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The discrete space, continuous time birth-death model is a key process for describing phylogenies in the absence of coalescent approximations. Extensively used in macroevolution for analysing diversification, and in epidemiology for estimating viral dynamics, the birth-death process (BDP) is an important null model for inferring the parameters of reconstructed phylogenies. In this paper we show how optimal, point process (Snyder) filtering techniques can be used for parametric inference on BDPs. Specifically, we introduce the Bayesian Snyder filter (SF) to estimate birth and death rate parameters, given a reconstructed phylogeny. Our estimation procedure makes use of the equivalent Markov birth process description for a reconstructed birth-death phylogeny (Nee et al, 1994). We first analyse the popular constant rate BDP and show that our method gives results consistent with previous work. Among these results is an analytic solution to the special case of the Yule-Furry model. We also find an equivalence between the SF Poisson likelihood and two standard conditioned birth-death model likelihoods. We then generalise our estimation problem to BDPs with time varying rates and numerically solve the SF for two illustrative cases. Our results compare well with a recent Markov chain Monte Carlo method by Hohna et al (2016) and we numericaly show that both methods are solving the same likelihood functions. Lastly we apply the SF to a model selection problem on empirical data. We use the Australian Agamid dataset and predict the same relative model fit as that of the original maximum likelihood technique developed and used by Rabosky (2006) for this dataset. While several capable parametric and non-parametric birth-death estimators already exist, ours is the first to take the Nee et al approach, and directly computes the posterior distribution of the parameters. The SF makes no approximations, beyond those required for parameter space discretisation and numerical integration, and is mean square error optimal. It is deterministic, easily implementable and flexible. We think SFs present a promising alternative parametric BDP inference engine.

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