pid-rs

Partial Information Decomposition & continuous mutual-information estimators in safe Rust.

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pid-rs implements the shared-exclusions partial information decomposition (I^sx_∩; Makkeh–Gutknecht–Wibral 2021) and the continuous k-nearest-neighbour estimators it builds on — KSG mutual information (Kraskov et al. 2004) and the continuous I^sx_∩ estimator (Ehrlich et al. 2024) — together with a Shannon-invariant screening layer, discrete I_min PID, geometry diagnostics, and dependence-aware uncertainty quantification.

It was built to diagnose how information from different sources (e.g. vision and language) is integrated in multimodal policies, but every estimator here is domain-agnostic: give it samples of sources S1, S2, … and a target T and it estimates how much of the information about T is redundant, unique, or synergistic.

                I(S1,S2; T)
              ┌──────┴───────┐
   Redundancy ·  Unique(S1) ·  Unique(S2) ·  Synergy

Highlights

• KSG mutual information for continuous variables — L∞ joint metric, strict-radius marginal counting, digamma reference table.
• Continuous I^sx_∩ (shared-exclusions redundancy) via the Ehrlich et al. 2024 disjunction-neighbourhood kNN estimator — not a min-of-pointwise heuristic.
• 2- and 3-source PID atoms whose Möbius identities (Red + Unq₁ + Unq₂ + Syn = I(S1,S2;T)) hold by construction and are asserted in tests.
• Discrete I_min PID over the full 18-antichain 3-source lattice (Williams & Beer 2010).
• Discrete shared-exclusions PID i^sx_∩ (Makkeh–Gutknecht–Wibral 2021) — pointwise and averaged signed atoms with informative/misinformative split, bit-faithful to the reference IDTxl wraps (Abzinger/SxPID) for 2- and 3-source, plus a general discrete_sxpid_n for 2–4 sources (the 166-antichain 4-source lattice). The discrete counterpart of the continuous I^sx_∩, so the library decomposes information with one measure across regimes.
• Shannon invariants — co-information, O-information (Rosas et al. 2019), and the average degrees of redundancy (r̄) and vulnerability (v̄) (Gutknecht et al. 2025) — as cheap screening statistics.
• Geometry diagnostics — intrinsic dimension (Levina–Bickel), distance concentration, Gromov hyperbolicity — to decide whether a continuous-kNN regime is even valid.
• Preprocessing — standardisation, PCA, hash projection, seeded jitter, and PLS.
• Honest uncertainty — block bootstrap and permutation tests that respect sample dependence.
• Reproducible by construction — content-addressed run-logs (pid-runlog; SHA-256 payload digests on action/intervention/bridge records + a whole-trace replay hash and whole-file manifest covering every record), seeded RNG, and an optional parallel feature whose results are bit-identical to the serial path.
• The estimator core (pid-core) is #![forbid(unsafe_code)], returns errors rather than panic!-ing on valid-but-degenerate input, and keeps a dependency-light tree. (pid-runlog is also unsafe-free; pid-python necessarily uses PyO3's unsafe internals.)

How pid-rs compares

A high-level orientation for readers who already know the established toolboxes (not a feature-parity scorecard — each tool leads in its own niche):

	pid-rs	IDTxl	dit	JIDT
Language	Rust (+ Python)	Python	Python	Java (+ wrappers)
KSG continuous MI	✅	✅	— (discrete-only)	✅
Continuous I^sx_∩ (Ehrlich 2024)	✅	—	—	—
Discrete SxPID i^sx_∩	✅ (bit-faithful to IDTxl)	✅ (reference impl.)	✅ (PID_SX)	—
Discrete I_min PID	✅	— (BROJA + SxPID only)	✅	—
Broad discrete PID/measure zoo	—	some	✅	—
Transfer entropy / network inference	—	✅	—	✅
Content-addressed, replayable run-logs	✅	—	—	—
Memory-safe, unsafe-free core	✅	n/a	n/a	n/a
Bit-identical serial↔parallel results	✅	—	—	—

Where the others lead: IDTxl for transfer entropy, full network inference, and its mature ecosystem (and it is the reference SxPID implementation this crate is validated against); dit for the sheer breadth of discrete information/PID measures; JIDT for its established JVM estimator suite. pid-rs's niche is a fast, memory-safe, reproducible implementation of the Wibral-group shared-exclusions PID unified across the continuous and discrete regimes.

Project status

pid-rs is at 0.4.0. The estimator core is validated against analytic ground truth (see Validation); the surrounding statistics, performance, and tooling layers are usable but have tracked follow-ups. This section is a quick honest map of where things stand — it does not repeat the per-claim detail in Conventions, Scientific cautions, or Known limitations.

What works today

Capability	Notes
KSG mutual information	Continuous variables, L∞ joint metric, strict-radius marginal counting; checked vs the closed-form Gaussian-channel MI.
Continuous I^sx_∩	Ehrlich et al. 2024 disjunction-neighbourhood kNN redundancy (IsxMethod::EhrlichKsg); checked against a closed-form additive-Gaussian oracle (tests/sxpid_gaussian_oracle.rs) and pinned by frozen fixed-data regression values.
2- & 3-source PID atoms	pid2_isx / pid3_isx; Möbius identities (Red + Unq₁ + Unq₂ + Syn = I(S1,S2;T)) hold by construction and are asserted in tests within 1e-10.
Discrete I_min PID	discrete_pid2 / discrete_pid3 over the full 18-antichain 3-source lattice (Williams & Beer 2010), with equal-width quantisation.
Discrete SxPID i^sx_∩	discrete_sxpid2 / discrete_sxpid3 + general discrete_sxpid_n (2–4 sources; Makkeh–Gutknecht–Wibral 2021); pointwise + averaged signed atoms with informative/misinformative split, bit-faithful to the Abzinger/SxPID + IDTxl reference values to 1e-12; MGW Theorem IV.2/IV.3 axiom tests.
Shannon invariants	Co-information, O-information (Rosas et al. 2019), and r̄/v̄ (Gutknecht et al. 2025) as cheap screening statistics.
Geometry diagnostics	Intrinsic dimension (Levina–Bickel), distance concentration, Gromov hyperbolicity — to decide whether a continuous-kNN regime is even valid.
Preprocessing / PLS	Standardisation, PCA, hash (CountSketch) projection, seeded jitter, and supervised PLS with CV component selection.
Uncertainty quantification	Moving-block bootstrap and permutation tests that respect sample dependence.
Run-logs	pid-runlog: versioned, content-addressed JSONL schema (SHA-256 payload hashes on action/intervention/bridge records; every record covered by the whole-trace replay hash + whole-file manifest) and replay/validate/compare/sidecar CLIs.
Python bindings	pid_core_rs (PyO3 + maturin, abi3 ≥ CPython 3.11) — 18 functions over C-contiguous float64 NumPy arrays. Bindings shipped in 0.2.0; discrete SxPID exports added in 0.3.0.
Reproducibility	Seeded RNG; the optional parallel feature is bit-identical to the serial path; #![forbid(unsafe_code)]; errors (not panics) on degenerate input.

What needs further work

• kNN is brute-force O(n²). kth_neighbor_distance_* and count_neighbors_within scan all pairs per query — there is no kd-tree / approximate-NN backend, so large n is slow.
• No multiple-comparison correction. Many atoms × sources × windows report raw per-atom p-values; apply your own FDR/FWER control.
• runlog --validate is per-record, not whole-trace integrity. It checks per-event invariants (payload/config-hash matches, monotone timestamps/steps, single run_started/run_ended, bridge causality, finite values). Whole-trace integrity is a separate path: the order-sensitive replay_trace_hash (--compare) and --verify-sidecars.
• exp0 is a diagnostic gate, not a pass/fail build step. It emits a GO/PIVOT/NO-GO verdict from monotonicity / invariant / geometry counters and exits 0 by default (its default sweep deliberately enters regimes where kNN MI is known to break down). --strict-gate does not gate that default sweep; it enforces GO (exit 3 otherwise) only on a curated, analytically grounded low-dimension band (see the exp0 section below).
• No crates.io release yet. Depend on the Git repository; the Python crate is publish = false by design (shipped as a wheel via maturin).
• External cross-validation of the continuous I^sx_∩ pending. The discrete SxPID is validated bit-faithfully against IDTxl/Abzinger reference values, and the continuous estimator against a closed-form Gaussian oracle — but no reproducible external csxpid cross-check exists yet: the fixed-data constants in tests/isx.rs / tests/pid3.rs are frozen regression pins of this implementation (their historical csxpid attribution left no dataset or invocation artifacts, so they must not be read as external validation).

Caveats

• kNN failure modes are real. Estimators assume i.i.d. samples (trajectory autocorrelation biases them — subsample or block-bootstrap); high ambient/intrinsic dimension causes distance concentration that degrades kNN geometry; and strong (near-deterministic) dependence can require prohibitive sample sizes (Gao et al. 2015). Run the geometry diagnostics and the exp0 gate before interpreting results.
• Negative atoms are real, not bugs. I^sx_∩ trades all-atom non-negativity for the target chain rule, so atoms (including redundancy) can be negative; atoms and the MI terms feeding them are never clamped (Allow is forced inside pid2_isx/pid3_isx/co-information). Note the standalone ksg_mi defaults to ClampToZero as a reporting convenience — set NegativeHandling::Allow when you need raw (possibly negative) MI estimates.
• Cross-estimator PID2 mixing. In pid2_isx, Unq/Syn combine KSG MI with Ehrlich I^sx redundancy (different bias profiles), so small near-zero atoms can be an estimator artefact rather than structure. Likewise, do not pool continuous I^sx_∩ atoms with discrete I_min atoms — they are different PID measures.

Install

[dependencies]
pid-core = { git = "https://github.com/sepahead/pid-rs" }

A crates.io release is planned; until then, depend on the Git repository.

Using Python? See the Python bindings below for pip install maturin and maturin develop.

Quickstart

use pid_core::{ksg_mi, pid2_isx, IsxConfig, KsgConfig, MatRef, NegativeHandling, Pid2Config};

// Columns are dimensions, rows are samples. Here: scalar S1, S2, T (n samples each).
// (s1_data/s2_data/t_data/n are your own `&[f64]` buffers; see examples/ksg_and_pid.rs for a runnable version.)
let s1 = MatRef::new(&s1_data, n, 1)?;
let s2 = MatRef::new(&s2_data, n, 1)?;
let t  = MatRef::new(&t_data,  n, 1)?;

// Mutual information (nats).
let ksg = KsgConfig { negative_handling: NegativeHandling::Allow, ..Default::default() };
let mi = ksg_mi(s1, t, &ksg)?;

// 2-source PID atoms via I^sx_∩.
let pid = pid2_isx(s1, s2, t, &Pid2Config { ksg, isx: IsxConfig::default() })?;
println!("Red={:.3}  Unq1={:.3}  Unq2={:.3}  Syn={:.3}",
         pid.redundancy, pid.unique_s1, pid.unique_s2, pid.synergy);
# Ok::<(), pid_core::PidError>(())

Run the worked examples end-to-end:

cargo run --release --example ksg_and_pid       # continuous KSG MI + I^sx_∩ PID
cargo run --release --example discrete_sxpid     # discrete shared-exclusions PID on logic gates

A Criterion benchmark suite tracks the cost of the brute-force O(n²) kNN backend and the discrete SxPID lattice across sample sizes (KSG MI, continuous I^sx_∩, 2-source PID, discrete SxPID):

cargo bench -p pid-core

(Common dev tasks are also codified as just recipes — run just to list them.)

Conventions

• Units: all information quantities are in nats (natural log).
• Co-information sign: for 2 sources CI₂ = Red − Syn, so negative ⇒ synergy-dominant. This does not carry over to 3 sources — CI₃ is parity-flipped (a pure 3-way synergy gives CI₃ > 0) and conflates atoms, so it is only a coarse screen.
• Negative atoms are real: I^sx_∩ trades all-atom non-negativity for the target chain rule, so atoms (including redundancy) can be negative. Atoms and the MI terms feeding them are never clamped (NegativeHandling::Allow is forced inside the PID/co-information paths); only the standalone ksg_mi defaults to ClampToZero, and that is a reporting choice you can override.

⚠️ Scientific cautions (read before trusting results)

kNN information estimators are powerful but have well-known failure modes. Validate before you interpret.

• i.i.d. assumption — trajectory/time-series autocorrelation biases kNN MI. Subsample or use the block bootstrap.
• Distance concentration — in high ambient/intrinsic dimension, kNN geometry degrades; check the geometry diagnostics first.
• Strong dependence — near-deterministic relationships (very large true MI) can need prohibitive sample sizes (Gao et al. 2015).
• Estimator ≠ truth — do not interpret a downstream result without passing a validation gate on synthetic systems whose information quantities are known analytically.

The exp0 binary is that diagnostic gate (synthetic systems with known MI, noise-dimension invariance, strong-dependence sweeps). It sweeps dimensions up to 256 at n=500 — a range that deliberately includes regimes where kNN MI is known to break down — so a PIVOT/NO-GO verdict on the full default sweep is the expected, informative outcome, not a build failure. It reports per-check counters (Monotonicity / Invariant / Geometry), and exits 0 by default.

--strict-gate is deliberately not allowed to gate that full sweep (a non-GO verdict there is expected, not a failure). Instead it enforces GO — exiting with code 3 otherwise — on a curated band where GO is legitimately expected and is checked against an analytic closed form: a small grid of jointly-Gaussian systems at d=1, n=4000 (the KSG estimator's validated regime), where the three mutual-information terms I(S1;T), I(S2;T), I(S1,S2;T) must match their Cover–Thomas Gaussian values within the scale-aware tolerance. --strict-gate implies --strict-band (run + report the band without enforcing). The four synthetic scenarios are still exercised at d ∈ {2,4,8} as a non-gating diagnostic alongside the band.

cargo run -p pid-core --bin exp0 -- --seeds 4 --summary-json summary.json --runlog run.jsonl
cargo run -p pid-runlog --bin pid-runlog-replay -- --validate run.jsonl

Validation

Correctness is checked against analytically known ground truth, not just self-consistency:

• KSG MI vs the closed-form Gaussian-channel MI I = −½ ln(1 − ρ²).
• Continuous I^sx_∩ against a closed-form additive-Gaussian oracle (the analytic continuous limit of the discrete i^sx; tests/sxpid_gaussian_oracle.rs), plus frozen fixed-data regression pins.
• Discrete SxPID bit-faithfully against the IDTxl/Abzinger reference values (to 1e-12), including its axioms (MGW Theorems IV.2/IV.3 in tests/sxpid_axioms.rs).
• Discrete PID against the independently re-derived I_min and the known structure of canonical gates (XOR = pure synergy ln 2; a redundant copy S1 = S2 = pure redundancy; AND's exact closed-form atoms; …).
• 2-/3-source PID identities (atoms reconstruct total MI) within 1e-10.
• parallel feature results are bit-identical to the serial path.

See crates/pid-core/tests for the suite.

Known limitations

This is a 0.4.0 release. The estimator core (KSG, continuous I^sx_∩, discrete I_min, and the PID identities) is validated against analytic ground truth, but the surrounding statistics/convenience layer has tracked follow-ups (see the issue tracker):

• No multiple-comparison correction. When testing many atoms × sources × windows, apply your own FDR/FWER control; the library reports raw per-atom p-values.
• Bootstrap/permutation caveats for kNN statistics. The block bootstraps are true moving-block (Künsch) resamplers, but with-replacement resampling necessarily duplicates rows, which distorts kNN local-density statistics even with tie-breaking jitter — prefer RowResampleScheme::Subsample for KSG-based statistics. permutation_pid3's full-row shuffle is not a block-permutation null; for autocorrelated trajectory data construct a block-aware null instead.
• Cross-estimator PID2 atoms. Unq/Syn combine KSG MI with Ehrlich I^sx redundancy (different bias profiles); small near-zero atoms can be an estimator artefact rather than structure.
• External cross-validation provenance. The discrete SxPID is validated against the IDTxl/Abzinger reference values; the continuous I^sx_∩ against a closed-form Gaussian oracle. A reproducible external csxpid cross-check of the continuous estimator (committed dataset + recorded invocation) is still pending; the fixed-data constants in tests/isx.rs / tests/pid3.rs are frozen regression pins, not external validation.

None of these affects a single point estimate of MI or a PID atom — they concern uncertainty quantification and convenience-API ergonomics.

Estimators & references

Component	Reference
KSG mutual information	Kraskov, Stögbauer & Grassberger (2004), Phys. Rev. E 69, 066138
Shared-exclusions redundancy i^sx_∩ (discrete discrete_sxpid2/3)	Makkeh, Gutknecht & Wibral (2021), Phys. Rev. E 103, 032149; reference impl. IDTxl pid_goettingen / Abzinger/SxPID
Parthood / formal-logic foundation of PID	Gutknecht, Wibral & Makkeh (2021), arXiv:2008.09535
Continuous I^sx_∩ kNN estimator	Ehrlich, Schick-Poland, Makkeh, Lanfermann, Wollstadt & Wibral (2024), Phys. Rev. E 110, 014115 (arXiv:2311.06373)
I_min redundancy & the PID lattice	Williams & Beer (2010), arXiv:1004.2515
Shannon invariants (r̄, v̄)	Gutknecht, Rosas, Ehrlich, Makkeh, Mediano & Wibral (2025), arXiv:2504.15779
O-information	Rosas, Mediano, Gastpar & Jensen (2019), Phys. Rev. E 100, 032305 (arXiv:1902.11239)
PID non-negativity / chain-rule / invariance trilemma	Matthias, Makkeh, Wibral & Gutknecht (2025), arXiv:2512.16662
kNN MI sample-complexity caveat	Gao, Ver Steeg & Galstyan (2015), arXiv:1411.2003

Workspace

Crate	Description
pid-core	The estimators, PID atoms, invariants, geometry, preprocessing, and the exp0 validation harness.
pid-runlog	Versioned, content-addressed run-log schema + replay/validation CLI for reproducible pipelines.
pid-python	Python bindings (PyO3 + maturin); the pid_core_rs module — 18 functions over NumPy arrays.

Python

The pid_core_rs bindings (added in 0.2.0) are built as a stable-ABI (abi3, CPython ≥ 3.11) wheel with maturin. Arrays are passed as C-contiguous float64 NumPy arrays (wrap transposed/order='F' arrays in np.ascontiguousarray first):

pip install maturin && maturin develop --release -m crates/pid-python/Cargo.toml
python -c "import numpy as np, pid_core_rs as p; print(p.compute_mi(np.random.randn(400,1), np.random.randn(400,1)))"

Minimum supported Rust version

1.80. The MSRV is treated as a semver-relevant property and is exercised in CI.

Contributing

Contributions are welcome — see CONTRIBUTING.md and the Code of Conduct. For anything security-sensitive, see SECURITY.md.

Citation

If you use pid-rs in academic work, please cite it via CITATION.cff (GitHub renders a “Cite this repository” button) and cite the underlying estimator papers above.

License

Licensed under either of

• MIT license (LICENSE-MIT), or
• Apache License, Version 2.0 (LICENSE-APACHE)

at your option. Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

Acknowledgements

The I^sx_∩ measure and its continuous estimator are due to the Wibral group (Göttingen); this is an independent, from-the-papers Rust implementation. Any errors are the maintainer's own.