The \Lambda-Wright--Fisher process describes the type-frequency evolution of an infinite  population. We model frequency-dependent selection pressure with a general polynomial drift vanishing at the boundary. An appropriate decomposition of the drift allows us to construct a series of Moran-type models that converge under suitable conditions to the solution of the associated stochastic differential equation  . The genealogical structure inherent in the graphical representation of these finite population models can be seen in the large population limit as a generalisation of the ancestral selection graph of Krone and Neuhauser. We introduce an ancestral process that keeps track of the sampling distribution along the ancestral structures and that satisfies a duality relation with the type-frequency process. We refer to it as Bernstein coefficient process and to the relation as Bernstein duality. The latter is a generalisation of the classic moment duality. Many classic results in the restricted setting of a moment duality generalise into our framework. In particular, we derive criteria for the accessibility of the boundary and determine the time to absorption. It turns out that multiple ancestral processes are associated to the same forward dynamics. We characterise the set of optimal ancestral structures and provide a recipe to construct them from the drift. In particular, this allows us to recover well-known ancestral structures of the literature .
 \Lambda-Wright--Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate frequency-dependent selection. A decomposition of the drift allows us to approximate the solution of the stochastic differential equation by a sequence of Moran models . The genealogical structure underlying the Moran model leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundaries; this extends a recent result by Gonz\'alez Casanova and Span\`o. An intriguing feature of the construction is that the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones .
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We introduce an ancestral process that keeps track of the sampling distribution along the ancestral structures and that satisfies a duality relation with the type-frequency process. We refer to it as Bernstein coefficient process and to the relation as Bernstein duality. The latter is a generalisation of the classic moment duality. Many classic results in the restricted setting of a moment duality generalise into our framework. In particular, we derive
Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and
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boundary and determine the time to absorption. It turns out that multiple ancestral processes are associated to
boundaries; this extends a recent result by Gonz\'alez Casanova and Span\`o. An intriguing feature of the construction is that
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process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones