Likelihood And Inference¶

Likelihood evaluates data under a tree, model, and parameters. Inference adds search or sampling to learn parameters, topology, or both. Keeping those layers separate makes numerical results and ownership claims precise.

flowchart LR
    data["Aligned characters"]
    tree["Tree and branch lengths"]
    model["State space and<br/>rate model"]
    likelihood["Conditional likelihood"]
    ml["Optimization and<br/>topology search"]
    bayes["Priors, proposals,<br/>posterior sampling"]

    data --> likelihood
    tree --> likelihood
    model --> likelihood
    likelihood --> ml
    likelihood --> bayes

Interpret Likelihood In The Correct Direction¶

For observed data D and a declared tree-model state S, likelihood is the quantity L(S; D) = P(D | S). The data are held fixed while candidate states are compared. It is not P(S | D), the probability that the tree or model is true, and it is not a measure of how biologically realistic the model is.

This distinction controls what a likelihood result can support:

Observation	Defensible reading	Reading to reject
one finite log-likelihood	the implementation evaluated these data under this complete state	the model is adequate or the tree is correct
a larger comparable log-likelihood	the data receive greater likelihood under one aligned candidate state	the improvement is meaningful without search, complexity, or comparison review
an optimized maximum	no evaluated state had a better objective under the recorded exploration	a global optimum was proven
a likelihood difference	two aligned calculations differ by the reported amount	the difference is a calibrated probability or support value by itself

Likelihood becomes evidence for selection only through a declared comparison rule and candidate population. It becomes part of a posterior only after priors and normalization over the modeled state space are introduced.

A Likelihood Value Has A Scientific Identity¶

A log-likelihood is not a portable property of a tree. It identifies one calculation under a complete data-and-model contract.

Identity component	Questions that must have one answer
observation population	Which taxa, sites, partitions, and ascertainment rule are included?
character process	Which state space, ambiguity policy, substitution model, rate mixture, and equilibrium frequencies apply?
tree state	Which topology, rooting convention, branch-length units, clock constraint, and fitted values apply?
numerical contract	Which transition calculation, scaling policy, tolerances, and invalid-state behavior apply?
objective	Is the value a fixed-state likelihood, an optimized maximum, a marginal likelihood, or part of a posterior calculation?
computation owner	Was the value produced natively, by an adapter, or by an external engine, and which version and invocation own it?

Compare scalar values only after these identities match. Otherwise, the difference may reflect a changed population, model, parameterization, or normalization rather than stronger support for a tree. A posterior probability is still further removed: it depends on priors and the sampled state space in addition to the likelihood contract.

Finite-State Likelihood¶

The native foundation covers nucleotide, protein, codon, and discrete Mk state spaces. A likelihood value is comparable only when data, tree, branch-length units, state encoding, model parameterization, site treatment, and numerical conventions agree.

Partial-likelihood pruning, transition probabilities, equilibrium frequencies, rate parameters, and scaling behavior belong to the computation contract. Underflow protection must preserve the reported log-likelihood meaning.

Layered Runtime Surfaces¶

Layer	Representative locator	Scientific question
compile	`compile_finite_state_likelihood_analysis` and model-specific compilers	Which state space, transition process, observations, partitions, and caches define the calculation?
evaluate	`evaluate_compiled_finite_state_likelihood_analysis` and named model evaluators	What likelihood is assigned to this fixed tree and parameter state?
optimize	branch-length, frequency, substitution, and joint-optimization reports	Which parameters moved, under what bounds and restart policy, and where did optimization stop?
search	NNI, SPR, TBR, annealing, ratchet, multi-start, and starting-tree reports	Which topologies were considered, accepted, refused, or left unexplored?
support	bootstrap, SH-like, site-likelihood, and topology-test reports	How stable is a split or topology under the named resampling/comparison rule?
result	`NucleotideMaximumLikelihoodResult`	Which model, final topology, fitted state, search evidence, support, warnings, and artifacts belong together?

The broad finite-state foundation does not imply equal inference depth for every state space. The documented end-to-end native tree-search result remains strongest for nucleotide models; protein, codon, and Mk likelihood surfaces must be described at the layer they actually implement.

Enter At The Layer Your Question Requires¶

Intended work	Begin with	Do not infer
score one declared tree and parameter state	model-specific evaluator or compiled finite-state analysis	that the tree or parameters are optimal
fit parameters on a fixed topology	optimizer surface with bounds and starts	that topology uncertainty was explored
infer a nucleotide tree	`NucleotideMaximumLikelihoodResult` workflow	that other state spaces have the same search depth
quantify split stability	named bootstrap or SH-like report	that support is model adequacy or posterior probability
sample a supported posterior	native Bayesian model and matching proposal schedule	that chain completion establishes convergence
invoke IQ-TREE2, FastTree, MrBayes, or BEAST	external-engine adapter	that Bijux owns the external numerical algorithm

Selecting a lower layer gives more control and more responsibility. Callers that assemble compile, evaluation, optimization, and search primitives must retain the same identity and diagnostic information that the unified result would otherwise preserve.

Maximum-Likelihood Inference¶

The documented native tree-inference result is strongest for nucleotide models. It retains model selection, fitted parameters, final tree, topology fingerprint, multi-start summaries, accepted search moves, optional bootstrap and SH-like support, and warnings.

NNI and other topology moves explore a non-convex search space. Multiple starts and repeated convergence provide search evidence; they do not prove a global optimum. Branch support quantifies resampling or local evidence under its named method, not model adequacy or historical certainty.

Bayesian Inference¶

Supported native DNA paths include fixed-topology and joint-topology models through an explicit dispatcher. Priors, proposal weights, tuning scales, iteration count, sampling interval, burn-in, and seed define the run. Posterior interpretation requires acceptance, autocorrelation, effective sample size, trace stability, and independent-chain review.

BEAST and MrBayes remain external-engine workflows. Their adapters preserve configuration, execution, parsing, and artifacts without relabeling the external computation.

Model Selection And Comparison¶

AIC-like criteria, likelihood-ratio tests, posterior summaries, and benchmark correspondence answer different questions. Always retain the compared models, parameter counts, nesting assumptions, objective convention, and decision rule.

Search, Model Selection, And Support defines the shared eligible/attempted/evaluated/selected populations and the record required for ties, failed candidates, and support denominators.

Comparison Admissibility¶

Two likelihoods are directly comparable only when they use the same observed sites and taxon set, tree interpretation, conditioning convention, partition accounting, state/missingness policy, root treatment, and log-likelihood scale. A likelihood-ratio test additionally requires a declared nested pair and valid degrees-of-freedom rule. Information criteria require consistent parameter counting. Posterior summaries require the same estimand and compatible prior, likelihood, and sampling definitions.

If those conditions are absent, retain both results as separate observations; do not manufacture a delta, ranking, or parity verdict.

Trace A Reported Likelihood¶

Every published likelihood should be recoverable through this chain:

flowchart LR
    input["alignment identity<br/>taxa and sites"]
    state["tree · branches<br/>model parameters"]
    convention["partitions · missingness<br/>root and scaling policy"]
    evaluation["log-likelihood"]
    selection["optimization or<br/>model comparison"]
    claim["bounded conclusion"]

    input --> evaluation
    state --> evaluation
    convention --> evaluation
    evaluation --> selection --> claim

If the tree, parameter state, or numerical convention cannot be recovered, the likelihood is an orphaned scalar. It may be displayed, but it cannot support reproducible model comparison.

For an optimized result, preserve both the final objective and the route that produced it: starting-state identity, bounds, termination condition, warnings, and search trace. For a posterior sample, preserve the corresponding prior and proposal identity as well.

Numerical Failure Modes¶

invalid transition matrices or state encoding;
underflow, overflow, or unstable scaling;
flat, multimodal, or boundary likelihood surfaces;
optimizer stagnation or start-dependent solutions;
poor proposal acceptance or posterior mixing;
topology labels or partitions inconsistent with the alignment;
comparison against a reference using different assumptions.