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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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. ...
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How to elicit prediction intervals from clients?
When I prepare probabilistic forecasts I am often left with a choice of what percentage highest-density region to choose for prediction intervals for clients. This matters for reporting uncertainty to ...
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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 ...
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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 $...
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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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