Introduction

In statistical phylogenetics, we are interested in learning the parameters of models in which evolutionary trees—phylogenies—play an important part. Such analyses have a surprisingly wide range of applications across the life sciences1,2,3. In fact, the research front in many disciplines is partly defined today by our ability to learn the parameters of realistic phylogenetic models.

Statistical problems are often analyzed using generic modeling and inference tools. Not so in phylogenetics, where empiricists are largely dependent on dedicated software developed by small teams of computational biologists3. Even though these software packages have become increasingly flexible in recent years, empiricists are still limited to a large extent by predefined model spaces and inference strategies. Venturing outside these boundaries typically requires the help of skilled programmers and inference experts.

If it were possible to specify arbitrary phylogenetic models in an easy and intuitive way, and then automatically learn the latent variables (the unknown parameters) in them, the full creativity of the research community could be unleashed, significantly accelerating progress. There are two major hurdles standing in the way of such a vision. First, we must find a formalism (a language) that can express phylogenetic models in all their complexity, while still being easy to learn for empiricists (the modeling language expressivity problem). Second, we need to be able to generate computationally efficient inference algorithms from such model descriptions, drawing from the full range of techniques available today (the automated inference problem).

In recent years, there has been significant progress toward solving the expressivity problem by adopting the framework of probabilistic graphical models (PGMs)4,5. PGMs can express many components of phylogenetic models in a structured way, so that efficient Markov chain Monte Carlo (MCMC) samplers—the current workhorse of Bayesian statistical phylogenetics—can be automatically generated for them5. Other, more novel inference strategies are also readily applied to PGM descriptions of phylogenetic model components, as exemplified by recent work using STAN6 or the new Blang framework7.

Unfortunately, PGMs cannot express the core of phylogenetic models: the stochastic processes that generate the tree, and anything dependent on those processes. This is because the evolutionary tree has variable topology, while a PGM expresses a fixed topology. The problem even occurs on a fixed tree, if we need to express the possible existence of unobserved side branches that have gone extinct or have not been sampled. There could be any number of those for a given observed tree, each corresponding to a separate PGM instance.

Similar problems occur when describing evolutionary processes occurring on the branches of the tree. Many of the standard models considered today for trait evolution, such as continuous-time discrete-state Markov chains, are associated with an infinite number of possible change histories on a given branch. It is not always possible to represent this as a single distribution with an analytical likelihood that integrates out all change histories. Thus, it is sometimes necessary to describe the model as an unbounded stochastic loop or recursion over potential PGMs (individual change histories).

PGM-based systems may address these shortcomings by providing model components that hide underlying complexity. For instance, a tree may be represented as a single stochastic variable in a PGM-based model description5. An important disadvantage of this approach is that it removes information about complex model components from the model description. This forces users to choose among predefined alternatives instead of enabling them to describe how these model components are structured. Furthermore, computers can no longer “understand” these components from the model description, making it impossible to automatically apply generic inference algorithms to them. Instead, special-purpose code has to be developed manually for each of the components. Finally, hiding a complex model component, such as a phylogenetic tree, also makes dependent variables unavailable for automated inference. In phylogenetics, for instance, a single stochastic tree node makes it impossible to describe branch-wise relations between the processes that generated the tree and other model components, such as the rate of evolution, the evolution of organism traits, or the dispersal of lineages across space.

Here, we show that the expressivity problem can be solved using universal probabilistic programming languages (PPLs). A “universal PPL” is an extension of a Turing-complete general-purpose language, which can express models with an unbounded number of random variables. This means that random variables are not fixed statically in the model (as they are in a finite PGM) but can be created dynamically during execution.

PPLs have a long history in computer science8, but until recently they have been largely of academic interest because of the difficulty of generating efficient inference machinery from model descriptions using such expressive languages. This is now changing rapidly thanks to improved methods of automated inference for PPLs9,10,11,12,13,14, and the increased interest in more flexible approaches to statistical modeling and analysis. Current PPL inference algorithms provide state-of-the-art performance for many models but they are still quite inefficient for others. Improving PPL algorithms so that they can compete with manually engineered solutions for more problem domains is currently a very active research area.

To demonstrate the potential of PPLs in statistical phylogenetics, we tackle a tough problem domain: models that accommodate variation across lineages in diversification rate. These include the recent cladogenetic diversification rate shift (ClaDS) (ClaDS0–ClaDS2)15, lineage-specific birth–death-shift (LSBDS)16, and Bayesian analysis of macroevolutionary mixtures (BAMM)17 models, attracting considerable interest among evolutionary biologists despite the difficulties in develo** good inference algorithms for some of them18.

Using WebPPL—an easy-to-learn PPL9—and Birch—a language with a more computationally efficient inference machinery14—we develop techniques that allow us to automatically generate efficient sequential Monte Carlo (SMC) algorithms from short descriptions of these models (~100 lines of code each). Although we found it convenient to work with WebPPL and Birch for this paper, we emphasize that similar work could have been done using other universal PPLs. Adopting the PPL approach allows us generate the first efficient SMC algorithms for these models, and the first asymptotically exact inference machinery for the full BAMM model. Among other benefits, SMC inference allows us to directly estimate the marginal likelihoods of the models, so that we can assess their performance in explaining empirical data using rigorous Bayesian model comparison. Taking advantage of this, we show that models with slowing diversification, constant turnover and many small shifts (all combined in ClaDS2) generally explain the data from 40 bird phylogenies better than alternative models. We end the paper by discussing a few problems, all seemingly tractable, which remain to be solved before PPLs can be used to address the full range of phylogenetic models. Solving them would facilitate the adoption of a wide range of novel inference strategies that have seen little or no use in phylogenetics before.

Results

Probabilistic programming

Consider one of the simplest of all diversification models, constant rate birth–death (CRBD), in which lineages arise at a rate λ and die out at a rate μ, giving rise to a phylogenetic tree τ. Assume that we want to infer the values of λ and μ given some phylogenetic tree τobs of extant (now living) species that we have observed (or inferred from other data). In a Bayesian analysis, we would associate λ and μ with prior distributions, and then learn their joint posterior probability distribution given the observed value of τ.

Let us examine a PGM description of this model, say in RevBayes5 (Algorithm 1). The first statement in the description associates an observed tree with the variable myTree. The priors on lambda and mu are then specified, and it is stated that the tree variable tau is drawn from a birth–death process with parameters lambda and mu and generating a tree with leaves matching the taxa in myTree. Finally, tau is associated with (clamped to) the observed value myTree.

Algorithm 1

PGM description of the CRBD model

1 myTree = readTrees("treefile.nex")

2

3 lambda ~ dnGamma(1, 1)

4 mu ~ dnGamma(1, 1)

5

6 tau ~ dnBirthDeath(lambda, mu, myTree.taxa)

7 tau.clamp(myTree)

There is a one-to-one correspondence between these statements and elements in the PGM graph describing the conditional dependencies between the random variables in the model (Fig. 1). Given that the conditional densities dnGamma and dnBirthDeath are known analytically, along with good samplers, it is now straightforward to automatically generate standard inference algorithms, such as MCMC, for this problem.

Fig. 1: A probabilistic graphical model describing constant rate birth–death (CRBD).
figure 1

The square boxes are fixed nodes (parameters of the gamma distributions) and the circles are random variables. The shaded variable (τ) is observed, and λ, μ are latent variables to be inferred.

Full size image

Unfortunately, a PGM cannot describe from first principles (elementary probability distributions) how the birth–death process produces a tree of extant species. The PGM has a fixed graph structure, while the probability of a surviving tree is an integral over many outcomes with varying topology. Specifically, the computation of dnBirthDeath requires integration over all possible ways in which the process could have generated side branches that eventually go extinct, each of these with a unique configuration of speciation and extinction events (Fig. 2). The integral must be computed by special-purpose code based on analytical or numerical solutions specific to the model. For the CRBD model, the integral is known analytically, but as soon as we start experimenting with more sophisticated diversification scenarios, as evolutionary biologists would want to do, computing the integral is likely to require dedicated numerical solvers, if it can be computed at all.

Fig. 2: Phylogenetic trees generated by a birth–death process.
figure 2

Two trees with extinct side branches (thin lines), each corresponding to the same observed phylogeny of extant species (thick lines). The trees illustrate just two examples of an infinite number of possible PGM expansions of the τ node in Fig. 1.

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Universal PPLs solve the expressivity problem by providing additional expressive power over PGMs. A PPL model description is essentially a simulation program (or generative model). Each time the program runs, it generates a different outcome. If it is executed an infinite number of times, we obtain a probability distribution over outcomes. Thus, a PPL provides “a programmatic model description”14.

A universal PPL provides two special constructs, one for drawing a random variable from a probability distribution, and one for conditioning a random variable on observed data. These special constructs are used by the PPL inference algorithms to manipulate executions of the program during inference. Many PPLs are embedded in existing programming languages, with these two special constructs added.

To use this approach, we need to write a PPL program so that the distribution over outcomes corresponds to the posterior probability distribution of interest. This is straightforward if we understand how to simulate from the model, and how to insert the constraints given by the observed data.

Assume, for instance, that we are interested in computing the probability of survival and extinction under CRBD for specific values of λ and μ, given that the process started at some time t in the past. We will pretend that we do not know the analytical solution to this problem; instead we will use a PPL to solve it. WebPPL9 is an easy-to-learn PPL based on JavaScript, and we will use it here for illustrating PPL concepts. WebPPL can be run in a web browser at http://webppl.org or installed locally (Supplementary Note Section 2.1). In WebPPL, the two special constructs mentioned above are: (1) the sample statement, which specifies the prior distributions from which random variables are drawn; and (2) the condition statement, conditioning a random variable on an observation. WebPPL provides a couple of alternatives to the condition statement, namely, the observe and factor statements. These are explained in Supplementary Note Section 3.3.

In WebPPL, we define a function goesExtinct, which takes the values of time, lambda, and mu corresponding to variables t, λ, and μ, respectively (Algorithm 2). It returns true if the process does not survive until the present (that is, goes extinct) and false otherwise (survives to the present).

Algorithm 2

Basic birth-death model simulation in WebPPL

1 var goesExtinct = function(time, lambda, mu) {

2 var waitingTime = sample(

3 Exponential({a: lambda + mu})

4 )

5

6 if (waitingTime > time) { return false }

7

8 var isSpeciation = sample(

9 Bernoulli({p: lambda / (lambda + mu)})

10 )

11

12 if (isSpeciation == false) { return true }

13

14 return goesExtinct(time - waitingTime, lambda, mu)

15 && goesExtinct(time - waitingTime, lambda, mu)

16 }

The function starts at some time > 0 in the past. The waitingTime until the next event is drawn from an exponential distribution with rate lambda + mu and compared with time. If waitingTime > time, the function returns false (the process survived). Otherwise, we flip a coin (the Bernoulli distribution) to determine whether the next event is a speciation or an extinction event. If it is a speciation, the process continues by calling the same function recursively for each of the daughter lineages with the updated time time - waitingTime. Otherwise the function returns true (the lineage went extinct).

If executed many times, the goesExtinct function defines a probability distribution on the outcome space {true, false} for specific values of t, λ, and μ. To turn this into a Bayesian inference problem, let us associate λ and μ with gamma priors, and then infer the posterior distribution of these parameters assuming that we have observed a group originating at time t = 10 and surviving to the present. To do this, we combine the prior specifications and the conditioning on survival to the present with the goesExtinct function into a program that defines the distribution of interest (Algorithm 3).

Algorithm 3

CRBD model description in WebPPL

1 var model = function() {

2 var lambda = sample(

3 Gamma({shape: 1, scale: 1})

4 )

5 var mu = sample(

6 Gamma({shape: 1, scale: 1})

7 )

8 var t = 10

9

10 condition(goesExtinct(t, lambda, mu) == false)

11

12 return [lambda, mu]

13 }

The goesExtinct function described above (Algorithm 2) uses unbounded stochastic recursion: the tree that we simulate in the program can in principle grow to infinite size. This effectively proves that the PPL defining this model, if it is to be used to simulate extinct side branches, must be a universal PPL. This, in turn, implies that a language that solves the expressivity problem in phylogenetics can also describe any phylogenetic model from which we can simulate using an algorithm. Adopting this approach thus allows a clean separation of model specification from inference. Of course, automated inference procedures now face the problem of executing complex universal PPL models on hardware with physical constraints, such as limited memory size. However, these challenges can typically be addressed using generic approaches that apply to arbitrary model descriptions, relieving both empiricists and algorithm developers from such concerns over technical implementation details.

Because of the popularity of PPLs in recent years, the term “probabilistic programming” is now often used to refer to the entire range of platforms, from universal PPLs to simple PGM frameworks. Unless we explicitly say otherwise, however, we will henceforth reserve “probabilistic programming” and “PPL” for platforms that implement universal modeling languages.

Inference in PPLs is typically supported by constructs that take a model description as input. Returning to the previous example, the joint posterior distribution is inferred by calling the built-in Infer function with the model, the desired inference algorithm, and the inference parameters as arguments (Algorithm 4).

Algorithm 4

Specifying inference strategy in WebPPL

1 Infer({model: model, method: ’SMC’, particles: 10000})

To develop this example into a probabilistic program equivalent to the RevBayes model discussed previously (Algorithm 1), we just need to describe the CRBD process along the observed tree, conditioning on all unobserved side branches going extinct (Supplementary Note Algorithms 2 and 3). The PPL specification of the CRBD inference problem is longer than the PGM specification because it does not use the analytical expression for the CRBD density. However, it exposes all the details of the diversification process, so it can be used as a template for exploring a wide variety of diversification models, while relying on the same inference machinery throughout. We will take advantage of this in the following.

Diversification models

The simplest model describing biological diversification is the Yule (pure birth) process19,20, in which lineages speciate at rate λ but never go extinct. For consistency, we will refer to it as constant rate birth (CRB). The CRBD model21 discussed in the examples above adds extinction to the process, at a per-lineage rate of μ.

An obvious extension of the CRBD model is to let the speciation and/or extinction rate vary over time instead of being constant22, referred to as the generalized birth–death process. Here, we will consider variation in birth rate over time, kee** turnover (μ/λ) constant, and we will refer to this as the time-dependent birth–death (TDBD) model, or the time-dependent birth (TDB) model when there is no extinction. Specifically, we will consider the function:

$$\lambda (t)={\lambda }_{0}{e}^{z({t}_{0}-t)},$$
(1)

where λ0 is the initial speciation rate at time t0, t is current time, and z determines the nature of the dependency. When z > 0, the birth rate grows exponentially and the number of lineages explodes. The case z < 0 is more interesting biologically; it corresponds to a niche-filling scenario. This is the idea that an increasing number of lineages leads to competition for resources and—all other things being equal—to a decrease in speciation rate. Other potential causes for slowing speciation rates over time have also been considered23.

The four basic diversification models—CRB, CRBD, TDB, and TDBD—are tightly linked (Fig. 3). When z = 0, TDBD collapses to CRBD, and TDB to CRB. Similarly, when μ = 0, CRBD becomes equivalent to CRB, and TDBD to TDB.

Fig. 3: Relations between the diversification models considered in the paper.
figure 3

Arrows and symbols mark the variable transformations needed to convert one diversification model into another.

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In recent years, there has been a spate of work on models that allow diversification rates to vary across lineages. Such models can accommodate diversification processes that gradually change over time. They can also explain sudden shifts in speciation or extinction rates, perhaps due to the origin of new traits or other factors that are specific to a lineage.

One of the first models of this kind to be proposed was BAMM17. The model is a lineage-specific, episodic TDBD model. A group starts out evolving under some TDBD process, with extinction (μ) rather than turnover (ϵ) being constant over time. A stochastic process running along the tree then changes the parameters of the TDBD process at specific points in time. Specifically, λ0, μ, and z are all redrawn from the priors at these switch points. In the original description, the switching process was defined in a statistically incoherent way; here, we assume that the switches occur according to a Poisson process with rate η, following a previous analysis of the model18.

The BAMM model has been implemented in dedicated software using a combination of MCMC sampling and other numerical approximation methods17,24. The implementation has been criticized because it can result in severely biased inference18. To date, it has not been possible to provide asymptotically exact inference machinery for BAMM.

In a recent contribution, a simplified version of BAMM was introduced: the LSBDS model16. LSBDS is an episodic CRBD model, that is, it is equivalent to BAMM when z = 0. Inference machinery for the LSBDS model has been implemented in RevBayes5 based on numerical integration over discretized prior distributions for λ and μ, combined with MCMC. The computational complexity of this solution depends strongly on the number of discrete categories used. If k categories are used for both λ and μ, computational complexity is multiplied by a factor k2. Therefore, it is tempting to simplify the model. We note that, in the empirical LSBDS examples given so far, μ is kept constant and only λ is allowed to change at switch points16. When z = 0, BAMM collapses to LSBDS, and when η → 0 it collapses to TDBD (Fig. 3). When η → 0, LSBDS collapses to CRBD.

A different perspective is represented by the ClaDS models15. They map diversification rate changes to speciation events, assuming that diversification rates change in small steps over the entire tree. After speciation, each descendant lineage inherits its initial speciation rate λi from the ending speciation rate λa of its ancestor through a mechanism that includes both a deterministic long-term trend and a stochastic effect. Specifically:

$$\mathrm{log}\,{\lambda }_{i} \sim {\mathcal{N}}\left(\mathrm{log}\,(\alpha {\lambda }_{a}),{\sigma }^{2}\right).$$
(2)

The α parameter determines the long-term trend, and its effects are similar to the z parameter of TDBD and BAMM. When α < 1, that is, \(\mathrm{log}\,\alpha\, <\, 0\), the speciation rate of a lineage tends to decrease over time. The standard deviation σ determines the noise component. The larger the value, the more stochastic fluctuation there will be in speciation rates.

The original ClaDS paper15 focuses on the rate multiplier \(m=\alpha \times \exp ({\sigma }^{2}/2)\) rather than on α, but we prefer the α parameterization mainly because it allows us to specify a conjugate prior that makes SMC inference more efficient (Supplementary Note Section 3). As pointed out elsewhere25, the dynamics of the ClaDS models is complex and differs considerably from superficially similar models, such as the BAMM, TDBD, and TDB models (for further discussion of this point, see Supplementary Note Section 3).

There are three different versions of ClaDS, characterized by how they model μ. In ClaDS0, there is no extinction, that is, μ = 0. In ClaDS1, there is a constant extinction rate μ throughout the tree. Finally, in ClaDS2, it is the turnover rate ϵ = μ/λ that is kept constant over the tree. All ClaDS models collapse to CRB or CRBD models when α = 1 and σ → 0 (Fig. 3). The ClaDS models were initially implemented in the R package RPANDA26, using a combination of advanced numerical solvers and MCMC simulation15. A new implementation of ClaDS2 in Julia instead relies on data augmentation25.

In contrast to previous work, where these models are implemented independently in complex software packages, we used PPL model descriptions (~100 lines of code each) to generate efficient and asymptotically correct inference machinery for “all” diversification models described above. The machinery we generate relies on SMC algorithms which, unlike classical MCMC, can also estimate the marginal likelihood (the normalization constant of Bayes theorem).

Estimating the marginal likelihood of a probability distribution that is only known up to a constant of proportionality is a hard problem in general. However, if we know how to sample from a similar distribution, classical importance sampling can provide a good estimate. The SMC algorithm is based on consecutive importance sampling from a series of probability distributions that change slowly toward the posterior distribution of interest. Thus, by piecing together the normalization constant estimates obtained in each of these steps, a good estimate of the marginal likelihood of the model is obtained essentially as a byproduct in the SMC algorithm27,28. Such series of similar probability distributions are not available naturally in the MCMC algorithm, but have to be constructed in more involved, computationally complex procedures, such as thermodynamic integration29,30, annealed importance sampling31, or step**-stone sampling

Methods

PPL software and model scripts

All PPL analyses described here used WebPPL version 0.9.15, Node version 12.13.19 and the most recent development version of Birch (as of June 12, 2020)14. We implemented all models (CRB, CRBD, TDB, TDBD, ClaDS0–ClaDS2, LSBDS, and BAMM) as explicit simulation scripts that follow the structure of the CRBD example discussed in the main text (Supplementary Note Section 5). We also implemented compact simulations for the four simplest models (CRB, CRBD, TDB, and TDBD) using the analytical equations for specific values of λ, μ, and z to compute the probability of the observed trees.

In the PPL model descriptions, we account for incomplete sampling of the tips in the phylogeny based on the ρ sampling model63. That is, each tip is assumed to be sampled with a probability ρ, which is specified a priori. To simplify the presentation in this paper, we usually assume ρ = 1. However, the model scripts we developed support ρ < 1, and in the empirical analyses we set ρ for each bird tree to the proportion of the known species included in the tree.

We standardized prior distributions across models to facilitate model comparisons (Supplementary Note Section 4, Supplementary Note Fig. 2). To simplify the scripts, we simulated outcomes on ordered but unlabeled trees, and reweighted the particles so that the generated density was correct for labeled and unordered trees (Supplementary Note Section 3.2). We also developed an efficient simulation procedure to correct for survivorship bias, that is, the fact that we can only observe trees that survive until the present (Supplementary Note Section 5.3).

Inference strategies

To make SMC algorithms more efficient on diversification model scripts, we applied three new PPL inference techniques: alignment, delayed sampling, and the alive particle filter. “Alignment”36,64 refers to the synchronization of resampling points across simulations (particles) in the SMC algorithm. The SMC algorithms previously used for PPLs automatically resample particles when they reach observe or condition statements. Diversification simulation scripts will have different numbers and placements of hidden speciation events on the surviving tree (Fig. 2), each associated with a condition statement in a naive script. Therefore, when particles are compared at resampling points, some may have processed a much larger part of the observed tree than others. Intuitively, one would expect the algorithm to perform better if the resampling points were aligned, such that the particles have processed the same portion of the tree when they are compared. This is indeed the case; alignment is particularly important for efficient inference on large trees (Supplementary Note Fig. 3). Alignment at code branching points (corresponding to observed speciation events in the diversification model scripts) can be generated automatically through static analysis of model scripts36. Here, we manually aligned the scripts by replacing the statements that normally trigger resampling with code that accumulate probabilities when they did not occur at the desired locations in the simulation (Supplementary Note Section 6.1).

“Delayed sampling”13 is a technique that uses conjugacy to avoid sampling parameter values. For instance, the gamma distribution we used for λ and μ is a conjugate prior to the Poisson distribution, describing the number of births or deaths expected to occur in a given time period. This means that we can marginalize out the rate, and simulate the number of events directly from its marginal (gamma-Poisson) distribution, without having to first draw a specific value of λ or μ. In this way, a single particle can cover a portion of parameter space, rather than just single values of λ and μ. Delayed sampling is only available in Birch; we extended it to cover all conjugacy relations relevant for the diversification models examined here.

The “alive particle filter”37 is a technique for improving SMC algorithms when some particles can “die” because their likelihood becomes 0. This happens when SMC is applied to diversification models because simulations that generate hidden side branches surviving to the present need to be discarded. The alive particle filter is a generic improvement on SMC, and it collapses to standard SMC with negligible overhead when no particles die. This improved version of SMC, partly inspired by our work on state-dependent speciation-extinction models37, is only available in Birch.

Verification

To verify that the model scripts and the automatically generated inference algorithms are correct, we performed a series of tests focusing on the normalization constant (Supplementary Note Section 7). First, we checked that the model scripts for simple models (CRB(D) and TDB(D)) generated normalization constant estimates that were consistent with analytically computed likelihoods for specific model parameter values (Supplementary Note Fig. 4). Second, we used the fact that all advanced diversification models (ClaDS0–2, LSBDS, and BAMM) collapse to the CRBD model under specific conditions, and verified that we obtained the correct likelihoods for a range of parameter values (Supplementary Note Fig. 6). Third, we verified for the advanced models that the independently implemented model scripts and the inference algorithms generated for them by WebPPL and Birch, respectively, estimated the same normalization constant for a range of model parameter values (Supplementary Note Fig. 7). Fourth, we checked that our normalization constant estimates were consistent with the RPANDA package15,26 for ClaDS0–ClaDS2, and with RevBayes for LSBDS5,16. For these tests, we had to develop specialized PPL scripts emulating the likelihood computations of RPANDA and RevBayes. The normalization constant estimates matched for LSBDS (Supplementary Note Fig. 9) and for ClaDS0 (Supplementary Note Fig. 8); for ClaDS1 and ClaDS2, they matched for low values of λ and μ (or ϵ) but not for larger values (Supplementary Note Fig. 8). Our best-effort interpretation at this point is that the PPL estimates for ClaDS1 and ClaDS2 are more accurate than those obtained from RPANDA for these values (Supplementary Note Section 7.4). Finally, as there is no independent software that computes BAMM likelihoods correctly yet, we checked that our BAMM scripts gave the same normalization constant estimates as LSBDS under settings where the former model collapses to the latter (Supplementary Note Fig. 10).

Data

We applied our PPL scripts to 40 bird clades derived from a previous analysis of divergence times and relationships among all bird species65. The selected clades are those with more than 50 species (range 54–316) after outgroups had been excluded (Supplementary Note Table 6). We followed the previous ClaDS2 analysis of these clades15 in converting the time scale of the source trees to absolute time units. The clade ages range from 12.5 to 66.6 Ma.

Bayesian inference

Based on JavaScript, WebPPL is comparatively slow, making it less useful for high-precision computation of normalization constants or estimation of posterior probability distributions using many particles. WebPPL is also less efficient than Birch because it does not yet support delayed sampling and the alive particle filter. Delayed sampling, in particular, substantially improves the quality of the posterior estimates obtained with a given number of particles. Therefore, we focused on Birch in computing normalization constants and posterior estimates for the bird clades.

For each tree, we ran the programs implementing the ClaDS, BAMM, and LSBDS models using SMC with delayed sampling and the alive particle filter as the inference method. We ran each program 500 times and collected the estimates of \(\mathrm{log}\,Z\) from each run together with the information needed to estimate the posterior distributions. The quality of the normalization constant estimates (on the log scale) from these 500 runs was estimated using the standard deviation, as well as the relative effective sample size and the conditional acceptance rate (Supplementary Note Section 9). We initially set the number of SMC particles to 5000, which was sufficient to obtain high-quality estimates for all models except BAMM (Supplementary Note Table 7). We increased the number of particles to 20000 for BAMM to obtain estimates of acceptable quality for this model.

For CRB, CRBD, TDB, and TDBD, we exploited the closed form for the likelihood in the programs. We used importance sampling with 10,000 particles as the inference method, and ran each program 500 times. This was sufficient to obtain estimates of very high accuracy for all models (Supplementary Note Table 7). The computational resources we used to obtain the results are specified in Supplementary Note Table 8.

Visualization

Visualizations were prepared with Matplotlib66. We used the collected data from all runs to draw violin plots for \(\mathrm{log}\,\widehat{Z}\) as well as the posterior distributions for λ, μ (for all models), z (for TDB, TDBD, and BAMM), \(\mathrm{log}\,\alpha\) and σ2 (for the ClaDS models), and η (for LSBDS and BAMM). By virtue of delayed sampling, the posterior distributions for λ and μ for all ClaDS models, as well as for BAMM and LSBDS, were calculated as mixtures of gamma distributions, the posterior distribution for \(\mathrm{log}\,\alpha\) and σ2 for all ClaDS models as mixtures of normal inverse gamma and inverse gamma distributions, and the posterior distribution for η for BAMM and LSBDS as a mixture of gamma distributions. For the remaining model parameters, we used the kernel density estimation method. Exact plot settings and plot data are provided in the code repository accompanying the paper.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.