Bayesian Inference for Retrospective Population Genetics Models Using Markov Chain Monte Carlo Methods

Genetics, the science of heredity and variation in living organisms, has a central role in medicine, in breeding crops and livestock, and in studying fundamental topics of biological sciences such as evolution and cell functioning. Currently the field of genetics is under a rapid development because of the recent advances in technologies by which molecular data can be obtained from living organisms. In order that most information from such data can be extracted, the analyses need to be carried out using statistical models that are tailored to take account of the particular genetic processes. In this thesis we formulate and analyze Bayesian models for genetic marker data of contemporary individuals. The major focus is on the modeling of the unobserved recent ancestry of the sampled individuals (say, for tens of generations or so), which is carried out by using explicit probabilistic reconstructions of the pedigree structures accompanied by the gene flows at the marker loci. For such a recent history, the recombination process is the major genetic force that shapes the genomes of the individuals, and it is included in the model by assuming that the recombination fractions between the adjacent markers are known. The posterior distribution of the unobserved history of the individuals is studied conditionally on the observed marker data by using a Markov chain Monte Carlo algorithm (MCMC). The example analyses consider estimation of the population structure, relatedness structure (both at the level of whole genomes as well as at each marker separately), and haplotype configurations. For situations where the pedigree structure is partially known, an algorithm to create an initial state for the MCMC algorithm is given. Furthermore, the thesis includes an extension of the model for the recent genetic history to situations where also a quantitative phenotype has been measured from the contemporary individuals. In that case the goal is to identify positions on the genome that affect the observed phenotypic values. This task is carried out within the Bayesian framework, where the number and the relative effects of the quantitative trait loci are treated as random variables whose posterior distribution is studied conditionally on the observed genetic and phenotypic data. In addition, the thesis contains an extension of a widely-used haplotyping method, the PHASE algorithm, to settings where genetic material from several individuals has been pooled together, and the allele frequencies of each pool are determined in a single genotyping.

Inference on Cointegration in Vector Autoregressive Models (summary section only)

In the thesis we consider inference for cointegration in vector autoregressive (VAR) models. The thesis consists of an introduction and four papers. The first paper proposes a new test for cointegration in VAR models that is directly based on the eigenvalues of the least squares (LS) estimate of the autoregressive matrix. In the second paper we compare a small sample correction for the likelihood ratio (LR) test of cointegrating rank and the bootstrap. The simulation experiments show that the bootstrap works very well in practice and dominates the correction factor. The tests are applied to international stock prices data, and the .nite sample performance of the tests are investigated by simulating the data. The third paper studies the demand for money in Sweden 1970—2000 using the I(2) model. In the fourth paper we re-examine the evidence of cointegration between international stock prices. The paper shows that some of the previous empirical results can be explained by the small-sample bias and size distortion of Johansen’s LR tests for cointegration. In all papers we work with two data sets. The first data set is a Swedish money demand data set with observations on the money stock, the consumer price index, gross domestic product (GDP), the short-term interest rate and the long-term interest rate. The data are quarterly and the sample period is 1970(1)—2000(1). The second data set consists of month-end stock market index observations for Finland, France, Germany, Sweden, the United Kingdom and the United States from 1980(1) to 1997(2). Both data sets are typical of the sample sizes encountered in economic data, and the applications illustrate the usefulness of the models and tests discussed in the thesis.

On probability-based inference under data missing by design

Whether a statistician wants to complement a probability model for observed data with a prior distribution and carry out fully probabilistic inference, or base the inference only on the likelihood function, may be a fundamental question in theory, but in practice it may well be of less importance if the likelihood contains much more information than the prior. Maximum likelihood inference can be justified as a Gaussian approximation at the posterior mode, using flat priors. However, in situations where parametric assumptions in standard statistical models would be too rigid, more flexible model formulation, combined with fully probabilistic inference, can be achieved using hierarchical Bayesian parametrization. This work includes five articles, all of which apply probability modeling under various problems involving incomplete observation. Three of the papers apply maximum likelihood estimation and two of them hierarchical Bayesian modeling. Because maximum likelihood may be presented as a special case of Bayesian inference, but not the other way round, in the introductory part of this work we present a framework for probability-based inference using only Bayesian concepts. We also re-derive some results presented in the original articles using the toolbox equipped herein, to show that they are also justifiable under this more general framework. Here the assumption of exchangeability and de Finetti's representation theorem are applied repeatedly for justifying the use of standard parametric probability models with conditionally independent likelihood contributions. It is argued that this same reasoning can be applied also under sampling from a finite population. The main emphasis here is in probability-based inference under incomplete observation due to study design. This is illustrated using a generic two-phase cohort sampling design as an example. The alternative approaches presented for analysis of such a design are full likelihood, which utilizes all observed information, and conditional likelihood, which is restricted to a completely observed set, conditioning on the rule that generated that set. Conditional likelihood inference is also applied for a joint analysis of prevalence and incidence data, a situation subject to both left censoring and left truncation. Other topics covered are model uncertainty and causal inference using posterior predictive distributions. We formulate a non-parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure, and apply the model in the context of optimal sequential treatment regimes, demonstrating that inference based on posterior predictive distributions is feasible also in this case.

Statistical analysis of the associations of hereditary factors with the risk of Type 1 diabetes and Diabetic Nephropathy based on the familial data collected through ascertaiment from population-based registers

In genetic epidemiology, population-based disease registries are commonly used to collect genotype or other risk factor information concerning affected subjects and their relatives. This work presents two new approaches for the statistical inference of ascertained data: a conditional and full likelihood approaches for the disease with variable age at onset phenotype using familial data obtained from population-based registry of incident cases. The aim is to obtain statistically reliable estimates of the general population parameters. The statistical analysis of familial data with variable age at onset becomes more complicated when some of the study subjects are non-susceptible, that is to say these subjects never get the disease. A statistical model for a variable age at onset with long-term survivors is proposed for studies of familial aggregation, using latent variable approach, as well as for prospective studies of genetic association studies with candidate genes. In addition, we explore the possibility of a genetic explanation of the observed increase in the incidence of Type 1 diabetes (T1D) in Finland in recent decades and the hypothesis of non-Mendelian transmission of T1D associated genes. Both classical and Bayesian statistical inference were used in the modelling and estimation. Despite the fact that this work contains five studies with different statistical models, they all concern data obtained from nationwide registries of T1D and genetics of T1D. In the analyses of T1D data, non-Mendelian transmission of T1D susceptibility alleles was not observed. In addition, non-Mendelian transmission of T1D susceptibility genes did not make a plausible explanation for the increase in T1D incidence in Finland. Instead, the Human Leucocyte Antigen associations with T1D were confirmed in the population-based analysis, which combines T1D registry information, reference sample of healthy subjects and birth cohort information of the Finnish population. Finally, a substantial familial variation in the susceptibility of T1D nephropathy was observed. The presented studies show the benefits of sophisticated statistical modelling to explore risk factors for complex diseases.

Bayesian fisheries stock assessment: integrating and updating knowledge

In this thesis the use of the Bayesian approach to statistical inference in fisheries stock assessment is studied. The work was conducted in collaboration of the Finnish Game and Fisheries Research Institute by using the problem of monitoring and prediction of the juvenile salmon population in the River Tornionjoki as an example application. The River Tornionjoki is the largest salmon river flowing into the Baltic Sea. This thesis tackles the issues of model formulation and model checking as well as computational problems related to Bayesian modelling in the context of fisheries stock assessment. Each article of the thesis provides a novel method either for extracting information from data obtained via a particular type of sampling system or for integrating the information about the fish stock from multiple sources in terms of a population dynamics model. Mark-recapture and removal sampling schemes and a random catch sampling method are covered for the estimation of the population size. In addition, a method for estimating the stock composition of a salmon catch based on DNA samples is also presented. For most of the articles, Markov chain Monte Carlo (MCMC) simulation has been used as a tool to approximate the posterior distribution. Problems arising from the sampling method are also briefly discussed and potential solutions for these problems are proposed. Special emphasis in the discussion is given to the philosophical foundation of the Bayesian approach in the context of fisheries stock assessment. It is argued that the role of subjective prior knowledge needed in practically all parts of a Bayesian model should be recognized and consequently fully utilised in the process of model formulation.

Discovering hidden structures in molecular data using a Bayesian partition model approach

Advancements in the analysis techniques have led to a rapid accumulation of biological data in databases. Such data often are in the form of sequences of observations, examples including DNA sequences and amino acid sequences of proteins. The scale and quality of the data give promises of answering various biologically relevant questions in more detail than what has been possible before. For example, one may wish to identify areas in an amino acid sequence, which are important for the function of the corresponding protein, or investigate how characteristics on the level of DNA sequence affect the adaptation of a bacterial species to its environment. Many of the interesting questions are intimately associated with the understanding of the evolutionary relationships among the items under consideration. The aim of this work is to develop novel statistical models and computational techniques to meet with the challenge of deriving meaning from the increasing amounts of data. Our main concern is on modeling the evolutionary relationships based on the observed molecular data. We operate within a Bayesian statistical framework, which allows a probabilistic quantification of the uncertainties related to a particular solution. As the basis of our modeling approach we utilize a partition model, which is used to describe the structure of data by appropriately dividing the data items into clusters of related items. Generalizations and modifications of the partition model are developed and applied to various problems. Large-scale data sets provide also a computational challenge. The models used to describe the data must be realistic enough to capture the essential features of the current modeling task but, at the same time, simple enough to make it possible to carry out the inference in practice. The partition model fulfills these two requirements. The problem-specific features can be taken into account by modifying the prior probability distributions of the model parameters. The computational efficiency stems from the ability to integrate out the parameters of the partition model analytically, which enables the use of efficient stochastic search algorithms.

Bayesian statistical analysis of bacterial diversity

Bacteria play an important role in many ecological systems. The molecular characterization of bacteria using either cultivation-dependent or cultivation-independent methods reveals the large scale of bacterial diversity in natural communities, and the vastness of subpopulations within a species or genus. Understanding how bacterial diversity varies across different environments and also within populations should provide insights into many important questions of bacterial evolution and population dynamics. This thesis presents novel statistical methods for analyzing bacterial diversity using widely employed molecular fingerprinting techniques. The first objective of this thesis was to develop Bayesian clustering models to identify bacterial population structures. Bacterial isolates were identified using multilous sequence typing (MLST), and Bayesian clustering models were used to explore the evolutionary relationships among isolates. Our method involves the inference of genetic population structures via an unsupervised clustering framework where the dependence between loci is represented using graphical models. The population dynamics that generate such a population stratification were investigated using a stochastic model, in which homologous recombination between subpopulations can be quantified within a gene flow network. The second part of the thesis focuses on cluster analysis of community compositional data produced by two different cultivation-independent analyses: terminal restriction fragment length polymorphism (T-RFLP) analysis, and fatty acid methyl ester (FAME) analysis. The cluster analysis aims to group bacterial communities that are similar in composition, which is an important step for understanding the overall influences of environmental and ecological perturbations on bacterial diversity. A common feature of T-RFLP and FAME data is zero-inflation, which indicates that the observation of a zero value is much more frequent than would be expected, for example, from a Poisson distribution in the discrete case, or a Gaussian distribution in the continuous case. We provided two strategies for modeling zero-inflation in the clustering framework, which were validated by both synthetic and empirical complex data sets. We show in the thesis that our model that takes into account dependencies between loci in MLST data can produce better clustering results than those methods which assume independent loci. Furthermore, computer algorithms that are efficient in analyzing large scale data were adopted for meeting the increasing computational need. Our method that detects homologous recombination in subpopulations may provide a theoretical criterion for defining bacterial species. The clustering of bacterial community data include T-RFLP and FAME provides an initial effort for discovering the evolutionary dynamics that structure and maintain bacterial diversity in the natural environment.

Computational Techniques for Haplotype Inference and for Local Alignment Significance

This thesis which consists of an introduction and four peer-reviewed original publications studies the problems of haplotype inference (haplotyping) and local alignment significance. The problems studied here belong to the broad area of bioinformatics and computational biology. The presented solutions are computationally fast and accurate, which makes them practical in high-throughput sequence data analysis. Haplotype inference is a computational problem where the goal is to estimate haplotypes from a sample of genotypes as accurately as possible. This problem is important as the direct measurement of haplotypes is difficult, whereas the genotypes are easier to quantify. Haplotypes are the key-players when studying for example the genetic causes of diseases. In this thesis, three methods are presented for the haplotype inference problem referred to as HaploParser, HIT, and BACH. HaploParser is based on a combinatorial mosaic model and hierarchical parsing that together mimic recombinations and point-mutations in a biologically plausible way. In this mosaic model, the current population is assumed to be evolved from a small founder population. Thus, the haplotypes of the current population are recombinations of the (implicit) founder haplotypes with some point--mutations. HIT (Haplotype Inference Technique) uses a hidden Markov model for haplotypes and efficient algorithms are presented to learn this model from genotype data. The model structure of HIT is analogous to the mosaic model of HaploParser with founder haplotypes. Therefore, it can be seen as a probabilistic model of recombinations and point-mutations. BACH (Bayesian Context-based Haplotyping) utilizes a context tree weighting algorithm to efficiently sum over all variable-length Markov chains to evaluate the posterior probability of a haplotype configuration. Algorithms are presented that find haplotype configurations with high posterior probability. BACH is the most accurate method presented in this thesis and has comparable performance to the best available software for haplotype inference. Local alignment significance is a computational problem where one is interested in whether the local similarities in two sequences are due to the fact that the sequences are related or just by chance. Similarity of sequences is measured by their best local alignment score and from that, a p-value is computed. This p-value is the probability of picking two sequences from the null model that have as good or better best local alignment score. Local alignment significance is used routinely for example in homology searches. In this thesis, a general framework is sketched that allows one to compute a tight upper bound for the p-value of a local pairwise alignment score. Unlike the previous methods, the presented framework is not affeced by so-called edge-effects and can handle gaps (deletions and insertions) without troublesome sampling and curve fitting.

On the Origin of Ideas : An Abductivist Approach to Discovery

The purpose of this study is to analyze and develop various forms of abduction as a means of conceptualizing processes of discovery. Abduction was originally presented by Charles S. Peirce (1839-1914) as a "weak", third main mode of inference -- besides deduction and induction -- one which, he proposed, is closely related to many kinds of cognitive processes, such as instincts, perception, practices and mediated activity in general. Both abduction and discovery are controversial issues in philosophy of science. It is often claimed that discovery cannot be a proper subject area for conceptual analysis and, accordingly, abduction cannot serve as a "logic of discovery". I argue, however, that abduction gives essential means for understanding processes of discovery although it cannot give rise to a manual or algorithm for making discoveries. In the first part of the study, I briefly present how the main trend in philosophy of science has, for a long time, been critical towards a systematic account of discovery. Various models have, however, been suggested. I outline a short history of abduction; first Peirce's evolving forms of his theory, and then later developments. Although abduction has not been a major area of research until quite recently, I review some critiques of it and look at the ways it has been analyzed, developed and used in various fields of research. Peirce's own writings and later developments, I argue, leave room for various subsequent interpretations of abduction. The second part of the study consists of six research articles. First I treat "classical" arguments against abduction as a logic of discovery. I show that by developing strategic aspects of abductive inference these arguments can be countered. Nowadays the term 'abduction' is often used as a synonym for the Inference to the Best Explanation (IBE) model. I argue, however, that it is useful to distinguish between IBE ("Harmanian abduction") and "Hansonian abduction"; the latter concentrating on analyzing processes of discovery. The distinctions between loveliness and likeliness, and between potential and actual explanations are more fruitful within Hansonian abduction. I clarify the nature of abduction by using Peirce's distinction between three areas of "semeiotic": grammar, critic, and methodeutic. Grammar (emphasizing "Firstnesses" and iconicity) and methodeutic (i.e., a processual approach) especially, give new means for understanding abduction. Peirce himself held a controversial view that new abductive ideas are products of an instinct and an inference at the same time. I maintain that it is beneficial to make a clear distinction between abductive inference and abductive instinct, on the basis of which both can be developed further. Besides these, I analyze abduction as a part of distributed cognition which emphasizes a long-term interaction with the material, social and cultural environment as a source for abductive ideas. This approach suggests a "trialogical" model in which inquirers are fundamentally connected both to other inquirers and to the objects of inquiry. As for the classical Meno paradox about discovery, I show that abduction provides more than one answer. As my main example of abductive methodology, I analyze the process of Ignaz Semmelweis' research on childbed fever. A central basis for abduction is the claim that discovery is not a sequence of events governed only by processes of chance. Abduction treats those processes which both constrain and instigate the search for new ideas; starting from the use of clues as a starting point for discovery, but continuing in considerations like elegance and 'loveliness'. The study then continues a Peircean-Hansonian research programme by developing abduction as a way of analyzing processes of discovery.

The effect of temperature on height growth of Scots pine in northern Finland

The effect of temperature on height growth of Scots pine in the northern boreal zone in Lapland was studied in two different time scales. Intra-annual growth was monitored in four stands in up to four growing seasons using an approximately biweekly measurement interval. Inter-annual growth was studied using growth records representing seven stands and five geographical locations. All the stands were growing on a dry to semi-dry heath that is a typical site type for pine stands in Finland. The applied methodology is based on applied time-series analysis and multilevel modelling. Intra-annual elongation of the leader shoot correlated with temperature sum accumulation. Height growth ceased when, on average, 41% of the relative temperature sum of the site was achieved (observed minimum and maximum were 38% and 43%). The relative temperature sum was calculated by dividing the actual temperature sum by the long-term mean of the total annual temperature sum for the site. Our results suggest that annual height growth ceases when a location-specific temperature sum threshold is attained. The positive effect of the mean July temperature of the previous year on annual height increment proved to be very strong at high latitudes. The mean November temperature of the year before the previous had a statistically significantly effect on height increment in the three northernmost stands. The effect of mean monthly precipitation on annual height growth was statistically insignificant. There was a non-linear dependence between length and needle density of annual shoots. Exceptionally low height growth results in high needle-density, but the effect is weaker in years of average or good height growth. Radial growth and next year s height growth are both largely controlled by current July temperature. Nevertheless, their growth variation in terms of minimum and maximum is not necessarily strongly correlated. This is partly because height growth is more sensitive to changes in temperature. In addition, the actual effective temperature period is not exactly the same for these two growth components. Yet, there is a long-term balance that was also statistically distinguishable; radial growth correlated significantly with height growth with a lag of 2 years. Temperature periods shorter than a month are more effective variables than mean monthly values, but the improvement is on the scale of modest to good when applying Julian days or growing-degree-days as pointers.