Inverse gamma - #53
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`Gamma(α, θ)` takes a shape and a scale, so its mean is `α * θ`, and `Gamma(α)` sets the scale to one. The density is closed form and traces; the distribution functions are not. `loggammap` and `loggammaq` give the regularized incomplete gamma integrals in log space. Returning logarithms is what keeps `logcdf` and `logccdf` finite where the probabilities underflow, and costs nothing: each tail is already built from a logarithmic prefactor. They sum a series below `x = a + 1` and run a continued fraction above it, both in the type they are given, so `BigFloat` keeps its precision. `quantile` then inverts the tail with Newton's method on `log(x)`, which is what lets the deep lower tail return a subnormal rather than zero. All of them loop until their terms stop changing the result, so none can run in traced or device-side code. Relative accuracy sits at the rounding error of the argument type for shapes up to about 1000 and falls off roughly in proportion to the shape after that, since the prefactor's terms grow while their sum does not. Sampling uses Marsaglia and Tsang's rejection method, boosted by `Gamma(α, θ) = Gamma(α + 1, θ) · U^(1/α)` below a unit shape. The accept step reads `basevalue(α)`, a new interface function returning the plain floating-point value inside a wrapped number, with methods in the ForwardDiff and ReverseDiff extensions and an Enzyme inactivity rule. The loop therefore runs on plain numbers whatever type the parameters carry, and the accepted noise enters the draw through arithmetic on `α` and `θ`, which leaves the draw differentiable with respect to both. The README lists of implemented measures also gain `Poisson`, which they had been missing. Assisted-by: Claude Opus 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01HiwLngD494i4dSLJ2DGPBG
`Wishart(ν, L)` is the first matrix-variate measure, so it brings the `Matrixvariate` variate form, the `ContinuousMatrixvariateMeasure` alias it dispatches on, and the `PositiveDefiniteMatrices` support. `L` is the lower-triangular factor of the scale matrix, the convention `MvNormal` already uses. It keeps `log|S|` a sum over a diagonal and the trace term a triangular solve rather than an inversion, and it makes Bartlett's decomposition, `L A A' L'`, plain arithmetic in the parameters. Sampling needs one chi-squared draw per dimension and so inherits `Gamma`'s sampler along with its derivative. Draws are symmetrized, which is what puts them exactly in the support rather than a rounding error away from it. `mean`, `var` and `std` take the shape of a draw. `cov` covers every pair of entries and so is indexed the way `vec` orders them; the conformance suite gains a `matrixsummaries` group for that shape. `src/core/linalg.jl` collects the triangular linear algebra the two factored measures share: `rowdot`, moved from `MvNormal`, alongside `rowsdot`, `forwardsolve`, `logdetdiag` and `cholfactor`. Each builds new arrays instead of writing into one, so reverse-mode backends, which reject array mutation, can follow them. `cholfactor` takes its pivots through the new `sqrtt`, so an indefinite argument yields a non-finite factor and the log-density reports `NaN` instead of throwing. Assisted-by: Claude Opus 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01HiwLngD494i4dSLJ2DGPBG
`InverseGamma(α, θ)` is the measure of `1/X` for `X` distributed as `Gamma(α, 1/θ)`. Its mean is infinite at or below a unit shape and its variance at or below a shape of two, both of which it reports rather than returning a negative number. Its distribution functions read the upper incomplete gamma integral at `θ/x`, which puts its lower tail on the integral's upper one. Inverting that through the lower tail would mean forming `1 - p`, and a probability of `1e-300` has no complement in floating point. So `gammaquantile` now takes which tail its probability measures, and `quantile` inverts the upper one directly, keeping small probabilities down to the point where the quantile itself underflows. `valuetype(d, x)` moves to `core/interface.jl`, where both gamma measures reach it, and `masstype` becomes its discrete case rather than a second copy of the same promotion. Assisted-by: Claude Opus 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01HiwLngD494i4dSLJ2DGPBG
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Closes #29.
Same situation as for #52. I'd have liked to point this onto the
wishartbranch but can't do so because I was working from my fork. Merge #51 and #52 first, then this one.