterpretation of a 95% confidence interval to say that that 95% of the time the true value lies within that interval. However, in Bayesian statistics, the 95% credible interval ...
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Computing Bayesian Credible Intervals for Bayesian Regression

I understand the Bayesian regression approach, in which I compute the posterior distribution of the parameters $\theta_i$. Now I am interested in how to compute the credible intervals of the ...
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Credible intervals after Expectation Maximization

We are applying EM to a problem where we have N items to classify (binary classification, 0 or 1 ) for which a correct classification exists but is unknown, and a set J of "judges" that performs the ...
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Bayesian model selection and credible interval

I have a dataset with three variables, where all variables are quantitatives. Let call it $y$, $x_1$ and $x_2$. I'm fitting a regression model in a Bayesian perspective via MCMC with ...
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1answer
517 views

Likelihood Ratio Confidence Intervals and Credible Intervals for MCMC

I am fitting a 5 parameters model to some data using Maximum Likelihood and Non Informative Bayesian Inference using Metropolis-Hastings algorithm. From the maximum likelihood fit, i calculated the ...