algebraic.dist

NEWS code{white-space: pre-wrap;} span.smallcaps{font-variant: small-caps;} span.underline{text-decoration: underline;} div.column{display: inline-block; vertical-align: top; width: 50%;} div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;} ul.task-list{list-style: none;} algebraic.dist 0.9.1 New Generics as_dist() — S3 generic for converting objects (e.g., fitted models) into dist objects. Identity method for dist ; designed as an extension point for downstream packages Unified Fallback Layer realized_dist subclass of empirical_dist — preserves provenance (source distribution, sample count) when materializing via Monte Carlo ensure_realized() internal memoized entry point — all MC fallback methods now share cached samples (calling cdf(e) + density(e) on the same edist no longer draws independent samples) conditional.dist and rmap.dist now route through ensure_realized() for consistent provenance New Simplification Rules Uniform(a,b) + c → Uniform(a+c, b+c) (location shift) Uniform(a,b) - c → Uniform(a-c, b-c) (location shift) c * Uniform(a,b) → Uniform(min(ca,cb), max(ca,cb)) for c ≠ 0 c * Weibull(k,λ) → Weibull(k, c*λ) for c > 0 c * ChiSq(df) → Gamma(df/2, 1/(2c)) for c > 0 c * LogNormal(μ,σ) → LogNormal(μ+log(c), σ) for c > 0 LogNormal * LogNormal → LogNormal(μ₁+μ₂, √(σ₁²+σ₂²)) -Uniform(a,b) → Uniform(-b, -a) (unary negation) New Operators /.dist — Division operator: dist / scalar delegates to scalar multiplication rules; scalar / dist and dist / dist create edist MVN Algebra conditional.mvn — closed-form Schur complement conditioning with given_indices / given_values , or predicate-based MC fallback affine_transform(x, A, b) — compute AX + b for normal/MVN distributions (exact) Mixture Multivariate marginal.mixture — marginal of mixture is mixture of marginals (exact) conditional.mixture — Bayes’ rule weight update for mixture-of-MVN conditioning, with predicate-based MC fallback Limiting Distributions clt(base_dist) — CLT limiting distribution: Normal(0, Var) or MVN(0, Σ) lln(base_dist) — LLN degenerate limit: Normal(μ, 0) or MVN(μ, 0) delta_clt(base_dist, g, dg) — delta method with user-supplied derivative/Jacobian normal_approx(x) — moment-matching normal approximation of any distribution algebraic.dist 0.2.0 New Distributions gamma_dist(shape, rate) — Gamma distribution with hazard/survival functions weibull_dist(shape, scale) — Weibull distribution with closed-form hazard chi_squared(df) — Chi-squared distribution with hazard/survival functions uniform_dist(min, max) — Uniform distribution on [min, max] beta_dist(shape1, shape2) — Beta distribution on (0, 1) lognormal(meanlog, sdlog) — Log-normal distribution with hazard/survival poisson_dist(lambda) — Poisson distribution with exact expectation via truncated summation mixture(components, weights) — Mixture distributions with law of total variance New Operators and Algebra *.dist — Scalar multiplication ( c * dist , dist * c , dist * dist ) ^.dist — Power operator ( dist ^ n ) Math.dist — Group generic for exp() , log() , sqrt() , abs() , etc. Summary.dist — Group generic for sum() , prod() , min() , max() Extended +.dist and -.dist for numeric location shifts Simplification Rules c * Normal(mu, v) simplifies to Normal(c*mu, c^2*v) c * Gamma(a, r) simplifies to Gamma(a, r/c) for c > 0 c * Exponential(r) simplifies to Gamma(1, r/c) for c > 0 Normal(mu, v) + c simplifies to Normal(mu+c, v) Gamma(a1, r) + Gamma(a2, r) simplifies to Gamma(a1+a2, r) (same rate) Exp(r) + Exp(r) simplifies to Gamma(2, r) (same rate) ChiSq(d1) + ChiSq(d2) simplifies to ChiSq(d1+d2) Poisson(l1) + Poisson(l2) simplifies to Poisson(l1+l2) Normal(0,1)^2 simplifies to ChiSq(1) exp(Normal(mu, v)) simplifies to LogNormal(mu, sqrt(v)) log(LogNormal(ml, sl)) simplifies to Normal(ml, sl^2) min(Exp(r1), ..., Exp(rk)) simplifies to Exp(sum(r)) New Infrastructure realize() generic — materialize any distribution to empirical_dist by sampling Auto-fallback methods for edist : cdf , density , sup , conditional , rmap , inv_cdf countable_set R6 class for countably infinite support (Poisson) inv_cdf.empirical_dist — quantile function for empirical distributions Improvements Informative error messages in all constructors (replaced stopifnot ) format() methods for all distribution types Standardized print() methods delegating to format() Fixed vcov.exponential — was returning rate instead of 1/rate^2 Fixed sampler.edist crash when n=1 conditional.empirical_dist gives informative error on zero matches Zero-variance guard in expectation_data() CI computation algebraic.dist 0.1.0 Initial CRAN release. Features Core distribution types: normal , mvn , exponential , empirical_dist Expression distributions ( edist ) for lazy composition of distributions Algebraic operations ( + , - ) on distributions with automatic simplification Support classes: finite_set , interval for representing distribution domains Generic methods: sampler , mean , vcov , density , cdf , params Monte Carlo estimation for expectation , conditional , and rmap operations

README code{white-space: pre-wrap;} span.smallcaps{font-variant: small-caps;} span.underline{text-decoration: underline;} div.column{display: inline-block; vertical-align: top; width: 50%;} div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;} ul.task-list{list-style: none;} pre > code.sourceCode { white-space: pre; position: relative; } pre > code.sourceCode > span { display: inline-block; line-height: 1.25; } pre > code.sourceCode > span:empty { height: 1.2em; } .sourceCode { overflow: visible; } code.sourceCode > span { color: inherit; text-decoration: inherit; } div.sourceCode { margin: 1em 0; } pre.sourceCode { margin: 0; } @media screen { div.sourceCode { overflow: auto; } } @media print { pre > code.sourceCode { white-space: pre-wrap; } pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; } } pre.numberSource code { counter-reset: source-line 0; } pre.numberSource code > span { position: relative; left: -4em; counter-increment: source-line; } pre.numberSource code > span > a:first-child::before { content: counter(source-line); position: relative; left: -1em; text-align: right; vertical-align: baseline; border: none; display: inline-block; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; padding: 0 4px; width: 4em; color: #aaaaaa; } pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; } div.sourceCode { } @media screen { pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; } } code span.al { color: #ff0000; font-weight: bold; } /* Alert */ code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */ code span.at { color: #7d9029; } /* Attribute */ code span.bn { color: #40a070; } /* BaseN */ code span.bu { color: #008000; } /* BuiltIn */ code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */ code span.ch { color: #4070a0; } /* Char */ code span.cn { color: #880000; } /* Constant */ code span.co { color: #60a0b0; font-style: italic; } /* Comment */ code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */ code span.do { color: #ba2121; font-style: italic; } /* Documentation */ code span.dt { color: #902000; } /* DataType */ code span.dv { color: #40a070; } /* DecVal */ code span.er { color: #ff0000; font-weight: bold; } /* Error */ code span.ex { } /* Extension */ code span.fl { color: #40a070; } /* Float */ code span.fu { color: #06287e; } /* Function */ code span.im { color: #008000; font-weight: bold; } /* Import */ code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */ code span.kw { color: #007020; font-weight: bold; } /* Keyword */ code span.op { color: #666666; } /* Operator */ code span.ot { color: #007020; } /* Other */ code span.pp { color: #bc7a00; } /* Preprocessor */ code span.sc { color: #4070a0; } /* SpecialChar */ code span.ss { color: #bb6688; } /* SpecialString */ code span.st { color: #4070a0; } /* String */ code span.va { color: #19177c; } /* Variable */ code span.vs { color: #4070a0; } /* VerbatimString */ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */ R package: algebraic.dist Installation Overview Distribution types Automatic simplification Multivariate operations Limiting distributions Quick example Documentation R package: algebraic.dist An algebra over probability distributions: compose, transform, and automatically simplify distribution expressions in R. Tags : probability distributions, distribution algebra, automatic simplification, multivariate normal, mixture models, CLT, delta method, Monte Carlo, R, statistics Installation Install the stable release from CRAN: install.packages ( "algebraic.dist" ) Or install the development version from GitHub: # install.packages("devtools") devtools :: install_github ( "queelius/algebraic.dist" ) Overview algebraic.dist lets you build and manipulate probability distributions as first-class R objects. Algebraic operations ( + , - , * , / , ^ , exp , log , min , max , …) on distribution objects automatically simplify to closed-form distributions when mathematical identities apply, and fall back to lazy Monte Carlo expressions ( edist ) otherwise. Distribution types Normal, exponential, gamma, Weibull, chi-squared, uniform, beta, log-normal, Poisson, multivariate normal (MVN), mixture, and empirical distributions. Automatic simplification Over 20 built-in rules, including: Normal + Normal → Normal Gamma + Gamma (same rate) → Gamma exp(Normal) → LogNormal Normal(0,1)^2 → ChiSq(1) min(Exp, ..., Exp) → Exp c * Uniform(a,b) → Uniform When no rule matches, the result is a lazy edist that samples from its components and evaluates the expression on demand. Multivariate operations MVN conditioning : closed-form Schur complement via conditional() Affine transforms : affine_transform(x, A, b) for exact linear maps Mixture conditioning : Bayesian weight updates via Bayes’ rule Marginals : exact for MVN and mixture distributions Limiting distributions clt() — Central Limit Theorem lln() — Law of Large Numbers delta_clt() — delta method for transformed means normal_approx() — moment-matching normal approximation Quick example library (algebraic.dist) # Sum of normals simplifies to a normal x <- normal ( 1 , 4 ) y <- normal ( 2 , 5 ) z <- x + y z #> Normal distribution (mu = 3, var = 9) # exp of a normal simplifies to log-normal w <- exp ( normal ( 0 , 1 )) w #> Log-normal distribution (meanlog = 0, sdlog = 1) # Gamma addition with matching rates g <- gamma_dist ( 3 , 2 ) + gamma_dist ( 4 , 2 ) g #> Gamma distribution (shape = 7, rate = 2) # CLT: the standardized sample mean converges to N(0, 1) clt ( normal ( 5 , 4 )) #> Normal distribution (mu = 0, var = 4) Documentation The full documentation is available at https://queelius.github.io/algebraic.dist/ . Vignettes: Getting started — core distribution objects, sampling, and basic operations Distribution algebra — simplification rules, limiting distributions, and the CLT/LLN/delta method Multivariate operations — MVN conditioning, affine transforms, and Gaussian mixture models

Help for package algebraic.dist const macros = { "\\R": "\\textsf{R}", "\\mbox": "\\text", "\\code": "\\texttt"}; function processMathHTML() { var l = document.getElementsByClassName('reqn'); for (let e of l) { katex.render(e.textContent, e, { throwOnError: false, macros }); } return; } Package {algebraic.dist} Contents algebraic.dist-package *.dist +.dist -.dist /.dist Math.dist Summary.dist ^.dist affine_transform as_dist beta_dist cdf cdf.beta_dist cdf.chi_squared cdf.edist cdf.empirical_dist cdf.exponential cdf.gamma_dist cdf.lognormal cdf.mixture cdf.mvn cdf.normal cdf.poisson_dist cdf.uniform_dist cdf.weibull_dist chi_squared clt conditional conditional.dist conditional.edist conditional.empirical_dist conditional.mixture conditional.mvn countable_set delta_clt density.beta_dist density.chi_squared density.edist density.empirical_dist density.exponential density.gamma_dist density.lognormal density.mixture density.mvn density.normal density.poisson_dist density.uniform_dist density.weibull_dist dim.beta_dist dim.chi_squared dim.countable_set dim.edist dim.empirical_dist dim.exponential dim.finite_set dim.gamma_dist dim.interval dim.lognormal dim.mixture dim.mvn dim.normal dim.poisson_dist dim.uniform_dist dim.weibull_dist edist empirical_dist ensure_realized expectation expectation.dist expectation.empirical_dist expectation.poisson_dist expectation.univariate_dist expectation_data exponential finite_set format.beta_dist format.chi_squared format.edist format.empirical_dist format.exponential format.gamma_dist format.lognormal format.mixture format.mvn format.normal format.poisson_dist format.realized_dist format.uniform_dist format.weibull_dist gamma_dist has has.countable_set has.finite_set has.interval hazard.chi_squared hazard.continuous_dist hazard.exponential hazard.gamma_dist hazard.lognormal hazard.weibull_dist infimum infimum.countable_set infimum.finite_set infimum.interval interval inv_cdf inv_cdf.beta_dist inv_cdf.chi_squared inv_cdf.edist inv_cdf.empirical_dist inv_cdf.exponential inv_cdf.gamma_dist inv_cdf.lognormal inv_cdf.normal inv_cdf.poisson_dist inv_cdf.uniform_dist inv_cdf.weibull_dist is_beta_dist is_chi_squared is_dist is_edist is_empirical_dist is_exponential is_gamma_dist is_lognormal is_mixture is_mvn is_normal is_poisson_dist is_realized_dist is_uniform_dist is_weibull_dist lln lognormal marginal marginal.empirical_dist marginal.mixture marginal.mvn mean.beta_dist mean.chi_squared mean.edist mean.empirical_dist mean.exponential mean.gamma_dist mean.lognormal mean.mixture mean.mvn mean.normal mean.poisson_dist mean.uniform_dist mean.univariate_dist mean.weibull_dist mixture mvn nobs.empirical_dist normal normal_approx nparams nparams.empirical_dist nparams.mixture obs obs.empirical_dist params params.beta_dist params.chi_squared params.edist params.empirical_dist params.exponential params.gamma_dist params.lognormal params.mixture params.mvn params.normal params.poisson_dist params.uniform_dist params.weibull_dist poisson_dist print.beta_dist print.chi_squared print.edist print.empirical_dist print.exponential print.gamma_dist print.interval print.lognormal print.mixture print.mvn print.normal print.poisson_dist print.realized_dist print.summary_dist print.uniform_dist print.weibull_dist realize realized_dist rmap rmap.dist rmap.edist rmap.empirical_dist rmap.mvn sample_mvn_region sampler sampler.beta_dist sampler.chi_squared sampler.default sampler.edist sampler.empirical_dist sampler.exponential sampler.gamma_dist sampler.lognormal sampler.mixture sampler.mvn sampler.normal sampler.poisson_dist sampler.uniform_dist sampler.weibull_dist simplify simplify.dist simplify.edist summary.dist summary_dist sup sup.beta_dist sup.chi_squared sup.edist sup.empirical_dist sup.exponential sup.gamma_dist sup.lognormal sup.mixture sup.mvn sup.normal sup.poisson_dist sup.uniform_dist sup.weibull_dist supremum supremum.countable_set supremum.finite_set supremum.interval surv.chi_squared surv.continuous_dist surv.exponential surv.gamma_dist surv.lognormal surv.weibull_dist uniform_dist vcov.beta_dist vcov.chi_squared vcov.default vcov.edist vcov.empirical_dist vcov.exponential vcov.gamma_dist vcov.lognormal vcov.mixture vcov.mvn vcov.normal vcov.poisson_dist vcov.uniform_dist vcov.univariate_dist vcov.weibull_dist weibull_dist Title: Algebra over Probability Distributions Version: 1.0.0 Description: Provides an algebra over probability distributions enabling composition, sampling, and automatic simplification to closed forms. Supports normal, exponential, gamma, Weibull, chi-squared, uniform, beta, log-normal, Poisson, multivariate normal, empirical, and mixture distributions with algebraic operators (addition, subtraction, multiplication, division, power, exp, log, min, max) that automatically simplify when mathematical identities apply. Includes closed-form MVN conditioning (Schur complement), affine transformations, mixture marginals/conditionals (Bayes rule), and limiting distribution builders (CLT, LLN, delta method). Uses S3 classes for distributions and R6 for support objects. License: GPL (≥ 3) Encoding: UTF-8 RoxygenNote: 7.3.3 Suggests: knitr, rmarkdown, testthat (≥ 3.0.0) Config/testthat/edition: 3 Imports: stats, mvtnorm, R6 Depends: R (≥ 3.5.0) VignetteBuilder: knitr URL: https://github.com/queelius/algebraic.dist , https://queelius.github.io/algebraic.dist/ BugReports: https://github.com/queelius/algebraic.dist/issues NeedsCompilation: no Packaged: 2026-03-17 17:48:01 UTC; spinoza Author: Alexander Towell [aut, cre] Maintainer: Alexander Towell <lex@metafunctor.com> Repository: CRAN Date/Publication: 2026-03-18 06:13:20 UTC algebraic.dist: Algebra over Probability Distributions Description Provides an algebra over probability distributions enabling composition, sampling, and automatic simplification to closed forms. Supports normal, exponential, gamma, Weibull, chi-squared, uniform, beta, log-normal, Poisson, multivariate normal, empirical, and mixture distributions with algebraic operators (addition, subtraction, multiplication, division, power, exp, log, min, max) that automatically simplify when mathematical identities apply. Includes closed-form MVN conditioning (Schur complement), affine transformations, mixture marginals/conditionals (Bayes rule), and limiting distribution builders (CLT, LLN, delta method). Uses S3 classes for distributions and R6 for support objects. Author(s) Maintainer : Alexander Towell lex@metafunctor.com ( ORCID ) See Also Useful links: https://github.com/queelius/algebraic.dist https://queelius.github.io/algebraic.dist/ Report bugs at https://github.com/queelius/algebraic.dist/issues Multiplication of distribution objects. Description Handles scalar * dist, dist * scalar, and dist * dist. Usage ## S3 method for class 'dist' x * y Arguments x first operand y second operand Value A simplified distribution or edist Examples # Scalar multiplication simplifies for normal z <- 2 * normal(0, 1) z # Normal(mu = 0, var = 4) # Product of two distributions yields an edist w <- normal(0, 1) * exponential(1) is_edist(w) # TRUE Method for adding dist objects, or shifting a distribution by a scalar. Description Creates an expression distribution and automatically simplifies to closed form when possible (e.g., normal + normal = normal, normal + scalar = normal with shifted mean). Usage ## S3 method for class 'dist' x + y Arguments x A dist object or numeric scalar y A dist object or numeric scalar Value A simplified distribution or edist if no closed form exists Examples # Sum of two normals simplifies to a normal z <- normal(0, 1) + normal(2, 3) z # Normal(mu = 2, var = 4) # Shift a distribution by a constant normal(0, 1) + 5 # Normal(mu = 5, var = 1) Method for negation or subtraction of dist objects. Description Unary: returns negated distribution (e.g., -N(mu, var) = N(-mu, var)) Binary: creates expression distribution and simplifies to closed form when possible (e.g., normal - normal = normal,