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Let $E_q$ be the event that all eight selected subjects carry two copies of the same specified ancestral copy $q$, and that no other generation-4 subject is autozygous at this genetic locus. The selected subjects occur in every sibship, so this event forces both generation-2 transmissions and all six relevant generation-3 transmissions. Applying the preceding conditional probabilities,Once all six parents carry $q$, their other copies come from the two unrelated outside pedigree founders, one on each side of the pedigree. A child not receiving two $q$ copies therefore cannot become autozygous for a different founding copy. This verifies the “only” condition, rather than merely excluding homozygosity for $q$.

For the event $E$ without specifying which of the original four copies is shared, the four alternatives $E_q$ are disjoint: a selected child cannot simultaneously have both its copies descended from two different ancestral copies. HenceAll selected individuals are then autozygous for the same copy and mutually 2-IBD. The distinction from the $3/4$ probability in the alternative interpretation of part (a) is precisely this disjointness.

Write $D$ for the disease observations and $M$ for the genetic marker observations. The LOD score for complete linkage isThe disease marginal cancels because the single-locus disease segregation law does not depend on the genetic marker–disease recombination fraction. The ten genetic markers form one nonrecombining haplotype; they do not supply ten independent copies of the pedigree's segregation evidence.

There is a distinction between identity by state and identity by descent. The reported observation that affected subjects all have $h/h$ establishes the former. An exact genetic marker likelihood requires pedigree founder haplotype frequencies, or the actual pedigree founder genotypes if conditioning on them. Here is an explicit expression using independent pedigree founder haplotypes, with frequency $p$ for the observed haplotype $h$, and linkage equilibrium between pedigree founder genetic marker and disease states. There are eight genetic marker copies in the four pedigree founders: the original couple and the two generation-2 outside spouses. For genetic marker inheritance configuration $v$, let $K(v)$ be the number of distinct pedigree founder copies ancestral to the sixteen chromosome copies in the eight selected subjects. Thenis the unlinked probability that all those copies have state $h$. At complete linkage the affected subjects all inherit the disease pedigree founder copy twice, and the probability that this copy has genetic marker state $h$ is $p$. Therefore

The denominator can be evaluated without an unspecified inheritance sum. Let $B_a=\left(\genfrac{}{}{0ex}{}{2}{a}\right)p^a(1-p{)}^{2-a}$ for $a=0,1,2$. If parental genetic marker genotypes contain $a,b$ copies of $h$, the child count distribution isFor a generation-2 couple defineFor a particular generation-3 parent with $j$ genetic marker copies, the chance it transmits $h$ to all its $r$ selected offspring is $(j/2{)}^r$, which explains each factor. Conditional on the original founding couple, its two generation-2 siblings and their independent outside spouses give independent left and right contributions. ThusExpanding givesIt satisfies $R(1)=1$, as it must for a monomorphic haplotype. This expression supplies a numerical LOD once $p$ is specified; the problem does not specify that frequency.

In the idealized limit where a matching rare haplotype identifies one ancestral copy, the leading coefficient is particularly simple. Dropping restrictions on the eight unaffected subjects, the probability that all selected subjects are autozygous for one common copy isAccordingly $R(p)/p\to 2^{-22}$ as $p\to 0$, and the affected-only IBD calculation gives $Z(0)=22{\log }_{10}2\simeq 6.623$.

If the genetic marker observations additionally establish the exact pattern $E$ of part (d), including that the eight unaffected subjects are not autozygous, its idealized IBD likelihood ratio is insteadThis latter value requires the additional negative genetic marker observations. It cannot be inferred merely from homozygosity in the affected subjects, and neither simplified IBD score is an exact identity-by-state likelihood for an unspecified common haplotype.

behavioural plasticity can slow inherited change by buffering selection, or accelerate it by permitting persistence and exposing organisms to new selection. There is no general ranking without specifying the trait, environment and timescale. The immediate adjustment of an individual's animal behaviour is a change in expression. evolution of that response requires a change in inherited properties of a biological population. A learned innovation spreading through cultural inheritance is another form of transmitted change, which should be distinguished from a change in allele frequencies.

A reaction norm makes this distinction explicit. Suppose a behavioural measurement in environment $E$ is $z=a+bE$, with inherited variation in baseline $a$ and responsiveness $b$. For illustration, let the favoured value be $E$, and let relative genotype fitness be:The first term penalizes mismatch; the second represents a possible cost of responsiveness. This is a deliberately simple model, not a claim that all animal behaviour has one optimum. Its phenotypic selection gradient on baseline, holding $b$ fixed, is:Immediately after a change to $E=1$, individuals with $a=0,b=1$ match the optimum and experience no mismatch selection on $a$; otherwise identical individuals with $b=0$ experience a gradient $2k$. Thus phenotypic buffering can reduce selection for inherited baseline change even when the response is highly beneficial. If $c>0$, selection on responsiveness can nevertheless persist. Buffering one dimension of the phenotype does not stop all evolution.

This impeding effect is biologically plausible when animals can avoid adverse conditions. Moving to a cooler microhabitat can reduce exposure to heat and therefore weaken selection on heat tolerance. Switching food can similarly reduce selection on digestive performance for an avoided resource. These are conditional mechanisms: the relevant evidence would be reduced differences in survival or reproduction among inherited physiological variants, rather than mere observation that animals move or switch foods. learning can also allow several inherited variants to reach similar behavioural performance, weakening selection among them. Such buffering may preserve genetic variation, which can become consequential after later environmental change; slow immediate change can therefore coexist with greater long-term evolutionary potential.

Acceleration begins with the demographic problem. An inherited advantage cannot spread in a lineage that disappears before it is expressed or transmitted. A flexible response can allow individuals to survive and reproduce after an environmental shift, providing time for subsequent inherited adaptation and possibly evolutionary rescue. This need not increase the selection coefficient of a particular variant. Increasing the number of surviving individuals or generations available for selection can matter even while the per-generation gradient becomes smaller. A comparison of change per generation and total adaptation over a fixed number of years can consequently give different answers.

behavioural plasticity can also change which environments impose selection. A newly learned foraging technique may expose animals to a new resource, making inherited differences in handling, digestion or sensory discrimination important. habitat preference can alter encounters with competitors and mates; repeated use of a new host can influence gene flow when mating occurs there. The host fidelity of Rhagoletis pomonella provides evidence that host-associated actions can constitute a premating barrier. It demonstrates a route from animal behaviour to evolutionary consequences, but does not by itself establish that the initiating host shift was a plastic response. ^[ref] Establishing that historical sequence is necessary before counting the case as plasticity-led evolution.

In the Baldwin effect, acquired adjustments change survival and the selection subsequently experienced by inherited variants. genetic accommodation can then refine the threshold, specificity or efficiency of an environmentally induced response. Its outcome may be greater phenotypic plasticity, a differently tuned response, or genetic assimilation, in which expression becomes less dependent on the original inducing condition. No step requires a learned action to be copied directly into DNA.

Experiments on spadefoot toad tadpoles provide relevant, though not exclusively behavioural, evidence. Populations that expressed the carnivore ecomorph more frequently showed refinement of associated resource-use performance and morphology under laboratory comparison. Such correspondence is predicted if recurrent induction exposes the expressed phenotype to selection. The comparison supports inherited refinement following environmental responsiveness, but contemporary populations are not a direct film of the ancestral transition, and their prior ecological histories could differ. ^[ref] Classic selection on heat-induced crossveinless wings in Drosophila melanogaster also showed that a previously induced phenotype could eventually appear without the inducing treatment. That is evidence for genetic assimilation, not direct evidence that behavioural innovations generally accelerate evolution. ^[ref]

The balance depends on cue reliability, environmental persistence and costs of phenotypic plasticity. A response based on a formerly reliable cue can become maladaptive after environmental change. A persistent new environment may favour a cheaper constitutive solution, whereas recurring variation can favour continued flexibility. Response delays, limited sensory discrimination and trade-offs between alternative activities can leave substantial residual selection. Conversely, sufficiently effective and inexpensive flexibility can remove much of the benefit of further inherited change. The availability of relevant genetic variation and its genetic covariance with other traits also constrain the response.

A decisive study would separate immediate adjustment from inherited divergence. Replicated populations could experience stable and fluctuating environments, with variation in access to a learned response or environmental cue. Descendants should then be assayed in a common garden experiment, both with and without that cue. Repeated measures estimate within-individual change; pedigrees or crosses estimate inherited reaction norm differences. Survival, reproduction, population persistence and changes in other traits must all be measured. Manipulating responsiveness requires controls for sensory damage, stress and general incapacity: disabling a neural process and then observing extinction would not isolate an evolutionary benefit of flexibility. Comparisons across species additionally require control for phylogenetic nonindependence and ecological opportunity.

The strongest conclusion is conditional: effective adjustment often weakens immediate selection on the buffered trait, while persistence, new ecological exposure and inherited refinement can increase cumulative evolutionary change. Those mechanisms can operate together within the same lineage; their relative importance is an empirical question rather than an intrinsic property of flexible animal behaviour.

Knowing the molecular basis of a behaviour is not a prerequisite for understanding its evolution or many of its evolutionary consequences. Evidence about transmission is essential for claims of inherited change, while detailed genetic architecture becomes essential for particular mechanistic predictions. The claim therefore depends on what is meant by a genetic basis and on the explanatory question being asked.

Tinbergen's four questions distinguish mechanism, development, adaptive significance and evolutionary history. Identifying a neural or molecular mechanism helps explain how a response is produced; it does not automatically show why a variant spread, whether it currently increases reproductive success, or whether the same mechanism generated the ancestral transition. Conversely, a well-supported evolutionary explanation can begin with experimentally demonstrated biological fitness consequences and transmission without a complete molecular account. These explanatory levels are complementary rather than substitutes. ^[ref]

The phrase genetic basis can refer to several different achievements: identifying inherited variation; estimating the number and effects of contributing genetic loci; finding a causal nucleotide change; or describing every gene required for development of the relevant sensory and motor systems. These are not equivalent. Many genes can be necessary for an animal to perform an action while none explains variation in that action within the studied population. Knocking out a broadly required neural gene and eliminating courtship does not identify an evolutionary substitution for a naturally varying mate preference.

quantitative genetics can predict a response without naming the causal molecular variants. Under the appropriate additive, generational and environmental assumptions, the multivariate breeder's equation gives:where $\mathbf{G}$ is the additive genetic covariance matrix and $\mathbf{\beta }$ the vector of phenotypic selection gradients. Behavioural measurements, pedigrees, repeated observations and reproductive success can therefore estimate whether and in what direction a response is expected. The covariances matter: selection favouring exploration can produce a correlated response in aggression or constrain the response when their biological fitness effects conflict. Naming an exploration-related gene would not replace this population-level information.

These estimates require strong controls. behavioural repeatability is consistency among observations of the same individual; it can reflect persistent environment as well as inherited differences and is not equivalent to heritability. Parent–offspring resemblance can also reflect teaching, maternal provisioning or shared habitat. cross-fostering, common garden experiments and appropriately structured breeding designs help separate these alternatives. heritability is conditional on the sampled population and environment, not a measure of immutability. A response measured in one environment may fail elsewhere because of genotype-environment interaction or altered selection. Thus ignorance of exact molecular variants is compatible with informative prediction, but ignorance of transmission and its confounders is not.

Molecular work adds power when it establishes a causal link to naturally occurring variation. In Drosophila melanogaster, rover and sitter food-search variants were connected to the foraging gene, which encodes a cGMP-dependent protein kinase. Mapping, enzyme activity and transgenic rescue linked the molecular difference to behavioural performance; comparable locomotion without food helped exclude a general activity explanation. ^[ref] This is much stronger than finding a statistical association between a gene and a behavioural score. Nevertheless, the experiment alone does not establish which ecological conditions historically maintained the polymorphism. That requires biological fitness comparisons across resources, densities and frequencies, with genetic-background controls.

genetic architecture is particularly informative when animal behaviour changes gene flow. In Heliconius melpomene and Heliconius cydno, mapping identified several major regions associated with male mating preferences; one lay close to the wing-pattern regulator Optix. Tight genetic linkage can help maintain associations between a mating cue and preference despite exchange between lineages. It does not establish pleiotropy, and courtship assays must distinguish attraction from the female's response and subsequent mating. ^[ref] Here molecular localization helps evaluate whether a proposed route to reproductive isolation is genetically feasible. A preference could still have been shown to reduce heterospecific mating before its contributing regions were mapped.

The influence of one individual's animal behaviour on another's development further complicates a simple own-genotype explanation. Maternal provisioning in Onthophagus taurus was studied using a breeding design that separated direct and maternal genetic contributions to offspring traits. ^[ref] An indirect genetic effect occurs when another individual's inherited properties alter the focal individual's phenotype. Such effects change the environment in which selection acts and can alter expected evolutionary responses. Identifying only the offspring's molecular variants would miss an important part of the causal system. Conversely, an observed maternal effect is not automatically a maternal genetic effect; provisioning can vary for environmental reasons.

Transmission need not follow genetic parentage. In wild great tits, experimenters trained demonstrators to use alternative puzzle-box techniques and followed the spread of the introduced variants. Persistent local traditions and preferential adoption of the common local technique supported social learning and conformist transmission. ^[ref] The experiment established a route of behavioural inheritance without identifying a molecular difference between the techniques' users. It did not prove that learning ability has no inherited basis, or that the new tradition changed allele frequencies. Rather, it shows why an account of behavioural change needs to identify the relevant transmission process instead of assuming it.

Even a learned response with no demonstrated inherited differences can influence subsequent evolution. A new resource use can alter selection on digestion; a socially learned mating signal can alter assortative mating; habitat choice can change contact with other populations. The immediate causal effect can be tested by manipulating the action or its social transmission and measuring biological fitness or exchange. To claim ensuing genetic adaptation, one must additionally show inherited change. To explain exactly which variants can respond, how quickly, and with what correlated effects, information about genetic architecture becomes increasingly valuable.

The most productive programme therefore combines behavioural intervention, transmission studies, biological fitness measurement and molecular analysis when it addresses an unresolved mechanism. Candidate variants should be tested across natural genetic backgrounds and environments, preferably with targeted perturbation and rescue; historical claims also need comparative or temporal evidence. Exact molecular knowledge is neither sufficient nor universally necessary: the necessary evidence is the evidence that connects the particular behavioural difference, its transmission and its measured evolutionary consequence.

There is no general basis for assigning behaviour a smaller causal role in macroevolution. Its immediate effects are usually easier to measure within populations, but those effects can alter lineage formation and persistence. The comparison needs a definition of “contribution”: the proportion of evolutionary changes initiated by behaviour, their magnitude, their necessity, or the strength of available evidence would give different answers. Microevolution concerns change within biological populations; macroevolution concerns patterns at and above the level of species, including speciation, adaptive radiation and extinction. They are connected processes with different objects of measurement.

At the population scale, animal behaviour changes who encounters which resources, survives, mates and reproduces. Habitat preference changes exposure to ecological natural selection; mate choice changes reproductive success; dispersal changes the spatial distribution of alleles. Behaviour can therefore affect the evolution of other traits even if the behaviour itself does not evolve. Conversely, an environmentally induced change in behaviour is not automatically evolutionary change. Learning and phenotypic plasticity can occur within one lifetime, whereas an evolutionary claim requires persistence through inherited differences or an explicitly specified process of cultural inheritance.

A simple population genetics calculation makes the distinction concrete. Let $x_i$ be an individual's number of copies of a focal allele, $w_i$ its contribution to reproduction, and $p=\mathbb{E}(x)/2$. With unbiased Mendelian segregation and no other transmission change, the expected allele frequency after selection satisfies:Behaviour can change $w$ and thereby change the covariance, whether selection acts directly on a behavioural difference or on another trait whose consequences depend on behaviour. The equation does not imply that every correlation between a behaviour and reproductive success is genetic: common environment, age and condition can generate such correlations. A common garden experiment, a pedigree and measurements across environments help identify the inherited component.

Behaviour may expose a previously uncommon phenotype to selection. Foraging in a new habitat can make a different resource profitable, after which selection can refine feeding structures or behavioural response thresholds. Laboratory competitions between spadefoot tadpoles from populations differing in expression of a carnivore ecomorph found differences in resource capture and morphology consistent with adaptive refinement of a formerly plastic response. Expression without the inducing cue also supported genetic assimilation. ^[ref] This is a combination of behaviour and development, not proof that behaviour universally comes first. Genetic accommodation requires evolution of the response, rather than assuming that an acquired habit is copied directly into DNA.

The opposite effect is also possible. Through phenotypic buffering, movement or flexible foraging may keep individuals in suitable conditions and reduce selection on physiological or morphological differences. Thus flexibility can promote occupation of a new environment, delay genetic divergence, or prevent local extinction. The sign depends on whether behaviour changes exposure to selection or allows animals to avoid it. A short-term adaptive response is consequently not evidence for a universal acceleration of evolutionary change.

At the boundary between populations and species, the behavioural effects are particularly direct. Assortative mating, host choice and learned signals can reduce gene flow. In the Big Bird lineage of Darwin's finches, field observations and genomic ancestry documented an immigrant Geospiza conirostris breeding with a resident Geospiza fortis and the formation of an endogamous descendant lineage. Song and distinctive morphology contributed to mating within that lineage. ^[ref] The evidence links immigration, hybridization and mate recognition to rapid reproductive separation; it does not show that song alone caused isolation or establish the lineage's indefinite future persistence. Nevertheless, behaviour participates in an event conventionally classified as macroevolution.

Over longer periods, these mechanisms can influence the rates of lineage origination and disappearance. Behaviour that permits colonization can expose populations to ecological opportunity and eventual adaptive radiation. Specialization in host use or mating signals can strengthen reproductive barriers. Conversely, costly reproductive competition or inflexible resource use can increase vulnerability. In fossil ostracods, greater sexual dimorphism was associated with greater estimated extinction risk, using carapace differences as a proxy for male reproductive investment. ^[ref] This is unusually direct evidence about actual lineage loss, but an anatomical proxy does not preserve courtship or experimentally isolate sexual selection from every correlated attribute. It supports a possible connection rather than a general law that strong sexual selection causes extinction.

Evaluating a claim about relative importance also requires correcting unequal evidence. Contemporary behavioural experiments can manipulate cues and follow reproduction, whereas most ancient behaviour must be inferred from traces, anatomy or living relatives. Living-only comparative studies omit extinct lineages and can confuse a trait associated with diversification with a trait that evolved after diversification. Comparisons should allow for shared ancestry, environmental opportunity and incomplete sampling, and distinguish an increased speciation rate from a reduced extinction rate. Counting observed behavioural changes over years against anatomical changes over millions of years is not a meaningful common denominator.

Behaviour has demonstrable population-level effects and plausible, sometimes well-documented effects on speciation and extinction. The stronger conclusion is that its evolutionary influence is context-dependent and transmitted across scales, while the evidence becomes harder to identify over deep time. A qualified statement about evidential accessibility is defensible; a general ranking of causal importance is not.

I would combine quantitative-genetic cross-fostering with controlled transfer of maternal brood-chamber material in the dung beetle Onthophagus taurus. The hypothesis is that maternal genetic differences influence offspring development partly through the amount of food provided and partly through the composition of material transferred from the mother's body. These pathways must be separated: a larger brood provision, a beneficial microorganism and an inherited offspring allele can all increase offspring mass, but imply different mechanisms.

The system has three relevant precedents. A paternal half-sib study of Onthophagus taurus detected genetic variation in maternal brood provisioning and substantial maternal contributions to offspring phenotype. ^[ref] A separate study reared this species with sterile dung and detected maternal contributions and shared bacterial sequences in the brood chamber and offspring, supporting transmission through the provisioned environment. ^[ref] Experimental developmental benefits of pedestal-associated microbiota were demonstrated in Onthophagus gazella, including context dependence under environmental stress; those functional results should not be assumed automatically to hold in Onthophagus taurus. ^[ref]

The conceptual link is an indirect genetic effect: the provider's genotype affects the recipient's phenotype through a rearing environment. The transferred mediator would be maternal faecal or secreted material, including associated viable microorganisms and soluble products. This fits the socially transferred material framework because the mother contributes material processed in or produced by her body; collecting external dung alone would not establish such a transfer. ^[ref] The actual molecular mediator is an experimental question. Finding bacterial DNA or a protein in a brood chamber does not establish that it is alive, active, beneficial, or responsible for an indirect genetic effect.

First establish a common-garden breeding population with a recorded pedigree and genotype markers. Use a paternal half-sib design with several dams per sire and several adult daughter providers per dam, distributed among independently randomized rearing containers. Standardize donor age, mating history, developmental nutrition and adult diet, and block dung source, moisture and temperature. Because maternally inherited microbes can themselves produce family resemblance, rear replicate provider families with a standardized microbial inoculum in an additional common-rearing arm. Persistence of a family difference alone is insufficient evidence for host genetic causation.

For the initial cross-fostering experiment, allow each provider to construct several brood balls on weighed, standardized dung. Measure construction time, final dry-mass estimate from matched material, egg-chamber properties and maternal faecal pedestal quantity. Remove the original egg before hatching and replace it with an age-matched egg from a separately bred recipient family. Randomize recipient families across provider families, with reciprocal assignments and repeated providers; include handled own-egg and untouched controls. Record egg size and biological-mother identity so that egg provisioning and other pre-transfer maternal effects are not silently attributed to the foster provider.

One possible balanced allocation has 20 sire families, two dams per sire and two daughter providers per dam, giving 80 providers. Four independently prepared brood balls per provider yield 320 offspring-rearing units, allocated across multiple recipient families and collection blocks. This is an illustrative feasible structure, not a claim of adequate power. The effective replication for a sire-level genetic component is much smaller than 320; repeated balls estimate provider consistency and within-provider variation. A pilot-based simulation of the planned pedigree, variance components, mortality and missing samples should determine the final scale before outcomes are examined.

Measure larval growth trajectories, survival, development time and adult size; subsequently measure reproductive performance where feasible. Score provisioning behaviour blind to family and recipient treatment. Collect matched donor and provision samples for absolute microbial load, community composition and soluble chemical profiles. Include extraction blanks, reagent controls and bacterial-load standards, especially because low-biomass sequence samples are vulnerable to contamination. Shared 16S sequences provide limited discrimination among strains; culture or higher-resolution sequencing can test whether a donor-associated strain actually reaches and persists in a recipient.

For an offspring $i$ reared in a ball prepared by provider $j$, an animal model in quantitative genetics could use:Here $a_i^D$ is the direct genetic effect of the offspring, $a_j^I$ the additive provider contribution, $c_j$ a repeatable non-genetic provider effect, $b_{\mathrm{birth}(i)}$ a biological-mother effect, and $B_{ij}$ prespecified batch and rearing terms. Genetic effects use the pedigree or genomic relationship matrix; provider identity alone cannot identify $a_j^I$. If the pedigree contains informative relatives in both roles, direct–indirect genetic covariance can also be estimated. Otherwise that covariance is not identified by the first experiment and should not be assumed known. Crossed recipient and provider families distinguish a general provider effect from matching between a particular recipient genotype and provider background.

The second experiment tests the transferred pathway rather than merely adding a microbial measurement to that model. Construct uniform artificial brood balls with the same treated dung, food mass and chamber geometry, and transfer equal donor-equivalent pedestal material. Compare intact material, material separated into a cell-free soluble fraction, a preparation containing non-viable microbial cells, and a live-cell add-back or rescue preparation. Include material-free, medium and sham-processing controls, together with untreated natural balls as a benchmark. Test viability rather than inferring it from a sterilization label. All processed-dung groups must share the same processing, because sterilization can also change nutritional and chemical properties.

Within a subset, exchange washed live-cell fractions and soluble fractions independently between donor families. A crossed donor-cell-source by donor-soluble-source design asks whether offspring differences track microbial cells, soluble maternal products, or their interaction. Hold total food quantity and transfer volume constant. Include both a standardized live-cell dose, testing compositional differences, and a naturally varying dose arm, testing a possible maternal-genotype effect on delivered quantity. An equal-dose experiment alone could erase a real dose-mediated pathway. Test a candidate cultured community or identified molecule in a dose range anchored to measured natural transfers, rather than interpreting a large artificial supplement as evidence for normal function.

The predicted pattern is specific. If food mass explains the provider effect, equalizing mass should greatly reduce it. If viable maternal microbes mediate an additional effect, removing viability at equal food quantity should reduce the relevant donor-associated growth difference, and a compatible live add-back should restore it. An effect retained in a cell-free fraction instead motivates identifying soluble maternal or microbial products. A failed rescue can reflect processing damage or a missing microbial partner, rather than absence of any natural pathway. If all provider differences disappear after standardized rearing, a non-genetic maternal or microbial inheritance explanation becomes more plausible than the proposed host genetic effect.

Analyse the cross-fostering data first for total provider genetic contributions, then analyse the randomized material manipulations for mechanism. Simply adjusting for a measured mediator can block part of the effect or introduce bias; causal mediation needs assumptions and experimental support. Prespecify the primary outcomes, appropriate survival analysis, family and donor clustering, planned contrasts and treatment-by-recipient-background interactions. Offspring within a maternal pedigree and repeated chemical measurements are not independent genetic replicates. Measure provider costs, such as future fecundity and provisioning effort, because faster-growing offspring alone do not establish the evolutionary advantage of the provisioning behaviour.

The decisive result would connect provider genotype to a measured transfer difference and then experimentally connect that difference to recipient performance, independently of offspring genotype and food quantity. Negative outcomes remain informative: maternal influence without additive provider variance is a maternal effect without the demonstrated genetic component, while heritable provisioning effects without a material-mediated mechanism leave a genuine indirect genetic effect whose pathway is still unresolved.

A pedigree records parent–offspring relationships. A founder in a pedigree has no recorded parents; all other individuals have a genotype conditional on their parental genotypes. Inbreeding can produce loops in the undirected diagram while the directed ancestry remains acyclic.

Assign one genotype random variable to each person in a pedigree. Independent founders in a pedigree carry population genotype laws, and each other genotype has its conditional distribution given its two parents from Mendelian segregation. The ancestry Directed acyclic graph is a Bayesian network for the joint genotype law. Its moral graph contains each parent-child edge and each coparent edge. Each founder factor and each child-with-parents factor belongs to a clique, giving an undirected graphical representation. The graph is an independence representation; equality with the minimal conditional independence graph needs additional faithfulness assumptions, particularly because inheritance can produce zero probabilities.

For a known-phase contribution $L_1(\theta )\propto {\theta }^{r_1}(1-\theta {)}^{m_1-r_1}$ and an independent phase-averaged contribution $L_2(\theta )$, an interior maximum satisfiesThis follows by differentiating $\log L_1+\log L_2$ and multiplying by $\theta (1-\theta )L_2$. The constrained maximum-likelihood estimate on $[0,1/2]$ must also be compared with the endpoints. An interior root cannot be asserted without the observed pedigree counts; complete nonrecombinant data, for example, have their maximum at zero.

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