Expectation Conditional Maximization Either
Algorithms AIMech:ExpectationConditionalMaximizationEither DRAFT
Aliases: ECME Algorithm, ECME
An extension of expectation conditional maximization that replaces selected final conditional maximizations of expected complete-data log likelihood with conditional maximizations of actual observed-data likelihood.
Identity Notes & Concept Scope
The either refers to the objective used for a conditional parameter update. The inspected construction performs its fixed-Q updates before final observed-likelihood updates; arbitrary interleaving with stale expectations is not asserted. It is distinct from ordinary ECM, unrestricted joint likelihood optimization, AECM changes of augmentation and PX-EM expansion/reduction. Primary 1995 and 1998 follow-ups support this construction; the original 1994 rate proof remains uninspected. No normalized identity match was found across all 442 pinned AIO labels and scoped synonyms and 463 preceding native IDs, names and aliases. Fixed-string searches of worktree and main included ignored and hidden files, yielding 52 and 53 matched lines respectively. All 41 distinct contextual passages across 105 line instances and nine checkout-qualified paths were read; these concern prior discovery references and factor-analysis evidence, plus an unrelated ignored URI-template parameter-expansion string. They are not existing identities for these three algorithms. All 223 semantic label/synonym rows were read. Fifteen exact annotated AIO CLASS nodes and outgoing edges were inspected: fourteen verified unchanged after prior reading and DataAugmentation newly read. Twenty-five contextual YAMLs comprise fifteen AIO and ten native records; twenty are unchanged verified prior readings, two are hash-verified final records fully read in the preceding cohort, and three were fully read here. General EM, tempered EM, Gaussian factor models, PPCA mixtures, Gibbs/Metropolis sampling, specialized deep Bayesian data augmentation and alternating Newton fitting do not identify these particular conditional/expanded EM procedures. Absence is bounded to the pinned snapshot, maintained corpus and inspected neighborhood.
Claims & Grounded Evidence
3 Claims
Mechanism
#ordered-either-objective-updates
Compute the E-step expectation at the current parameter. Carry out the selected conditional Q maximizations using that fixed expectation, followed by the selected conditional actual-likelihood maximizations at the current intermediate parameters. The Q sequence establishes likelihood ascent relative to its E-step anchor; the subsequent observed-likelihood steps preserve or improve that value. The complete ECME iteration need not increase the original Q and therefore need not be a generalized EM step relative to that Q.
Mechanism
#t-model-example
For the multivariate t normal/gamma augmentation, compute expected latent weights from the old parameters. Update location and scatter through Q at the old degrees of freedom, then conditionally maximize actual observed likelihood over positive degrees of freedom using the new location and scatter. In this final step, terms depending on degrees of freedom must vary with each trial value; they are not frozen old E-step statistics. The fully observed scatter update centers weighted residuals at the new mean and divides by sample count. The parameter is scatter, not the marginal t covariance.
Scope
#factor-model-and-safeguards
In the inspected Gaussian factor-analysis application, Q updates loadings and any free factor-covariance parameters, followed by actual-likelihood fitting of diagonal uniqueness variances at those updated values. Numerical Newton steps are acceptable only with the required likelihood increase; Newton iteration alone does not guarantee it. This changes the fitting procedure while retaining the same marginal factor model.
⚠️ Curated Limitations & Negative Results
- Observed-likelihood ascent is not global optimality or unconditional parameter convergence. Conditional numerical solves require valid parameter constraints and objective safeguards; a stationary root need not be the intended conditional maximum.
- The scheduling statement is the final-likelihood-step construction supported by the inspected sources. An earlier likelihood update followed by an unrefreshed Q update can decrease the full-iteration likelihood, as the saved curator Gaussian example shows.
- The t examples assume ignorable missingness and correct conditional first and second moments. The source contains inconsistent displayed conditional-mean signs and old/new mean subscripts; the record uses the model-consistent moment interpretation and does not reproduce those expressions.
- The original 1994 publisher PDF returned HTTP 403. Its general rate proof is not claimed as read. Two complete primary follow-ups support the stated method; historical iteration/time comparisons, clinical examples and source software were not reproduced. No universal acceleration or compute-cost guarantee is asserted.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MachineLearningTask
Machine Learning Task
|
related_match | This is an explicit likelihood-fitting algorithm used to learn parameters from data. The pinned target denotes a learning process/task, so the link is algorithm-to-task relatedness, not equivalence or a subclass assertion. Latent-data augmentation and auxiliary parameter expansion do not identify the pinned transformation-based DataAugmentation term. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Chuanhai Liu, Donald B. Rubin
|
primary paper
full_text
|
ID: https://www3.stat.sinica.edu.tw/statistica/oldpdf/a5n12.pdf
sha256:52999e1b82ef…
|
factor-source
|
Chuanhai Liu, Donald B. Rubin
|
primary paper
full_text
|
ID: https://www3.stat.sinica.edu.tw/statistica/oldpdf/A8n36.pdf
sha256:0d03448be6ec…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-12T03:11:01Z
Updated: 2026-10-12T03:11:01Z