# 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".

25 questions
Filter by
Sorted by
Tagged with
23 views

### 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. ...
• 23
471 views

### 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\...
1 vote
16 views

### 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 ...
• 2,536
1 vote
15 views

### 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 ...
• 133
1 vote
26 views

### 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 ...
• 772
47 views

### 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 ...
• 4,223
72 views

### 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 ...
• 96.1k
104 views

### 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 ...
• 21
108 views

### 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 ...
• 31.2k
1 vote
43 views

### 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 ...
• 11
152 views

### 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 (...
• 891
40 views

### 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 ...
• 141
7k views

### 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 ...
• 155
1 vote
63 views

### 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 ...
57 views

### 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 ...
• 2,706
1k views

### 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 ...
• 147
671 views

### 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 ...
• 85
881 views

### 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. ...
• 85
205 views

### 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 ...
• 116
25k views

### 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? ...
• 321
1k views

### 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 ...
• 646
81 views

### 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 ...
• 148
26k views

### 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 ...
• 912
364 views

### 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})/(\...
• 31