Sequence alignment is one of the most intensely studied problems in bioinformatics, and is an important step in a wide range of analyses. An issue that has gained much attention in recent years is the fact that downstream analyses are often highly sensitive to the specific choice of alignment. One way to address this is to jointly sample alignments along with other parameters of interest. In order to extend the range of applicability of this approach, the first chapter of this thesis introduces a probabilistic evolutionary model for protein structures on a phylogenetic tree; since protein structures typically diverge much more slowly than sequences, this allows for more reliable detection of remote homologies, improving the accuracy of the resulting alignments and trees, and reducing sensitivity of the results to the choice of dataset. In order to carry out inference under such a model, a number of new Markov chain Monte Carlo approaches are developed, allowing for more efficient convergence and mixing on the high-dimensional parameter space. The second part of the thesis presents a directed acyclic graph (DAG)-based approach for representing a collection of sampled alignments. This DAG representation allows the initial collection of samples to be used to generate a larger set of alignments under the same approximate distribution, enabling posterior alignment probabilities to be estimated reliably from a reasonable number of samples. If desired, summary alignments can then be generated as maximum-weight paths through the DAG, under various types of loss or scoring functions. The acyclic nature of the graph also permits various other types of algorithms to be easily adapted to operate on the entire set of alignments in the DAG. In the final part of this work, methodology is introduced for alignment-DAG-based sequence annotation using hidden Markov models, and RNA secondary structure prediction using stochastic context-free grammars. Results on test datasets indicate that the additional information contained within the DAG allows for improved predictions, resulting in substantial gains over simply analysing a set of alignments one by one.

Tato disertační práce představuje derivační stromy několika různých typů gramatik ve zobecněné Kurodově normální formě; jmenovitě obecné a regulárně řízené gramatiky, gramatiky s rozptýleným kontextem a spolupracující distribuované gramatické systémy. Definuje jednoduché stromové rysy založené na kontextových vlastnostech jednotlivých diskutovaných gramatik a dokazuje, že pokud existuje limitující konstanta k taková, že každá věta generovaného jazyka L odpovídá řetězci listových uzlů derivačního stromu, ve kterém je výskyt definovaných stromových rysů omezen konstantou k, jazyk L je ve skutečnosti bezkontextový. Tato práce dále ukazuje, že dosažený výsledek představuje silný nástroj důkazu bezkontextovosti jazyka. Vše je doplněno příklady praktického využití nástroje.