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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 $100(1-\alpha)\%$ posterior probability. Credible intervals treat the interval as fixed and the parameter as random.

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confidence intervals for proportions containing a theoretically impossible value (zero)

This is really a hypothetical question not related to an actual issue I have, so this question is just out of curiosity. I'm aware of this other related question What should I do when a confidence ...
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Experimental Design: Selecting value of $n$ given desired width of credible interval

Note that this is a cross post from here. I realize this is probably a better space Suppose I have $n$ IID Bernoulli trials with $k$ successes. Assume that as a prior we are assuming that $P(\theta)$ ...
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Extract credible/confidence interval of a threshold in a Bayesian posterior draws distribution [closed]

I have a Bayesian model created through bayer package in R on which I need to calculate confidence/credible intervals for a ...
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Highest-density vs equal-tailed confidence interval

When a sampling distribution is symmetric (and I'm okay assuming unimodal too, if necessary), it's natural to center confidence intervals around the point estimate. But for a skewed distribution (e.g. ...
Quinn Culver's user avatar
6 votes
1 answer
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Gaussian Process: confidence interval vs prediction interval vs credible interval

Let a distribution over functions be described by a Gaussian Process (GP) prior, following the notation of Rasmussen and Williams: $$ f(\mathbf{x})\sim\mathcal{GP}(m(\mathbf{x}), k(\mathbf{x},\mathbf{...
abc's user avatar
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Region of Practical Equivalence (ROPE)

I seek to understand how to employ the Region of Practical Equivalence (ROPE), so I conducted a basic Bayesian analysis of disease incidence rates on a public data set. Is my rationale sound regarding ...
Tavaro Evanis's user avatar
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1 answer
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Can we report a credibility or confidence interval for a quantity measured only once, but whose distribution is obtained by Bayesian methods?

Suppose you use Bayesian methods to calculate the probability density function (pdf) of a quantity of interest $X$, given its measured value $x$ (measured only once) and some other assumptions/...
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How to determine if there is a statistically significant difference between the predictions of two Bayesian models

I would like to determine if there is a statistically significant difference between the predictions of two Bayesian models. Model1 predicts the distance travelled on foot during daylight based on ...
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What's a sensible way to put a credible interval on samples in [0, 1]?

Say I take a bunch of measurements of a discrete random variable in {0, 1} (coin flips of a weighted coin for instance). I understand that a reasonable way to report on the mean and credible interval ...
Alexander Soare's user avatar
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Correct interpretation of Bayesian credible intervals

I'm starting to learn about Bayesian statistics for my own personal education (this is not a homework question). Something puzzles me a bit about the interpretation of credible intervals. From the few ...
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Discontiguous credible interval (take out the middle)

As far as I know, credible intervals usually involve a single contiguous interval and allow one to make statements such as: "Based on the model, data, and priors, $X$ is within the bounds $(x_1, ...
Isaac's user avatar
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Highest posterior density [duplicate]

How is to find the value of $k$ for the following equation which gives $100(1-\alpha)\%$ highest posterior density? $$\int_{\theta:\pi(\theta|\mathbf x)>k}\pi(\theta|\mathbf x)d\theta=1-\alpha,$$ ...
user149054's user avatar
2 votes
1 answer
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Is it possible to merge credible intervals from different Bayesian prediction models into a single estimate?

The situation Imagine an archaeological site, 10.0m deep. For my study, I construct an "age-profile" for this site, i.e., I produce a model of age as a function of depth. There are various ...
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Simulate posterior from a linear mixed model with categorical and continuous variables and their interactions

Currently, I am working on a data set containing isotopic ratios. To understand differences in the ratios I am fitting a linear mixed model in R using the lme4 package. I am then using the arm package ...
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Credible interval for a quantile from log-normally distributed data

Both reviewers of my latest article suggested better to work with credible intervals instead confidence intervals. Unfortunately, I am not familiar with Bayesian statistics. Until today, I was (...
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Best way to show one Bayesian model is more certain and accurate than another, based on simulated data?

I'm trying to compare performance of two bayesian models $A$ and $B$ on simulated data. It's a recruitment curve fitting problem and I'm interested in how accurate these models are in estimating only ...
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Estimating posterior of proportion of positives in population from per-observation probabilities

I have a sample from some population of 0s and 1s and need to estimate the posterior of the proportion of 1s in this population. But the catch is: for each observation in the sample I only have ...
nikoliazekter's user avatar
1 vote
1 answer
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Credible interval estimation for a difference of paired binomials

Assume one is given two series of binomially distributed data: $$X_{1k} \sim Binom(n_{1k}, p_{1k})$$ $$X_{2k} \sim Binom(n_{2k}, p_{2k})$$ $$k\in \{1\ldots K\}$$ The goal is to estimate $p_2 - p_1$ ...
Nathaniel Bubis's user avatar
3 votes
1 answer
156 views

Comparing three posterior distributions

I estimated a variable for three different species and have a posterior distribution of 4000 estimates for each species. Now I want to know if the distribution between species differs significantly. ...
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1 answer
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Differences between HPDI and PI intervals

In Bayesian statistics, we may want to determine at what interval for example 95% of the posterior probability exists. For this we may want to use the Highest Posterior Density Interval (HPDI) which ...
Quinten's user avatar
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How to compute a rectangular credible region from samples

Given high-dimensional Monte Carlo samples $\bf{X_1},...,\bf{X_N}$ from a probability distribution $p({\bf x})$ in $\mathbb{R^d}$, I want to estimate a rectangular highest-density credible region for $...
iLikeBayes's user avatar
12 votes
1 answer
399 views

Why do Bayesians care about the frequentist properties of Bayesian credible intervals?

I've been doing some reading on the topic of credible vs confidence intervals but unfortunately it feels like the more I read the more I'm confused. There seems to be a general sense or consensus that ...
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May we choose the confidence interval's width or credible interval'swidth based on a loss-function or cross-validation?

May we choose the confidence interval or credible interval based on an empirical risk loss-function? I think we blindly generate the width intervals without any optimization or rely on some ...
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1 vote
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Closed form credible region/marginal credible interval for Gaussian distribution

Suppose we have a multivariate Gaussian distribution $X \sim \mathcal{N}(\mu,\Sigma)$ where $\mu=(\mu_1,\mu_2,\dots,\mu_n) \in \mathbb{R}^n$ (the posterior of a conjugate Gaussian prior perhaps). Is ...
123 456's user avatar
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2 votes
1 answer
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How to calculate Bayesian marginal credible interval?

I have a Bayesian model that I've fit using Stan, and I'm trying to figure out the best way to calculate the correct credible interval that I am interested in. The model is a hierarchical GLM with a ...
wzbillings's user avatar
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How is highest posterior density interval estimated in this code snippet?

I found the following (Julia) implementation for estimating the highest posterior density interval from a posterior sample (link). Below, I turn it into pseudocode for simplicity. ...
Alex Ten's user avatar
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131 views

Wide prediction intervals for short time series: how to fix that?

I have very small time series data(24 points) for sales for different-different regions. I need to build Range Forecast (Confidence/Credible Intervals) for sales around it for every region. I tried FB'...
Amit Kumar's user avatar
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120 views

Limit of $\frac{\lambda}{\chi _{2Y}^{2}}$ as $Y \sim \textrm{Poisson}(n\lambda)$ and $\lambda\to\infty.$

There are the following lines in Casella & Berger on page 438, before the equation (9.2.22): ..., write $$\lambda = \frac{\lambda}{\chi _{2Y}^{2}}\chi _{2Y}^{2}$$ where $\chi _{2Y}^{2}$ is a chi ...
Hooman's user avatar
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3 votes
2 answers
140 views

How to fix hypothesis testing: MBF versus "directional correctness"

A common misinterpretation of a p-value is that it represents the probability of a false positive in the context of hypothesis testing. Here a "positive" means rejecting the null. There are ...
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What is the relationship between model uncertianty and model parameter count?

I'm looking for references, information and/or existing theory behind the relationship between the uncertainty in a given model vs its complexity/parameter count. The situation I have in mind is using ...
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Using the terminology "Bayesian confidence interval" in place of "Bayesian credible interval."

In Peter Hoff's "A first course in Bayesian statistical methods," he states: "Most authors refer to intervals of high probability as 'credible intervals' as opposed to confidence ...
damarsh's user avatar
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How to find 95% credible interval of a posterior predictive distribution?

I obtained a posterior predictive with 1,000 samples using MCMC, and I need to quantify the 95% credible intervals. I know that the difference between confidence intervals and credible intervals. One ...
Leslie's user avatar
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Why do we trust credibility intervals to contain the true parameter?

I understand confidence intervals and to what extent they can be trusted (and why). However, I’m not so sure how to motivate why I should trust credibility intervals except insofar as they can also ...
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4 votes
1 answer
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D&D - Confidence Interval for enemy armor class

In dungeons and dragons, characters and monsters have two properties called Attack Bonus($AB \, \in \, \mathbb{Z}$) and Armor Class($AC \, \in \, \mathbb{N}$). Let $AB_c$ be the character attack bonus ...
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Practical consequences of wrong interpretation of confidence intervals

Can you give me a simple (but preferably common) example (or R/Python simulation) of what are practical consequences of wrong interpretation of frequentist confidence intervals? Especially when they ...
3 votes
0 answers
621 views

mcmc vs the bootstrap

What is more accurate, the mcmc derive 95% credible interval or the bootstrap derived 95% confidence interval? Can this be proved mathematically? the emphasis of the Bayesian approach is that one is ...
StatsBio's user avatar
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1 answer
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Statistical conclusions from incompatible measurement results

I assume that the following situation is not uncommon in scientific practice: Two research groups analyse two samples. The reported results concern the same physical quantity at the same location ...
Hans-Peter Stricker's user avatar
1 vote
0 answers
242 views

Showing calibration of Bayesian credible intervals

I'd like to try and understand how one can prove that a particular strategy for assessing correctness of computational methods for Bayesian inference is sound. For a number $M$ of simulations, ...
Luiz Max Carvalho's user avatar
6 votes
2 answers
1k views

When does a Bayesian 95 % credibility interval have 95% coverage?

Consider the generative model $$ \begin{align} \theta &\sim \pi(\cdot \mid \phi),\\ y \mid \theta &\sim f(\cdot \mid \theta). \end{align} $$ Compute the posterior distribution $p(\theta \mid y)...
Luiz Max Carvalho's user avatar
6 votes
2 answers
719 views

Converting a confidence interval into a credible interval

The problem of correctly interpreting confidence intervals has been discussed at length here. I have a related question which I believe may be a useful contribution: Frequentist probabilities by ...
chicken_game's user avatar
4 votes
1 answer
225 views

Are HPD intervals invariant to reparameterization?

Suppose I have a parameter (or just any random variable) $\theta$, which we can assume to be absolutely continuous. If I compute a Highest Posterior Probability $I$ for $\theta$ and another one (let's ...
PedroSebe's user avatar
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2 votes
1 answer
143 views

Bayesian Credible interval in Poisson Regression

Assume that $\pmb{y} = (y_1,...,y_n)$ are count data which depends on a dummy predictor $\pmb{x}$ (x = 0 if male, x = 1 if female) such that : \begin{align} \pmb{y} \sim Poisson(\lambda) \end{align} ...
outofthegreen's user avatar
1 vote
0 answers
92 views

How to compute a 95% credible interval for a small sample size with little prior information?

Say I have the following empirically measured data describing a parameter of interest: x = c(12, 4, 6, 8, 5, 12, 12, 10, 6) where x is a continuous random variable....
Dan Scott's user avatar
3 votes
0 answers
629 views

Finding changepoints in a GAM?

I'm using generalized additive modeling to investigate the relationship between two variables, X and Y. I want to find changepoints--i.e., X values at which the slope changes direction. I can get ...
gam-changepoints's user avatar
3 votes
0 answers
42 views

Convergence of credible regions on simple Bayesian model

Consider a basic Bayesian model : $$ \begin{array}{rcl} \theta &\sim &\pi(\theta)\\ X_1, \cdots, X_n&\overset{IID}{\sim} &\mathcal{N}(\theta, I_d)\\ \end{array} $$ where $d$ is the (...
Pohoua's user avatar
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1 vote
0 answers
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How to find the credible interval on the difference between the means of two proportions in R?

I have two proportions of the beta-binomial family. x1=41, n1=88, x2=41, n2=90 (x= no. of successes; n=no. of trials). The beta prior shape parameters for the two proportions are alpha1= 20, beta1=26, ...
Rohan Sarkar's user avatar
1 vote
0 answers
53 views

How to find credible interval for multiple bayesian distributions that disagree?

Feel free to suggest changes to my terminology here, I don't think this is proper Bayesian analysis. I'm a scientist trying to produce a credible interval for an unknown experimental value X. The ...
thepenguin77's user avatar
20 votes
4 answers
2k views

How to correctly word a frequentist confidence interval

I am aware that there are many, many threads on this (e.g. this excellent thread). I may have missed it but I can't seem to find one that actually explains how to accurately report a frequentist ...
llewmills's user avatar
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2 votes
5 answers
239 views

Interval Estimation for a Change in a Binomial Proportion

I'm not sure how to estimate the confidence interval (CI) for a change in a small sample size binomial proportion using the same sample set both times. I have two methods that I would like to compare (...
R. Cox's user avatar
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1 vote
1 answer
117 views

In estimating a posterior mean, should one use the entire density or the HPDI?

Recognizing that HPDIs are analogous to confidence intervals in frequentist statistics, if one calculates an HPDI for a posterior distribution and then wants to report a posterior mean, does it make ...
TSP's user avatar
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