# 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 ...
26 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 ...
57 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 ...
67 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 ...
37 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 ...
95 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 (...
32 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 ...
3k 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 ...
53 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 ...
27 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 ...
734 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 ...
378 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 ...
489 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. ...
145 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 ...
18k 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? ...
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 ...
18k 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 ...
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})/(\...