Discrete Ancestral States And Transitions

Discrete ancestral analysis has three separate outputs: a fitted transition model, conditional state probabilities at internal nodes, and branch-level transition summaries derived from those states. A preferred state at a node is not an observed history, and a branch transition is not certain merely because its endpoint preferences differ.

flowchart LR
    observations["Tip states<br/>and exclusions"]
    model["State order · rate matrix<br/>constraints · root prior"]
    fit["Likelihood and<br/>fitted rates"]
    nodes["Node-state<br/>probabilities"]
    branches["Branch transition<br/>observations"]
    sensitivity["Tree, model and<br/>ambiguity sensitivity"]
    claim["Bounded history claim"]

    observations --> fit
    model --> fit --> nodes --> branches --> sensitivity --> claim

Distinguish Four Transition Claims

The word “transition” can describe four materially different records:

Claim	What is known	What remains unknown
endpoints prefer different states	parent and child node summaries differ under one fit	whether, where, or how often change occurred on the branch
a change has posterior or likelihood weight	model-integrated evidence favors a change	a single realized history
a sampled history contains an event	one stochastic history realizes the event	whether the event is stable across samples, trees, and models
an observed state pair differs across taxa	tip observations differ	the ancestral direction and branch on which change occurred

Reports must name which record is being counted. A preferred-endpoint change is a deterministic summary of reconstructed probabilities; a sampled event is a draw conditional on a model; and neither is an observed historical event. Transition frequencies become interpretable only with their branch or tree-sample opportunity denominator.

Direction Begins At The Root

An unrooted topology can define splits but not ancestors, descendants, or a directed transition. Root placement is therefore part of the scientific model, not a display preference applied after reconstruction.

Root contract	Consequence for transition evidence
biologically justified outgroup root	direction depends on the admitted outgroup and its placement
clock or dated-tree root	direction inherits the clock, calibration, and time-tree assumptions
midpoint or other algorithmic root	direction is conditional on that rooting heuristic
root prior over states	basal node probabilities and early transitions depend on the declared prior
alternative plausible roots	node identity and transition direction must be recomputed, not relabeled

Retain the rooted tree identity with every node and branch row. A structural branch key must identify the descendant clade under that root; traversal numbers and drawing orientation cannot provide durable correspondence. When a plausible root changes a headline transition, publish the conclusion as root-sensitive rather than selecting the visually convenient history.

Establish State Identity

Record the complete state vocabulary, order, aliases, ambiguity symbols, missing-state policy, and every tip exclusion. State order determines the layout of probabilities and rate parameters. Alphabetical display sorting must not silently change that layout.

Input condition	Required treatment
unobserved but allowed state	retain in the declared state space and explain identifiability limits
ambiguous tip state	preserve the allowed-state set; do not coerce to one preferred value
missing tip state	distinguish missingness from a biologically meaningful unknown state
state alias or spelling variant	apply a declared mapping and retain source plus normalized value
duplicate taxon observation	refuse or apply an explicit aggregation/error model

The admitted taxon population and state mapping belong to every downstream fit, node row, transition row, and figure.

Identify The Transition Model

The native discrete likelihood surface supports fitted Mk-family rate structures and declared transition constraints. The model record includes the state order, rate-matrix parameterization, fixed or estimated values, allowed and forbidden transitions, root prior, bounds, starts, objective convention, optimizer state, and warnings.

ER, SYM, and ARD labels are not enough by themselves. Two fits are comparable only when their state spaces, parameter layouts, root treatment, likelihood convention, and constraints align. Ordered and irreversible summaries add scientific restrictions that must survive into the report even when the best node states remain unchanged.

Read Node Probabilities Without Inventing Certainty

Use stable structural node keys, such as descendant-tip identity under the declared rooted tree, rather than traversal order. Retain the full probability vector, not only the maximum state.

Node result	Safe statement	Unsafe statement
concentrated probability under one model/tree	one state is preferred conditionally	the ancestor was observed in that state
broad probability vector	the reconstruction is ambiguous	choose the maximum and omit ambiguity
state changes across models or trees	inference is model- or topology-sensitive	report only the preferred configuration
node absent after taxon/tree change	result is not structurally comparable	align rows by node number

Probability ties and near-ties are substantive results. Preserve the tie rule and every state within the declared materiality threshold.

Derive Branch Transitions Explicitly

summarize_ancestral_transitions and the tree-set transition review surfaces turn node reconstructions into branch-keyed observations and count ledgers. The record must distinguish:

• preferred-state change between parent and child;
• probability-weighted evidence for a change;
• a transition allowed or forbidden by the model;
• a simulated or sampled event history;
• an observed opportunity with no supported transition.

These are not interchangeable. Endpoint preferences can suggest a change but do not locate an event within a branch or establish the number of changes.

Use Opportunity-Aware Denominators

Count transitions relative to eligible branches, time, or another declared exposure. Retain branches with ambiguous endpoints, unavailable nodes, forbidden transitions, exclusions, and failed tree-set members. Reporting only branches classified as changes makes rates and burdens uninterpretable.

Summary	Denominator to retain
transition count by state pair	all eligible directed branch opportunities under the state model
clade transition burden	eligible branches in the structurally identified clade
tree-set transition frequency	admitted trees in which the branch/clade identity is resolvable
temporal transition count	branches and exposure assigned to each declared time bin

Propagate Across A Tree Set

Tree-set review repeats reconstruction and transition summarization for an identified tree population. Preserve per-tree fit state, branch observations, unresolvable clades, failures, and aggregation rules before computing a frequency or stable conclusion.

flowchart TB
    trees["Admitted tree set"] --> fits["Per-tree fit and<br/>node probabilities"]
    fits --> branch_rows["Per-tree branch<br/>transition rows"]
    branch_rows --> aggregate["Structurally keyed<br/>frequencies and conflicts"]
    aggregate --> outcome["Stable · qualified<br/>unstable · not assessable"]

A transition supported on one selected tree is conditional evidence. A tree-set frequency is meaningful only over members where the relevant structural identity and states were eligible for evaluation.

Publish The Complete Record

Retain the tree or tree-set identity, tip-state table and mapping, admitted and excluded taxa, model and parameter layout, root prior, optimizer diagnostics, full node probabilities, branch-keyed transition rows, opportunity denominators, sensitivity results, and structured artifact inventory. Rendered state trees and migration diagrams are projections of those records.

Refuse an unqualified historical claim when node identity is unstable, the state model is unidentified, endpoint ambiguity is substantial, or plausible trees and constraints change the transition conclusion.