evtools-dst 0.8.0

Evidence Theory Tools — belief functions, combination rules and contextual correction mechanisms for the Dempster-Shafer / Transferable Belief Model

evtools

Evidence Theory Tools — a Python library for working with belief functions in the Dempster-Shafer theory / Transferable Belief Model. Version 0.8.0.

Modules

Module	Description
evtools.dsvector	DSVector — unified container for any belief function representation
evtools.conversions	Low-level conversions via the Fast Möbius Transform
evtools.combinations	Combination rules: CRC, Dempster, DRC, Cautious, Bold, and decombinations
evtools.corrections	Correction mechanisms: discounting, reinforcement, negating
evtools.display	Display formats: ANSI terminal, plain text, HTML, LaTeX
evtools.constants	Numerical tolerance constants

---

evtools.dsvector

DSVector is the central object of evtools. It represents any belief function as a vector on 2^Ω, in both sparse (dict) and dense (numpy array) forms. The sparse representation is the master; the dense array is computed on demand and cached.

Kind enum

Kind	Symbol	Name
Kind.M	m	Basic Belief Assignment (mass function)
Kind.BEL	bel	Belief function
Kind.PL	pl	Plausibility function
Kind.B	b	Commonality function
Kind.Q	q	Implicability function
Kind.V	v	Disjunctive weight function
Kind.W	w	Conjunctive weight function

Constructors

from evtools.dsvector import DSVector, Kind

# Human-friendly: name focal elements as strings
# Missing mass is automatically assigned to Ω
m = DSVector.from_focal(["a", "b", "c"], {"a": 0.3, "b,c": 0.5})

# From a dense numpy array (binary index ordering, Smets 2002)
m = DSVector.from_dense(["a", "b", "c"], np.array([0, 0.3, 0, 0, 0.5, 0, 0, 0.2]))

# From a sparse dict of frozensets
m = DSVector.from_sparse(["a", "b", "c"], {
    frozenset({"a"}):          0.3,
    frozenset({"b", "c"}):    0.5,
    frozenset({"a","b","c"}): 0.2,
})

Simple MF constructors

Simple MFs are the elementary building blocks of correction mechanisms.

# Simple MF A^β — focal sets Ω (mass β) and A (mass 1−β)
# Used in Contextual Reinforcement (CR), CdR, CN
s = DSVector.simple(["a", "b", "c"], frozenset({"a"}), beta=0.6)

# Negative simple MF A_β — focal sets ∅ (mass β) and A (mass 1−β)
# Used in Contextual Discounting (CD), CdD
ns = DSVector.negative_simple(["a", "b", "c"], frozenset({"a"}), beta=0.4)

Conversions

pl  = m.to(Kind.PL)   # returns a new DSVector with kind=Kind.PL
bel = m.to_bel()      # shortcut
b   = m.to_b()        # commonality
q   = m.to_q()        # implicability
v   = m.to_v()        # disjunctive weights (requires subnormal BBA)
w   = m.to_w()        # conjunctive weights (requires subnormal BBA)

Accessing values

m.sparse                     # dict[frozenset, float]
m.dense                      # np.ndarray of length 2^n
m.is_valid                   # True if all masses ≥ 0 and sum = 1 (Kind.M only)
m[frozenset({"a"})]          # value for a given subset (0.0 if absent)
for subset, value in m: ...  # iterate over non-zero focal elements

Display

m.display("ansi")    # colored terminal (default __repr__)
m.display("plain")   # plain text, no colors
m.display("html")    # HTML table (Jupyter renders this automatically)
m.display("latex")   # LaTeX tabular for papers

---

evtools.combinations

Combination rules for aggregating beliefs from multiple sources.

from evtools.combinations import crc, dempster, drc, cautious, bold
from evtools.combinations import decombine_crc, decombine_drc

m12 = crc(m1, m2)        # m1 & m2  — Conjunctive Rule (TBM), distinct reliable sources
m12 = dempster(m1, m2)   # m1 @ m2  — Dempster's normalized rule
m12 = drc(m1, m2)        # m1 | m2  — Disjunctive Rule, at least one reliable
m12 = cautious(m1, m2)   # Cautious rule, nondistinct reliable sources (idempotent)
m12 = bold(m1, m2)       # Bold disjunctive rule, nondistinct possibly unreliable (idempotent)

# Decombination — inverse operations (result may not be valid, check .is_valid)
m1 = decombine_crc(m12, m2)  # m12 6∩ m2 — removes m2 from a conjunctive combination
m1 = decombine_drc(m12, m2)  # m12 6∪ m2 — removes m2 from a disjunctive combination

Choice of rule:

	All sources reliable	At least one reliable
Distinct sources	crc / dempster	drc
Nondistinct sources	cautious	bold

Both crc and drc support method="sparse" (default) or method="dense".

---

evtools.corrections

Correction mechanisms for adjusting a BBA based on knowledge about the quality of a source (reliability, truthfulness).

Notation:

• A^β — simple MF: focal sets Ω (mass β) and A (mass 1−β)
• A_β — negative simple MF: focal sets ∅ (mass β) and A (mass 1−β)

from evtools.corrections import (
    discount,
    contextual_discount,
    theta_contextual_discount,
    contextual_reinforce,
    contextual_dediscount,
    contextual_dereinforce,
    contextual_negate,
)

# Classical discounting — source reliable with degree β ∈ [0,1]
# β=1: unchanged; β=0: vacuous BBA
m_disc = discount(m, beta=0.6)

# Contextual discounting (CD) — reliability per singleton context
# Uses negative simple MFs A_β and the DRC
betas = {frozenset({"a"}): 0.6, frozenset({"h"}): 1.0, frozenset({"r"}): 1.0}
m_cd = contextual_discount(m, betas)

# Θ-contextual discounting — reliability per coarsening partition
betas_theta = {frozenset({"a"}): 0.4, frozenset({"h","r"}): 0.9}
m_theta = theta_contextual_discount(m, betas_theta)

# Contextual Reinforcement (CR) — dual of CD, uses simple MFs A^β and the CRC
m_cr = contextual_reinforce(m, betas)

# Inverse operations (result may not be valid — check .is_valid)
m_cdd = contextual_dediscount(m_cd, betas)    # reverses CD
m_cdr = contextual_dereinforce(m_cr, betas)   # reverses CR

# Contextual Negating (CN) — source non-truthful with probability 1−β
m_cn = contextual_negate(m, {frozenset({"a"}): 0.7})

Hierarchy of discounting:

discount(m, β)
  └── theta_contextual_discount(m, {Ω: β})

contextual_discount(m, β)
  └── theta_contextual_discount(m, β)   [Θ = singletons]

theta_contextual_discount(m, β)         [general Θ partition]

---

evtools.display

Four output formats, all adapting the column header to the kind (m, bel, pl, ...). In Jupyter notebooks, DSVector._repr_html_() is called automatically.

from evtools.display import repr_plain, repr_html, repr_latex

print(repr_plain(m))   # plain text, no colors
print(repr_latex(m))   # LaTeX tabular for papers
m.display("ansi")      # colored terminal (default)
m.display("html")      # HTML table

---

evtools.conversions

Low-level conversion functions operating on plain numpy arrays (length 2^n), using the Fast Möbius Transform (Smets 2002). Every conversion is available as <source>to<target>, e.g. mtob, pltom, qtow, beltov, etc.

from evtools.conversions import mtob, mtopl, mtobel, mtoq

m = np.array([0.0, 0.5, 0.0, 0.0, 0.5, 0.0, 0.0, 0.0])
print(mtoq(m))    # commonality function
print(mtopl(m))   # plausibility function

Array indices follow the binary ordering of Smets (2002): index i corresponds to the subset whose members are the frame atoms at the bit positions set in i.

---

Installation

pip install evtools-dst

Or from source:

git clone https://github.com/daviddavkanmercier/evtools.git
cd evtools
pip install -e .

Running tests

pip install -e ".[dev]"
pytest tests/

References

• P. Smets. The application of the matrix calculus to belief functions, International Journal of Approximate Reasoning, 31(1–2):1–30, 2002.
• T. Denœux. Conjunctive and disjunctive combination of belief functions induced by non-distinct bodies of evidence, Artificial Intelligence, 172:234–264, 2008.
• D. Mercier, B. Quost, T. Denœux. Refined modeling of sensor reliability in the belief function framework using contextual discounting, Information Fusion, Vol. 9, Issue 2, pp 246-258, April 2008.
• F. Pichon, D. Mercier, É. Lefèvre, F. Delmotte. Proposition and learning of some belief function contextual correction mechanisms, International Journal of Approximate Reasoning, Vol. 72, pp 4-42, May 2016.

License

MIT