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.

On Games on Non-Wellfounded Sets and Stationary Sets

In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.

Psychologically-Based and Content-Oriented Experience in Entertainment Virtual Environments

In daily life, rich experiences evolve in every environmental and social interaction. Because experience has a strong impact on how people behave, scholars in different fields are interested in understanding what constitutes an experience. Yet even if interest in conscious experience is on the increase, there is no consensus on how such experience should be studied. Whatever approach is taken, the subjective and psychologically multidimensional nature of experience should be respected. This study endeavours to understand and evaluate conscious experiences. First I intro-duce a theoretical approach to psychologically-based and content-oriented experience. In the experiential cycle presented here, classical psychology and orienting-environmental content are connected. This generic approach is applicable to any human-environment interaction. Here I apply the approach to entertainment virtual environments (VEs) such as digital games and develop a framework with the potential for studying experiences in VEs. The development of the methodological framework included subjective and objective data from experiences in the Cave Automatic Virtual Environment (CAVE) and with numerous digital games (N=2,414). The final framework consisted of fifteen factor-analytically formed subcomponents of the sense of presence, involvement and flow. Together, these show the multidimensional experiential profile of VEs. The results present general experiential laws of VEs and show that the interface of a VE is related to (physical) presence, which psychologically means attention, perception and the cognitively evaluated realness and spatiality of the VE. The narrative of the VE elicits (social) presence and involvement and affects emotional outcomes. Psychologically, these outcomes are related to social cognition, motivation and emotion. The mechanics of a VE affect the cognitive evaluations and emotional outcomes related to flow. In addition, at the very least, user background, prior experience and use context affect the experiential variation. VEs are part of many peoples lives and many different outcomes are related to them, such as enjoyment, learning and addiction, depending on who is making the evalua-tion. This makes VEs societally important and psychologically fruitful to study. The approach and framework presented here contribute to our understanding of experiences in general and VEs in particular. The research can provide VE developers with a state-of-the art method (www.eveqgp.fi) that can be utilized whenever new product and service concepts are designed, prototyped and tested.