Likelihood Calculation Research Articles

A Likelihood Ratio Test for Hybridization Under the Multispecies Coalescent

Several methods have been developed to carry out a statistical test for hybridization at the species level, including the ABBA-BABA test and HyDe. Here, we propose a new method for detecting hybridization and quantifying the extent of hybridization. Our test computes the likelihood of a species tree that is possibly subject to hybridization using site pattern frequencies from genomic-scale datasets under the multispecies coalescent. To do this, we extend the calculation of the likelihood for site pattern frequency data for the 4-taxon symmetric and asymmetric species trees proposed in Chifman and Kubatko (2015) by incorporating an inheritance parameter, resulting in efficient computation of the likelihood under a scenario of hybridization. We use this likelihood computation to construct a likelihood ratio test that a given species is a hybrid of two parental species. Simulations demonstrate that our test is more powerful than existing tests of hybridization, including HyDe, and that it achieves the desired type I error rate. We apply the method to two empirical data sets, one for which hybridization is believed to have occurred and one for which previous methods have failed to detect hybridization.

On the Detection of Population Heterogeneity in Causation Between Two Variables: Finite Mixture Modeling of Data Collected from Twin Pairs.

Causal inference is inherently complex and relies on key assumptions that can be difficult to validate. One strong assumption is population homogeneity, which assumes that the causal direction remains consistent across individuals. However, there may be variation in causal directions across subpopulations, leading to potential heterogeneity. In psychiatry, for example, the co-occurrence of disorders such as depression and substance use disorder can arise from multiple sources, including shared genetic or environmental factors (common causes) or direct causal pathways between the disorders. A patient diagnosed with two disorders might have one recognized as primary and the other as secondary, suggesting the existence of different types of comorbidity. For example, in some individuals, depression might lead to substance use, while in others, substance use could lead to depression. We account for potential heterogeneity in causal direction by integrating the Direction of Causation (DoC) model for twin data with finite mixture modeling, which allows for the calculation of individual-level likelihoods for alternate causal directions. Through simulations, we demonstrate the effectiveness of using the Direction of Causation Twin Mixture (mixDoC) model to detect and model heterogeneity due to varying causal directions.

Bayesian Selection of Relaxed-clock Models: Distinguishing Between Independent and Autocorrelated Rates.

In Bayesian molecular-clock dating of species divergences, rate models are used to construct the prior on the molecular evolutionary rates for branches in the phylogeny, with independent and autocorrelated rate models being commonly used. The two classes of models, however, can result in markedly different divergence time estimates for the same dataset, and thus selecting the best rate model appears important for obtaining reliable in- ferences of divergence times. However, the properties of Bayesian rate model selection are not well understood, in particular when the number of sequence partitions analysed increases and when age calibrations (such as fossil calibrations) are misspecified. Further- more, Bayesian rate model selection is computationally expensive as it requires calculation of marginal likelihoods by MCMC sampling, and therefore methods that can speed up the model selection procedure without compromising its accuracy are desirable. In this study, we use a combination of computer simulations and real data analysis to investigate the sta- tistical behaviour of Bayesian rate model selection and we also explore approximations of the likelihood to improve computational efficiency in large phylogenomic datasets. Our simulations demonstrate that the posterior probability for the correct rate model converges to one as more molecular sequence partitions are analysed and when no calibrations are used, as expected due to asymptotic Bayesian model selection theory. Furthermore, we also show the model selection procedure is robust to slight misspecification of calibrations, and reliable inference of the correct rate model is possible in this case. However, we show that when calibrations are seriously misspecified, calculated model probabilities are com- pletely wrong and may converge to one for the wrong rate model. Finally, we demonstrate that approximating the phylogenetic likelihood under an arcsine branch-length transform can dramatically reduce the computational cost of rate model selection without compro- mising accuracy. We test the approximate procedure on two large phylogenies of primates (372 species) and flowering plants (644 species), replicating results obtained on smaller datasets using exact likelihood. Our findings and methodology can assist users in selecting the optimal rate model for estimating times and rates along the Tree of Life.

Efficient Large-Scale Nonstationary Spatial Covariance Function Estimation Using Convolutional Neural Networks

Spatial processes observed in various fields, such as climate and environmental science, often occur at large-scale and demonstrate spatial nonstationarity. However, fitting a Gaussian process with a nonstationary Matérn covariance is challenging, as it requires handling the complexity and computational demands associated with modeling the varying spatial dependencies over large and heterogeneous domains. Previous studies in the literature have tackled this challenge by employing spatial partitioning techniques to estimate the parameters that vary spatially in the covariance function. The selection of partitions is an important consideration, but it is often subjective and lacks a data-driven approach. To address this issue, in this study, we use the power of Convolutional Neural Networks (ConvNets) to derive subregions from the nonstationary data by employing a selection mechanism to identify subregions that exhibit similar behavior to stationary fields. We rely on the ExaGeoStat software for large-scale geospatial modeling to implement the nonstationary Matérn covariance for large scale exact computation of nonstationary Gaussian likelihood. We also assess the performance of the proposed method with synthetic and real datasets at large-scale. The results revealed enhanced accuracy in parameter estimations when relying on ConvNet-based partition compared to traditional user-defined approaches.

Cosmocnc: A fast, flexible, and accurate framework for galaxy cluster number count likelihood computation

We introduce cosmocnc, a new framework for computing the number count likelihood of galaxy cluster catalogues in a fast, flexible and accurate way. cosmocnc offers three types of likelihoods: an unbinned, a binned, and an extreme value likelihood. It also supports the addition of stacked cluster data, which is modelled consistently with the cluster catalogue. The unbinned likelihood, which is the main focus of the framework, can take an arbitrary number of mass observables as input and deal with several complexities in the data, such as variations in the properties of the cluster observable across the survey footprint, the possibility of different clusters having measurements for different combinations of mass observables, redshift measurement uncertainties, and the presence on unconfirmed detections in the catalogue. If there are more than one mass observables, the unbinned likelihood is computed with a novel approach, the backward convolutional approach. After introducing the framework in detail, we demonstrate its application with synthetic Simons-Observatory-like catalogues, finding excellent agreement between their properties and cosmocnc's predictions and obtaining constraints on cosmological and scaling relation parameters featuring negligible biases. A Python implementation of the cosmocnc framework is publicly available at https://github.com/inigozubeldia/cosmocnc.

FamLink2 – A comprehensive tool for likelihood computations in pedigrees analyses involving linked DNA markers accounting for genotype uncertainties

There is an increasing demand for software that can handle an arbitrary number of linked markers in forensic genetics; primarily with application to inference of relationships and direct matching but also in applications such as ancestry inference and mixture interpretation. With the emergence of sequencing technologies, denser sets of SNP markers are generated and analyzed. Additionally, sequence data of low quality and quantity DNA generate uncertainty about the underlying true genotype. We provide an efficient implementation of a general model for pedigree likelihood computations with genetic marker data using a three-layered approach. The top and first layer is the population model where allele frequencies and population substructure are accounted for. The second layer is the inheritance model which efficiently handles linked markers using an IBD model. The third and bottom layer is the observational level where we model the likelihood of the true genotype given underlying reads as well as parameters for errors. We exemplify the utility of our implementation as well as provide validation according to guidelines established by the ISFG using a combination of two published SNP panels. We demonstrate that computations are feasible for panels encompassing 10,000 markers and we argue that, due to the properties of the underlying algorithm, extending the number of markers will result in a linear increase in computation time. In addition we study the impact of parameters used in our model and suggest some guidelines pertaining to their values. The results demonstrate that a probabilistic model for low coverage sequence read data is needed instead of relying on an a threshold based genotype and applying our general model for inference of relationships on a real case can be superior, i.e. higher information content, to other methods relying on either fixed genotypes with low quality sequence data or simple pair wise relationship tests. In summary, the implementation, FamLink2 (freely available at https://famlink.se), can jointly handle genetic linkage, genotype uncertainty and population substructure for an arbitrary pedigree with data for any number of individuals. Whereas the current study will focus on calculations disregarding mutations, FamLink2 has the ability to model mutations for certain built-in pedigrees.

Beyond Time-Homogeneity for Continuous-Time Multistate Markov Models

–Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations—a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time- inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this article, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online.

Assessing superficial temporal artery-middle cerebral artery anastomosis patency using FLOW 800 hemodynamics.

The objective of this study was to investigate the use of indocyanine green videoangiography with FLOW 800 hemodynamic parameters intraoperatively during superficial temporal artery-middle cerebral artery (STA-MCA) bypass surgery to predict patency prior to anastomosis performance. A retrospective and exploratory data analysis was conducted using FLOW 800 software prior to anastomosis to assess four regions of interest (ROIs; proximal and distal recipients and adjacent and remote gyri) for four hemodynamic parameters (speed, delay, rise time, and time to peak). Medical records were used to classify patients into flow and no-flow groups based on immediate or perioperative anastomosis patency. Hemodynamic parameters were compared using univariate and multivariate analyses. Principal component analysis was used to identify high risk of no flow (HRnf) and low risk of no flow (LRnf) groups, correlated with prospective angiographic follow-ups. Machine learning models were fitted to predict patency using FLOW 800 features, and the a posteriori effect of complication risk of those features was computed. A total of 39 cases underwent STA-MCA bypass surgery with complete FLOW 800 data collection. Thirty-five cases demonstrated flow after anastomosis revascularization and were compared with 4 cases with no flow after revascularization. Proximal and distal recipient speeds were significantly different between the no-flow and flow groups (proximal: 238.3 ± 120.8 and 138.5 ± 93.6, respectively [p < 0.001]; distal: 241.0 ± 117.0 and 142.1 ± 103.8, respectively [p < 0.05]). Based on principal component analysis, the HRnf group (n = 10) was characterized by high-flow speed (> 75th percentile) in all ROIs, whereas the LRnf group (n = 10) had contrasting patterns. In prospective long-term follow-up, 6 of 9 cases in the HRnf group, including the original no-flow cases, had no or low flow, whereas 8 of 8 cases in the LRnf group maintained robust flow. Machine learning models predicted patency failure with a mean F1 score of 0.930 and consistently relied on proximal recipient speed as the most important feature. Computation of posterior likelihood showed a 95.29% chance of patients having long-term patency given a lower proximal speed. These results suggest that a high proximal speed measured in the recipient vessel prior to anastomosis can elevate the risk of perioperative no flow and long-term reduction of flow. With an increased dataset size, continued FLOW 800-based ROI metric analysis could be used to guide intraoperative anastomosis site selection prior to anastomosis and predict patency outcome.

Likelihood calculation in a multistate model of vocabulary evolution for linguistic dating

Abstract Computational methods of language dating make inferences about the divergence times of protolanguages by evaluating the patterns of inheritance in the vocabulary of modern languages, given the specification of a model of vocabulary evolution. We consider a model that describes vocabulary evolution as the replacement of traits by new traits from an infinite state space along a tree. This model has been introduced in previous literature but so far it has not been used in many applications. We give a general recursive algorithm for calculating likelihoods and argue that the model gives a more realistic representation of vocabulary evolution over time, compared to existing models like the Stochastic Dollo model. We also provide a case study demonstrating the model’s potential applications.

Fast mvSLOUCH: Multivariate Ornstein–Uhlenbeck‐based models of trait evolution on large phylogenies

Abstract The PCMBase R package is a powerful computational tool that enables efficient calculations of likelihoods for a wide range of phylogenetic Gaussian models. Taking advantage of it, we redesigned the R package mvSLOUCH. Here, we demonstrate how the new version of the package can be used to thoroughly examine the evolution and adaptation of traits in a large dataset of 1252 vascular plants through the use of multivariate Ornstein–Uhlenbeck processes. The results of our analysis demonstrate the ability of the modelling framework to distinguish between various alternative hypotheses regarding the evolution of functional traits in angiosperms.

Comparison of the ABC and ACMG systems for variant classification.

The ABC and ACMG variant classification systems were compared by asking mainly European clinical laboratories to classify variants in 10 challenging cases using both systems, and to state if the variant in question would be reported as a relevant result or not as a measure of clinical utility. In contrast to the ABC system, the ACMG system was not made to guide variant reporting but to determine the likelihood of pathogenicity. Nevertheless, this comparison is justified since the ACMG class determines variant reporting in many laboratories. Forty-three laboratories participated in the survey. In seven cases, the classification system used did not influence the reporting likelihood when variants labeled as "maybe report" after ACMG-based classification were included. In three cases of population frequent but disease-associated variants, there was a difference in favor of reporting after ABC classification. A possible reason is that ABC step C (standard variant comments) allows a variant to be reported in one clinical setting but not another, e.g., based on Bayesian-based likelihood calculation of clinical relevance. Finally, the selection of ACMG criteria was compared between 36 laboratories. When excluding criteria used by less than four laboratories (<10%), the average concordance rate was 46%. Taken together, ABC-based classification is more clear-cut than ACMG-based classification since molecular and clinical information is handled separately, and variant reporting can be adapted to the clinical question and phenotype. Furthermore, variants do not get a clinically inappropriate label, like pathogenic when not pathogenic in a clinical context, or variant of unknown significance when the significance is known.

Fractional gaussian noise: Spectral density and estimation methods

The fractional Brownian motion (fBm) process, governed by a fractional parameter , is a continuous‐time Gaussian process with its increment being the fractional Gaussian noise (fGn). This article first provides a computationally feasible expression for the spectral density of fGn. This expression enables us to assess the accuracy of a range of approximation methods, including the truncation method, Paxson's approximation, and the Taylor series expansion at the near‐zero frequency. Next, we conduct an extensive Monte Carlo study comparing the finite sample performance and computational cost of alternative estimation methods for under the fGn specification. These methods include two semi‐parametric methods (based on the Taylor series expansion), two versions of the Whittle method (utilising either the computationally feasible expression or Paxson's approximation of the spectral density), a time‐domain maximum likelihood (ML) method (employing a recursive approach for its likelihood calculation), and a change‐of‐frequency method. Special attention is paid to highly anti‐persistent processes with close to zero, which are of empirical relevance to financial volatility modelling. Considering the trade‐off between statistical and computational efficiency, we recommend using either the Whittle ML method based on Paxson's approximation or the time‐domain ML method. We model the log realized volatility dynamics of 40 financial assets in the US market from 2012 to 2019 with fBm. Although all estimation methods suggest rough volatility, the implied degree of roughness varies substantially with the estimation methods, highlighting the importance of understanding the finite sample performance of various estimation methods.

Single-cell multi-omics analysis identifies context-specific gene regulatory gates and mechanisms.

There is a growing interest in inferring context specific gene regulatory networks from single-cell RNA sequencing (scRNA-seq) data. This involves identifying the regulatory relationships between transcription factors (TFs) and genes in individual cells, and then characterizing these relationships at the level of specific cell types or cell states. In this study, we introduce scGATE (single-cell gene regulatory gate) as a novel computational tool for inferring TF-gene interaction networks and reconstructing Boolean logic gates involving regulatory TFs using scRNA-seq data. In contrast to current Boolean models, scGATE eliminates the need for individual formulations and likelihood calculations for each Boolean rule (e.g. AND, OR, XOR). By employing a Bayesian framework, scGATE infers the Boolean rule after fitting the model to the data, resulting in significant reductions in time-complexities for logic-based studies. We have applied assay for transposase-accessible chromatin with sequencing (scATAC-seq) data and TF DNA binding motifs to filter out non-relevant TFs in gene regulations. By integrating single-cell clustering with these external cues, scGATE is able to infer context specific networks. The performance of scGATE is evaluated using synthetic and real single-cell multi-omics data from mouse tissues and human blood, demonstrating its superiority over existing tools for reconstructing TF-gene networks. Additionally, scGATE provides a flexible framework for understanding the complex combinatorial and cooperative relationships among TFs regulating target genes by inferring Boolean logic gates among them.

Statistical inference for a two-parameter Rayleigh distribution under generalized progressive hybrid censoring scheme

The two-parameter Rayleigh distribution, as an extended distribution of the Rayleigh distribution, has been widely applied in reliability analysis. With the introduction of the location parameter, two-parameter Rayleigh distribution becomes more flexible in fitting real-data. In this paper, based on generalized progressively hybrid censored(GPHC) sample from the two-parameter Rayleigh distribution, Classical and Bayesian inferences are discussed. The Newton-Raphson(NR) and Expectation-Maximization(EM) algorithms are used to compute the maximum likelihood estimates(MLEs). As well as the asymptotic confidence interval(ACI) estimation is obtained through the asymptotic distribution theory of maximum likelihood estimation(MLE) and computation of the observed Fisher information matrix. In Bayesian frame, the estimation of unknown parameters and prediction of future observable are taken into consideration. Due to Bayesian estimation is challenging to compute precisely and for the purpose of comparison, the Lindley's approximation, the Tierney-Kadane(TK) approximation and Markov chain Monte Carlo(MCMC) method are employed to obtain Bayesian estimates. Then, combining the MCMC algorithm mentioned in the article, the one- and two- samples Bayesian prediction are obtained. Finally, the simulation results are provided and a real-life data set is used for illustration purpose.

Bilby in space: Bayesian inference for transient gravitational-wave signals observed with LISA

ABSTRACT The Laser Interferometer Space Antenna (LISA) is scheduled to launch in the mid-2030s, and is expected to observe gravitational-wave candidates from massive black hole binary mergers, extreme mass ratio inspirals, and more. Accurately inferring the source properties from the observed gravitational-wave signals is crucial to maximize the scientific return of the LISA mission. bilby, the user-friendly Bayesian inference library, is regularly used for performing gravitational-wave inference on data from existing ground-based gravitational-wave detectors. Given that Bayesian inference with LISA includes additional subtitles and complexities beyond its ground-based counterpart, in this work we introduce bilby_lisa , a python package that extends bilby to perform parameter estimation with LISA. We show that full nested sampling can be performed to accurately infer the properties of LISA sources from transient gravitational-wave signals in (a) zero noise and (b) idealized instrumental noise. By focusing on massive black hole binary mergers, we demonstrate that higher order multipole waveform models can be used to analyse a year’s worth of simulated LISA data, and discuss the computational cost and performance of full nested sampling compared with techniques for optimizing likelihood calculations, such as the heterodyned likelihood.

Improving likelihood calculation by mapping MITRE ATT&amp;CK to existing controls

Assessing the likelihood of threats is notoriously difficult for assessors. This paper will demonstrate a new, evidence-based approach to leverage existing security control assessments in determining likelihood of specific MITRE ATT&amp;CK adversarial tactics, techniques and procedures (TTPs). Through automation, we can develop organisation-specific threat profiles for known adversaries and assist in strategic security programme management.

Bayesian Phylogenetic Analysis on Multi-Core Compute Architectures: Implementation and Evaluation of BEAGLE in RevBayes With MPI.

Phylogenies are central to many research areas in biology and commonly estimated using likelihood-based methods. Unfortunately, any likelihood-based method, including Bayesian inference, can be restrictively slow for large datasets-with many taxa and/or many sites in the sequence alignment-or complex substitutions models. The primary limiting factor when using large datasets and/or complex models in probabilistic phylogenetic analyses is the likelihood calculation, which dominates the total computation time. To address this bottleneck, we incorporated the high-performance phylogenetic library BEAGLE into RevBayes, which enables multi-threading on multi-core CPUs and GPUs, as well as hardware specific vectorized instructions for faster likelihood calculations. Our new implementation of RevBayes+BEAGLE retains the flexibility and dynamic nature that users expect from vanilla RevBayes. In addition, we implemented native parallelization within RevBayes without an external library using the message passing interface (MPI); RevBayes+MPI. We evaluated our new implementation of RevBayes+BEAGLE using multi-threading on CPUs and 2 different powerful GPUs (NVidia Titan V and NVIDIA A100) against our native implementation of RevBayes+MPI. We found good improvements in speedup when multiple cores were used, with up to 20-fold speedup when using multiple CPU cores and over 90-fold speedup when using multiple GPU cores. The improvement depended on the data type used, DNA or amino acids, and the size of the alignment, but less on the size of the tree. We additionally investigated the cost of rescaling partial likelihoods to avoid numerical underflow and showed that unnecessarily frequent and inefficient rescaling can increase runtimes up to 4-fold. Finally, we presented and compared a new approach to store partial likelihoods on branches instead of nodes that can speed up computations up to 1.7 times but comes at twice the memory requirements.

Social, demographic and morbimortality characteristics of the cases treated for COVID-19 at the Ignacio Chávez National Institute of Cardiology. A descriptive cross-sectional study

The COVID-19 pandemic brought with it a large number of adverse consequences for public health with serious socioeconomic repercussions. In this study we characterize the social, demographic, morbidity and mortality conditions of individuals treated for COVID-19 in one of the SARS-CoV-2 reference hospitals in Mexico City. A descriptive cross-sectional study was carried out in 259 patients discharged from the Instituto Nacional de Cardiología Ignacio Chávez, between April 11, 2020 and March 14, 2021. A multivariate logistic regression model was used to identify the association between sociodemographic and clinical variables. An optimization was performed using maximum likelihood calculations to choose the best model compatible with the data. The maximum likelihood model was evaluated using ROC curves, goodnessof-fit estimators, and multicollinearity analysis. Statistically significant patterns of comorbidities were inferred by evaluating a hypergeometric test over the frequencies of co-occurrence of pairs of conditions. A network analysis was implemented to determine connectivity patterns based on degree centrality, between comorbidities and outcome variables. The main social disadvantages of the studied population are related to the lack of social security (96.5%) and the lag in housing conditions (81%). Variables associated with the probability of survival were being younger (p < 0.0001), having more durable material goods (p = 0.0034) and avoiding: pneumonia (p = 0.0072), septic shock (p < 0.0001) and acute respiratory failure (p < 0.0001); (AUROC: 91.5%). The comorbidity network for survival cases has a high degree of connectivity between conditions such as cardiac arrhythmias and essential arterial hypertension (Degree Centrality = 90 and 78, respectively). Given that among the factors associated with survival to COVID-19 there are clinical, sociodemographic and social determinants of health variables, in addition to age; It is imperative to consider the various factors that may affect or modify the health status of a population, especially when addressing emerging epidemic phenomena such as the current COVID-19 pandemic.

Making Likelihood Calculations Fast: Automatic Differentiation Applied to RooFit

With the growing datasets of current and next-generation HighEnergy and Nuclear Physics (HEP/NP) experiments, statistical analysis has become more computationally demanding. These increasing demands elicit improvements and modernizations in existing statistical analysis software. One way to address these issues is to improve parameter estimation performance and numeric stability using Automatic Differentiation (AD). AD’s computational efficiency and accuracy are superior to the preexisting numerical differentiation techniques, and it offers significant performance gains when calculating the derivatives of functions with a large number of inputs, making it particularly appealing for statistical models with many parameters. For such models, many HEP/NP experiments use RooFit, a toolkit for statistical modeling and fitting that is part of ROOT. In this paper, we report on the effort to support the AD of RooFit likelihood functions. Our approach is to extend RooFit with a tool that generates overheadfree C++ code for a full likelihood function built from RooFit functional models. Gradients are then generated using Clad, a compiler-based source-codetransformation AD tool, using this C++ code. We present our results from applying AD to the entire minimization pipeline and profile likelihood calculations of several RooFit and HistFactory models at the LHC-experiment scale. We show significant reductions in calculation time and memory usage for the minimization of such likelihood functions. We also elaborate on this approach’s current limitations and explain our plans for the future.

Position-specific evolution in transcription factor binding sites, and a fast likelihood calculation for the F81 model.

Transcription factor binding sites (TFBS), like other DNA sequence, evolve via mutation and selection relating to their function. Models of nucleotide evolution describe DNA evolution via single-nucleotide mutation. A stationary vector of such a model is the long-term distribution of nucleotides, unchanging under the model. Neutrally evolving sites may have uniform stationary vectors, but one expects that sites within a TFBS instead have stationary vectors reflective of the fitness of various nucleotides at those positions. We introduce 'position-specific stationary vectors' (PSSVs), the collection of stationary vectors at each site in a TFBS locus, analogous to the position weight matrix (PWM) commonly used to describe TFBS. We infer PSSVs for human TFs using two evolutionary models (Felsenstein 1981 and Hasegawa-Kishino-Yano 1985). We find that PSSVs reflect the nucleotide distribution from PWMs, but with reduced specificity. We infer ancestral nucleotide distributions at individual positions and calculate 'conditional PSSVs' conditioned on specific choices of majority ancestral nucleotide. We find that certain ancestral nucleotides exert a strong evolutionary pressure on neighbouring sequence while others have a negligible effect. Finally, we present a fast likelihood calculation for the F81 model on moderate-sized trees that makes this approach feasible for large-scale studies along these lines.