Delaporte Distribution

A discrete count distribution formed as the convolution of independent Poisson and negative-binomial components, equivalently a Poisson count with a mean containing fixed and gamma-random parts.

Version

v1 · 2026-08-30 · History

Domain-specific #

1636

Origin domain

probability and statistics

Subdomain

overdispersed count distributions

Aliases

Poisson–negative-binomial convolution distribution

Core Idea

Delaporte Distribution is a discrete count distribution formed as the convolution of independent Poisson and negative-binomial components, equivalently a Poisson count with a mean containing fixed and gamma-random parts. ^[1]

Let X be Poisson with mean lambda and Y be independent negative binomial with a declared shape/scale parameterization. Then N=X+Y has a Delaporte distribution. Equivalently, conditional on a random intensity Lambda=lambda+G with G gamma-distributed, N is Poisson(Lambda). Parameterizations differ across sources, so formulas must declare whether beta is a scale, rate, probability, or mean component.

The operative boundary is exact: The Poisson-plus-negative-binomial count decomposition and its intermediate dispersion regime remain uncovered. The abstraction is therefore not the topic named by its field, but the reusable role structure specified below.

Structural Signature

Sig role-phrases:

• the fixed Poisson component — baseline count intensity lambda
• the negative-binomial component — heterogeneous or clustered count variation
• the independence assumption — the condition making the sum distribution a convolution
• the nonnegative-integer support — counts 0,1,2,...
• the three-parameter family — baseline plus shape and dispersion parameters under a declared convention
• the convolution mass function — sum over every split of an observed count between components
• the mean–variance signature — mean combines components while variance adds extra negative-binomial dispersion
• the limiting cases — Poisson when the heterogeneous component vanishes and negative binomial when baseline vanishes

Recognition test. A case qualifies only when its roles can be mapped to the declared the fixed Poisson component, the negative-binomial component, the independence assumption, the nonnegative-integer support, and when the characteristic boundary conditions are preserved. Surface vocabulary or a loose analogy is insufficient.

What It Is Not

• Not a finite mixture distribution. The observed count is a sum of independent components, not a draw from one component selected by a mixing weight.
• Not a shifted Poisson. The added negative-binomial count is random, not a fixed displacement.
• Not one universal beta convention. Scale, rate, and probability parameterizations yield different formulas.
• Not automatically zero-inflated. Its variance flexibility can increase zeros, but no separate structural-zero mass is definitional.
• Not the negative binomial. That family is recovered only when the fixed Poisson component vanishes.
• Not proof of two causal processes. A good distributional fit does not identify the data-generating decomposition.

Scope of Application

The abstraction has a bounded but recurring habitat. These are literal applications of the same domain machinery, not cross-domain metaphors. ^[2]

• Actuarial claim counts. baseline events and heterogeneous risk can be represented in one frequency law.
• Epidemic offspring counts. fixed and variable transmissibility components create intermediate dispersion.
• Ecological abundance. background counts and clustered variation can coexist.
• Genomic count models. signal plus overdispersed biological component motivates convolution likelihoods.
• Queueing and reliability. independent baseline and environment-driven counts can yield the same law.
• Simulation studies. the additive construction gives a direct random-generation algorithm.

Clarity

The safest specification is constructive: name independent X and Y and their exact parameterizations, then set N=X+Y. Mean and variance follow by addition. Under a common gamma scale beta and shape alpha, E[N]=lambda+alpha beta and Var[N]=lambda+alpha beta(1+beta).

A useful audit proceeds in order: identify the candidate roles, verify their types and quantifiers, apply the recognition test, and then test every stated exclusion. If a case supplies only the broad parent pattern while dropping the domain accent, it is not Delaporte Distribution.

Manages Complexity

The family separates a constant Poisson contribution from heterogeneous overdispersion while retaining a closed count model. It interpolates between Poisson and negative binomial behavior and supplies a simulation story, probability-generating function, and moment structure from component laws.

The compression remains accountable because every simplification has a named validity condition. A user can ask which role is missing, which assumption fails, and which neighboring abstraction should replace the candidate instead of treating the label as an unanalyzed bundle.

Abstract Reasoning

R1. Lock the negative-binomial parameterization before copying a mass function.

R2. Use convolution only when the components are independent.

R3. Check limiting cases by setting each component to zero in turn.

R4. Compare mean and variance to identify, not prove, overdispersion structure.

R5. Distinguish parameter identifiability from numerical optimizer convergence.

The reasoning pattern is deliberately typed: definitions establish identity, calculations or constructions establish consequences, and empirical or institutional evidence establishes whether a real case instantiates the roles. One kind of support cannot silently substitute for another.

Knowledge Transfer

The law transfers literally wherever nonnegative counts plausibly decompose into independent Poisson and negative-binomial components. The broader convolution abstraction travels to other variables; the Delaporte name stays with this exact pair and parameter family.

The transfer boundary follows from the classification test: The family recurs in actuarial, epidemiological, and count-data models, but its three parameters, independence, convolution, support, overdispersion envelope, and limiting cases are constitutive. The safe portable move is to name the broader parent when the home-domain machinery is absent and to retain the domain name only when literal recognition succeeds.

Examples

Canonical: component moments

Let X be Poisson(2) and let Y have a negative-binomial parameterization with mean 3 and variance 6. For independent components, N=X+Y has mean 5 and variance 8. The variance exceeds the mean because only the Y component contributes extra-Poisson dispersion. Simulation draws X and Y separately and adds them. ^[2]

Mapped back: the fixed Poisson component; the negative-binomial component; the independence assumption; the mean–variance signature.

Applied / In Practice: infectious-disease secondary cases

A model separates a relatively fixed transmissibility contribution from gamma-varying individual infectiousness. Conditional Poisson counts integrate to a Delaporte marginal law. Comparing it with a negative-binomial-only model asks whether a stable component improves fit, but the fitted parameters should not be given a causal interpretation without external epidemiological evidence. ^[1]

Mapped back: the three-parameter family; the convolution mass function; the limiting cases; the diagnostic boundary.

Structural Tensions

T1: Flexibility versus identifiability. Three parameters fit intermediate dispersion but can trade off in finite samples. Diagnostic: Do likelihood profiles distinguish baseline from heterogeneous components?

T2: Constructive story versus causal claim. The convolution gives a generative representation without proving real-world mechanisms are independent. Diagnostic: What external evidence supports the component interpretation?

T3: Overdispersion versus zero inflation. Extra variance can improve zero counts while missing a separate structural-zero process. Diagnostic: Are zeros consistent with the fitted tail and mean or require another component?

T4: Parameter richness versus estimation stability. Greater flexibility can produce flat likelihoods and boundary estimates. Diagnostic: Are uncertainty intervals and limiting models reported?

T5: Formula portability versus convention drift. Different negative-binomial parameterizations use similar symbols for incompatible quantities. Diagnostic: Can every symbol be reconstructed from the stated mean and variance?

T6: Domain autonomy vs prime reduction. Convolution and probability distribution are generic, but the Poisson-plus-negative-binomial law with named limits is the Delaporte abstraction. Diagnostic: Would another component pair satisfy the same parent yet cease to be Delaporte? If yes, retain the domain node.

Structural–Framed Character

The five-criterion aggregate is 0.10 (structural). The classification is reasoned rather than cosmetic:

• Vocabulary travels — structural (0.25). The operative vocabulary retains the home-domain types named in the Structural Signature even when a thinner parent pattern travels.
• Evaluative weight — structural (0.00). The score records whether applying the abstraction requires a normative or interpretive judgment in addition to structural recognition.
• Institutional origin — structural (0.25). The score records whether the abstraction is constituted by a scholarly, legal, technical, or administrative convention rather than merely discovered in nature.
• Human-practice bound — structural (0.00). The score records how far the named roles depend on a human practice, measurement regime, language, or institution.
• Import versus recognize — structural (0.00). Beyond its home habitat, use of the name increasingly becomes import by analogy rather than recognition of the same mechanism.

The portable skeleton is: combine independent random contributions by convolution so moments and generating functions compose. That skeleton belongs to the related parent abstractions; it does not make the fully accented node a prime. Its character: structural, with a real structural core whose recognition remains bounded by domain-specific types and validity conditions.

Structural Core vs. Domain Accent

This section decides why Delaporte Distribution is a domain-specific abstraction rather than a prime.

Structural core: Combine independent random contributions by convolution so moments and generating functions compose. This relational skeleton can recur outside the home domain and is the part legitimately carried by broader primes.

Domain accent: Nonnegative counts, poisson baseline, negative-binomial heterogeneity, gamma mixing, three parameters, and actuarial dispersion. Remove those types and constraints and the result may still resemble the skeleton, but it is no longer recognized as this named abstraction.

Why it does not clear the prime bar: The convolution skeleton is generic; the exact component laws and parameter limits define a recognized statistical distribution. Cross-domain transfer is therefore routed through the parents, while the named entry remains available for precise in-domain diagnosis.

Instantiates / Related Primes

• Probability Distribution. is the taxonomic parent.
• Mixture Distribution. is a close but structurally different random-selection construction.
• Displaced Poisson Distribution. shares an added-count intuition but not the negative-binomial component.

These are prose relations only. They do not create structured DAG edges, and placement must still pass the live endpoint, redundancy, and cycle checks recorded in the bundle's placement memo.

Relationships to Other Abstractions

Current abstraction Delaporte Distribution Domain-specific

Parents (1) — more general patterns this builds on

• Delaporte Distribution is a kind of Probability Distribution Domain-specific

  The accepted reference-grade review places Delaporte Distribution under Probability Distribution because the child instantiates or depends on the parent's broader structure while retaining its own constitutive identity.

Hierarchy paths (5) — routes to 3 parentless roots

• Delaporte Distribution → Probability Distribution → Random Variable → Function (Mapping)

Show alternative paths (4)

Neighborhood in Abstraction Space

Delaporte Distribution sits in a sparse region of the domain-specific corpus (81^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Yule–Simon Distribution — 0.83
• Benford's Law — 0.82
• Polykay — 0.82
• Variance function — 0.82
• Empirical Measure — 0.81

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Negative-binomial distribution. a gamma–Poisson mixture with no fixed added Poisson component. Tell: Does lambda vanish?
• Poisson mixture. randomizes a Poisson intensity. Tell: Is the intensity specifically fixed plus gamma?
• Finite mixture model. selects among component laws. Tell: Are counts selected from or added across components?
• Zero-inflated negative binomial. adds a structural-zero gate. Tell: Is there a separate point mass at zero?
• Hermite distribution. another overdispersed count family from different Poisson components. Tell: Which component-generating function is present?

References

[1] Ryuichi Omori et al., “Characterizing Superspreading Potential of Infectious Disease”, PLOS Computational Biology 18(7) (2022). registry ↩a ↩b

[2] Avraham Adler, Delaporte: Statistical Functions for the Delaporte Distribution, CRAN package documentation. registry ↩a ↩b