All Questions
Tagged with credible-interval confidence-interval
74 questions
331
votes
10
answers
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What's the difference between a confidence interval and a credible interval?
Joris and Srikant's exchange here got me wondering (again) if my internal explanations for the difference between confidence intervals and credible intervals were the correct ones. How you would ...
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42
votes
5
answers
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What is a Highest Density Region (HDR)?
In statistical inference, problem 9.6b, a "Highest Density Region (HDR)" is mentioned. However, I didn't find the definition of this term in the book.
One similar term is the Highest Posterior ...
- 1,122
41
votes
6
answers
7k
views
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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39
votes
6
answers
4k
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What is the connection between credible regions and Bayesian hypothesis tests?
In frequentist statistics, there is a close connection between confidence intervals and tests. Using inference about $\mu$ in the $\rm N(\mu,\sigma^2)$ distribution as an example, the $1-\alpha$ ...
- 12.1k
21
votes
4
answers
2k
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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 ...
- 2,187
16
votes
3
answers
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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 ...
- 1,188
16
votes
2
answers
229
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Why would one use `random' confidence or credible intervals?
I was reading a paper recently that incorporated randomness in its confidence and credible intervals, and I was wondering if this is standard (and, if so, why it is a reasonable thing to do). To set ...
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12
votes
1
answer
1k
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Examples of when confidence interval and credible interval coincide
In the wikipedia article on Credible Interval, it says:
For the case of a single parameter and
data that can be summarised in a
single sufficient statistic, it can be
shown that the credible ...
- 21.6k
12
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3
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What does a confidence interval (vs. a credible interval) actually express? [duplicate]
Possible Duplicate:
What, precisely, is a confidence interval?
Yes, similar questions have been asked before, but many of the answers seem contradictory and don't address my issue. (Or my ...
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12
votes
1
answer
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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 ...
- 69.5k
12
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1
answer
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Should I report credible intervals instead of confidence intervals?
After stumbling upon the concept in a statistics textbook, I tried to wrap my head about it, and finally came to a conclusion which seems to fit all the explanations which I have seen so far: A ...
- 1,899
12
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1
answer
485
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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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10
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1
answer
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Interpretation of Bayesian 95% prediction interval
Assume the following bivariate regression model:
$$
y_i = \beta x_i + u_i,
$$
where $u_i$ is i.i.d $N(0, \sigma^2 = 9)$ for $i = 1,\ldots, n$.
Assume a noninformative prior $p(\beta) \propto \text{...
- 899
9
votes
1
answer
3k
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Taking into account the uncertainty of p when estimating the mean of a binomial distribution
I have a binomial distribution with parameters $N$ and $p$, and the estimate for the mean of my distribution is N$\times p$. The values of $N$ and $p$ are such that we can use the Gaussian ...
9
votes
1
answer
935
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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 ...
- 6,724
7
votes
2
answers
1k
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Bayesian analysis: Estimate whether a parameter is 0 or not
I have the following problem: I need to assess whether a given
parameter $B$ is equal to 0. Let's consider the following model (my problem is more complicated but I think that this example is ...
- 5,093
7
votes
0
answers
724
views
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 ...
6
votes
3
answers
747
views
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 ...
- 93
6
votes
2
answers
489
views
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. ...
- 53
6
votes
4
answers
3k
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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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6
votes
2
answers
778
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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 ...
- 93
6
votes
2
answers
807
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Should I have "Confidence" in Credibility Intervals?
Preliminaries
First, I know that the Bayesian/Frequentist debate is rather long in tooth at this point, but I hope my question is sufficiently different from the others I reviewed on this site before ...
user75138
6
votes
1
answer
651
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 ...
- 1,909
6
votes
1
answer
706
views
When do (and don't) confidence intervals and credible intervals coincide?
Yes, I know there are many questions on comparing these two types of intervals, but none of them appear to answer this exact question.
Here is a blog post demonstrating one case where the two ...
- 1,188
6
votes
1
answer
283
views
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:
$$
f(\mathbf{x})\sim\mathcal{GP}(m(\mathbf{x}), k(\mathbf{x},\mathbf{...
- 63
5
votes
2
answers
752
views
Bayesian and frequentist optimization and intervals
I realize the methodology pursued by the Frequentist and Bayesian camps generally differ. However, one method of estimation that they do share is optimization of a certain function:
Frequentists ...
- 852
5
votes
1
answer
343
views
Is there a Bayesian analogue to a simultaneous confidence band?
A simultaneous confidence band denotes the probability
$$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{...
- 6,724
5
votes
1
answer
1k
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Is there a radical difference in how bayesian and frequentist approaches treat nuisance parameters?
The wiki article on credible intervals has the following statement:
credible intervals and confidence intervals treat nuisance parameters in radically different ways.
What is the radical ...
user28
5
votes
0
answers
145
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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 ...
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4
votes
1
answer
723
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Is It Ever Appropriate to Treat a Bayesian Credible Interval as a Frequentist Confidence Interval?
I know that a bayesian credible interval and a frequentist confidence interval measure very different things, and have different interpretations. However, is it ever appropriate to treat a bayesian ...
- 241
4
votes
1
answer
95
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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 ...
- 1,233
4
votes
1
answer
428
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Why is the $1-\alpha$ Bayesian credible interval for $\lambda \sim \chi^2_{v}$ have endpoints $\chi^2_{v, 1-\alpha/2}$ and $\chi^2_{v,\alpha/2}$?
Suppose that a posterior distribution $\lambda$ has distribution $\lambda \sim \chi^2_{v}$.
Then, it is often written that a $1-\alpha$ Bayesian credible interval for $\lambda \sim \chi^2_{v}$ will ...
- 2,425
4
votes
1
answer
150
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 $...
- 321
4
votes
1
answer
1k
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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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4
votes
0
answers
374
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 ...
- 435
4
votes
0
answers
81
views
Confidence and credible interval: cases
I am having difficulties in understanding these two approaches.
Let's say given the data I compute both confidence and credible interval, then what is the intuition/interpretation of having:
Big CI ...
- 269
3
votes
1
answer
223
views
Mathematical proof that the posterior probability that a CI contains the true parameter is in $\{0,1\}$
There are great posts on confidence intervals, a subject that was brought up recently on this question, leading to an endogamous and circular surfing between CV classics, such as this one and this one ...
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3
votes
2
answers
613
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Different Confidence vs. Credible Interval (Continuous case, noninformative prior) [duplicate]
Okay, so, credible intervals aren't the same as confidence intervals. We all know that. In fact, they're only guaranteed to be the same when they're about a location or a scale parametre with a ...
- 535
3
votes
1
answer
2k
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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 ...
- 157
3
votes
0
answers
714
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 ...
- 103
3
votes
0
answers
699
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 ...
3
votes
0
answers
152
views
Confidence intervals converging to credible intervals when Bootstraping?
As explained already by Rasmus, the Bayesian Bootstrap and the non-parametric Bootstrap "converge" when n is sufficiently large.
But if taking the percentile interval in the Bayesian ...
- 217
3
votes
0
answers
171
views
A non-statistician reference for confidence-interval Vs credibility-interval interpretation
As a beginner statistician, discussing the need to be accurate with the interpretation of statistical results with non-statistician is not an easy task. In particular, I am trying to convince some ...
- 5,093
2
votes
1
answer
66
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 ...
- 1,256
2
votes
1
answer
526
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{...
2
votes
2
answers
903
views
credible interval equivalent of confint() for bayesglm() in Gelman et al's 'arm' package?
How do I extract a credible interval ala confint on a glm object when working with the object returned by bayesglm() in arm?
- 2,724
2
votes
5
answers
242
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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 (...
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2
votes
1
answer
528
views
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 ($\...
- 31
2
votes
1
answer
346
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 ...
- 770
2
votes
1
answer
601
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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