Questions tagged [highest-density-region]
The smallest region over which a density exceeds a threshold. More useful for summarizing multimodal distributions than other probability regions. Frequently used in Bayesian statistics ("Highest Posterior Density Regions"). The term in one dimension is "Highest Density Interval".
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Finding a HPD Interval for unknown population mean
I'm doing a question from a Bayesian stats textbook for study and am a bit confused with the result. Pretty sure I've messed something up but I'm not sure exactly what is going on here so any help is ...
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Quality measure for predictive Highest Density Regions
An alternative to point, interval and density forecasts/predictions would be "predictive highest density regions (pHDRs)", i.e., HDRs for the conditional density of a yet-unknown future ...
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How to determine the size of highest density region in high dimensions
I want to calculate the "size" of the highest density region (HDR) that contains p% of the total probability for multivariate samples of a Bayesian posterior obtained via MCMC.
In 1D this "size" is ...
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Highest probability set and density ratios equal to probability ratios
I came across a pretty result I had not seen before, and wondered if there were more examples
For a random variable with an exponential distribution, if you want the highest probability set to ...
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Find the CI for a given interval of HDI?
I'm working with big data that doesn't fit well to a distribution but often exhibits a peak (maybe two). I'm looking for a method to calculate the confidence for a given range around the mode.
For ...
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Highest Density Interval for the measure of central tendency
When samples are skewed, mean is not a good estimation of central tendency. But instead, median is a better choice.
For cauchy distribution, I heard that there's a completely different estimator (...
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Is the boundary of an HDR a region of the sample space with equal density value?
After reading this question, I read in the reference provided (Hyndman, 1996, The American Statistician) the following:
It follows inmediately from the definition that the boundary of an HDR ...
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How can I estimate the highest posterior density interval from a set of x,y values describing the PDF?
I'd like to estimate the Highest Posterior Density Interval (HPDI) of a calculated density function, rather than from empirical samples as is normally done (e.g., from an ...
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Bayesian hypotesis testing, can i use HPD?
I have a statistic question concerning the null hypotesis testing in bayesian inference.
I red that i can use HPD for testing the null value in a linear bayesian regression:
"We can then use the ...
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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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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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How to determine the cut off value of an hyperellipsoid in order to retrieve a single quantile of a multivariate normal distribution?
Introduction
My goal is to retrieve the $\alpha$ quantile of a N(0, H) (multivariate normal) random variable $X$ where H is a known d-dimensional positive definite matrix (with $d >3$). In other ...
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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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What would be the most appropriate method for constructing a bootstrap percentile confidence region?
The situation: let suppose we have a random variable that follows some parametric distribution $X\sim F_{\boldsymbol\theta}$, $\boldsymbol\theta\in\mathbb{R}^2$. We are provided with the bootstrap ...
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How to find 95% credible interval?
I am trying to compute the 95% credible interval of the following posterior distribution. I could not find the function in R for it but is the approach below correct?
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How to get multivariate credible interval estimate(s) / highest density regions (HDR) after MCMC
I'm estimating 15 parameters of my model using a Bayesian approach and a Markov Chain Monte Carlo (MCMC) method. My data after running a MCMC chain of 100000 samples is therefore a 100000×15 table of ...
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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 ...
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Confidence intervals for bootstrap and FPC
I have a population that I am technically sampling without replacement using a stratified design. The resultant ratio estimator of the sample mean in any stratum $i$ is $\hat{R}_{i} = (\sum y_{i})/(\...
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Find probability density intervals
I have the vector
x <- c(1,2,3,4,5,5,5,6,6,6,6,
7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
7,7,7,7,7,7,7,7,8,8,8,8,9,9,9,10)
(my actual vector has ...
