Highest Posterior Density Region Calculator

Example Data Table

Formula Used

The HPD region contains points with the highest posterior density until the required posterior mass is reached.

HPD set: R = { θ : p(θ | y) ≥ c }

Credible mass: ∫_R p(θ | y)dθ = 1 − α

Normal density: p(θ) = exp[-(θ − μ)² / (2σ²)] / (σ√(2π))

Beta density: p(θ) = θ^a−1(1−θ)^b−1 / B(a,b)

Gamma rate density: p(θ) = r^kθ^k−1exp(−rθ) / Γ(k)

For samples, the calculator selects the shortest interval containing the chosen posterior mass.

How to Use This Calculator

Select the posterior model that matches your chemistry problem. Use normal for symmetric concentration estimates. Use beta for fractions, purity, yield, or conversion. Use gamma for positive rates, decay constants, or variance-like parameters. Use empirical samples when you have MCMC output.

Enter the credible mass. A common value is 95%. Add units for clearer exported reports. Leave grid limits blank unless the automatic range misses important tail mass. Increase grid points for smoother numerical results.

Press the calculate button. The HPD result appears above the form and below the header. Use the CSV or PDF buttons to save the final result.

Highest Posterior Density Region in Chemistry

Bayesian Interval Meaning

A highest posterior density region is a Bayesian interval. It reports the most credible parameter values after data and prior information are combined. In chemistry, this can support assay validation, kinetic fitting, calibration, and uncertainty review. The method is useful when a single estimate is not enough. It describes where the parameter is most supported by the posterior distribution.

Why HPD Regions Help

A common credible interval may include low density values. An HPD region avoids that weakness. It keeps the densest values first. Then it expands until the selected posterior probability is reached. This gives a compact summary. It is often shorter than an equal tail interval. It also works well for skewed posteriors. Many chemical values are skewed because they cannot fall below zero.

Choosing a Posterior Model

The normal model is useful for repeated measurements with near symmetric error. Concentration, pH adjustment residuals, and instrument calibration slopes may fit this case. The beta model is better for bounded values. Examples include conversion, yield fraction, purity fraction, or probability of detection. The gamma model handles positive quantities. It is helpful for rate constants, scale values, and positive response parameters.

Grid and Sample Methods

This calculator uses two practical methods. For known posterior families, it builds a dense grid. It evaluates density at each grid point. It sorts points from high density to low density. Then it keeps enough points to match the chosen credible mass. For posterior samples, it sorts the samples and finds the shortest interval that contains the requested mass.

Reading the Output

The lower and upper bounds define the reported HPD region. The width shows uncertainty size. The mode gives the most dense value. The density cutoff is the boundary level for grid models. Values above that cutoff belong inside the HPD set. Values below it are outside. Use the captured mass to check numerical accuracy.

Reporting Results

Report the model, parameters, credible mass, and units. Include the grid range when limits were custom. For regulatory or laboratory records, keep exported CSV and PDF files with the analysis notebook. Always compare the interval with chemical knowledge. A result can be numerically correct but chemically unrealistic when the model is poor.

FAQs