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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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.
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Chi-squared confidence interval for variance
When constructing, for example, a $90\%$ confidence interval for the population variance using the chi-squared distribution, we have:
\begin{align}
& P\left(a<\frac{(n-1)S^2}{\sigma^2}<b\...
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Can we generate HPD regions from MCMC draws using convex hulls?
I thought of a procedure to generate high probability density regions with probability $1-\alpha$ from $n$ MCMC draws:
Find the $\lfloor(1-\alpha)\cdot n\rfloor$ draws with the largest probability ...
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Extraction of modes from a multi-modal density function
I am trying to extract modes from a multi-modal density function and not just peaks. For example, in the two density functions below (images), I would like to extract the curves contained in the black ...
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When can a winner of the election be called: estimating population proportion without the assumption of random sampling
While following a recent election, I wanted to estimate population proportion of people who voted for a certain candidate knowing the sample proportion, sample size (and population size).
I first ...
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Is there a theoretical motivation for how we construct confidence regions?
I've recently had to construct a confidence region for a vector of means $\theta \in \mathbb{R}^k$, and I realized my understanding of some concepts regarding the fundamentals of building confidence ...
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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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Significant support of non-central chi-quared distribution
I want to find the support of a non-central chi-squared distribution ($99.9 \%$ of the energy). For example, If I have a Gaussian distribution with parameters $\mu$ and $\sigma$, I know $99.9 \%$ 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 ...
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