AIMech AI

Fixed-Component Mixture-Weight EM

Algorithms AIMech:FixedComponentMixtureWeightEM DRAFT
Aliases: Mixture-Weight Expectation Maximization
An expectation-maximization specialization that keeps every component distribution fixed and updates only their simplex mixing weights by averaging current component responsibilities.
Identity Notes & Concept Scope The name is a descriptive AIMech label for the fixed-component coefficient iteration in Csiszar-Tusnady Theorem5 and its mixture-decomposition interpretation. Fixed component laws do not mean merely a fixed number of components. This retains a distinct convex weight-fitting problem beneath generic EM; joint component-parameter fitting in Gaussian mixtures or MPPCA is outside it, and a mixture density network predicts input-dependent parameters instead. No normalized identity collision across all 442 pinned AIO IDs, labels/scoped synonyms and 483 maintained IDs, names and aliases. Full-checkout searches in the worktree and main included ignored and hidden files: 8 and 9 matched lines, respectively, reduce to six distinct matched texts, all inspected. Hits concern existing discovery leads and the unrelated assembler mnemonic AAM. The fixed-family exact KL projection procedure is narrower than generalized alternating minimization; weight-only EM fixes component laws rather than fitting a full mixture model; rate-distortion BA minimizes information plus distortion with a fixed source, whereas the IM algorithm maximizes a variational information bound; adaptive alternating minimization changes the feasible families at each iteration. Qualified AAM and descriptive weight-only names avoid unsupported acronym or historical naming claims. Absence is bounded to this snapshot and inspected corpus. Identity-search counts: 8 worktree and 9 main matched lines, all six distinct texts read, including ignored and hidden files.

Claims & Grounded Evidence

3 Claims
Mechanism #responsibility-average
For fixed component probabilities mu_i(y) and a target probability p(y), let q_c(y)=sum_i c_i mu_i(y). With positive initial weights summing to one and q_c positive on target support, set r_i(y)=c_i mu_i(y)/q_c(y) and c_i_new=sum_y p(y) r_i(y). The update normalizes because the responsibilities sum to one on target support. For empirical frequencies, it is the average responsibility over observations. No component location, covariance or shape parameter is updated. The source gives the corresponding integral formula for general fixed finite measures.
Scope #convex-coefficient-result
Under Theorem5: a probability target, finitely many fixed component measures, strictly positive simplex initialization and a finite best divergence, the coefficient vector converges to a vector attaining the minimum divergence. For fixed probability components in a finite observation space this is the convex mixture-weight likelihood problem. The proof does not require identifiable component representation, so the limit can depend on initialization; identical components preserve their initial relative weights. This guarantee does not extend to jointly moving component parameters.
Limitation #certificate-and-support
The theorem bounds the current divergence gap by max_i log(c_i_(n+1)/c_i_n) where the ratios are defined. A coefficient driven to zero because its component has no target-support mass can be removed with an explicit support convention. Arbitrary zero initialization instead may permanently exclude useful components. With mu_1=(0.8,0.2), mu_2=(0.2,0.8) and target(0.6,0.4), an interior start tends to weights(2/3,1/3), while start(1,0) stays on a suboptimal face. Continuous-observation likelihood requires fixed density versions and the appropriate lifted measure; empirical point masses cannot naively be assigned finite KL to a continuous density.

⚠️ Curated Limitations & Negative Results

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes
AIO Target Class Mapping Relation Mapping Rationale
AIO:UnsupervisedLearning
Unsupervised Learning
related_match The procedure estimates latent component proportions from an observed distribution without component labels. This relates the fitting algorithm to the AIO unlabeled-learning task without asserting equivalence or subclass membership.

Evidence Sources & Primary Literature

1 Source
Source ID Publication / Document Type / Access Reference & Checksum
projection-source
Imre Csiszar, Gabor Tusnady
primary paper
full_text
sha256:96ef51b835fd…

Curation Provenance

Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-12T05:22:42Z
Updated: 2026-10-12T05:29:50Z
Source File: data/curated/FixedComponentMixtureWeightEM.yaml