For the applied statistician, data augmentation is a powerful tool for solving optimization problems. In this thesis, I address a problem in some data augmented Gibbs samplers. I show that although...Show moreFor the applied statistician, data augmentation is a powerful tool for solving optimization problems. In this thesis, I address a problem in some data augmented Gibbs samplers. I show that although introducing latent variables renders a sampling problem tractable, this comes at the price of raising the autocorrelation of the Markov chain, as the number of parameters increases, in this case the number of items in a test. By means of an example, I show that data augmentation is a powerful yet inefficient tool in cases of increased number of items, since the autocorrelation (and hence the rate of the convergence) of the addressed augmented Gibbs sampler is proved to be dependent on the number of item parameters. We wish to show that although most data-augmented samplers are well behaved, in this example the algorithm becomes really slow and faces the possibility of grinding to a haltShow less
The introduction of the Rényi entropy allowed a generalization of the Shannon entropy and unified its notion with that of other entropies. However, so far there is no generally accepted conditional...Show moreThe introduction of the Rényi entropy allowed a generalization of the Shannon entropy and unified its notion with that of other entropies. However, so far there is no generally accepted conditional version of the Rényi entropy corresponding to the one of the Shannon entropy. Different definitions proposed so far in the literature lacked central and natural properties one way or another. In this thesis we propose a new definition for the conditional case of the Rényi entropy. Our new definition satisfies all of the properties we deem natural. First and foremost, it is consistent with the existing, commonly accepted, definition of the conditional Shannon entropy as well as with the right notion of the conditional min entropy. Furthermore, and in contrast to previously suggested definitions, it satisfies the two natural properties that are monotonicity and (weak) chain rule and which we feel need to be satisfied by any ‘good’ entropy notion. Another characteristic of our new definition is that it can be formulated in terms of the Rényi divergence. Additionally, it enables the use of (entropy) splitting. We conclude with an application where we use our new entropy notion as a tool to analyze a particular quantum cryptographic identification scheme.Show less
The topic of this Master Thesis is quantitative methodologies for optional features and bundles. The frame is the one of Quantitative Marketing Research, a field whose goal is to give market...Show moreThe topic of this Master Thesis is quantitative methodologies for optional features and bundles. The frame is the one of Quantitative Marketing Research, a field whose goal is to give market intelligence in forms of, among others, market shares, population clustering and scenario simulations. The particular problem we have worked on is the one of optional features and bundles i.e. services that can be selected for an extra price when purchasing a product. The technique we have used in our analysis is a discrete choice model, Choicebased Conjoint. The content of this thesis is based on an internship at the international market research company SKIM. The internship was jointly supervised by Senior Methodologist Kees van der Wagt (SKIM) and Prof. Dr. Richard Gill (Mathematisch Instituut Leiden). The two most important results of the thesis are new methodologies to study products with optional features and bundles. These methodologies produce utilities that match the respondent’s observed choices. Only knowing the estimated utilities, we are able to answer the questionnaire producing answers similar to the observed ones. The methodologies enjoy all typical properties of conjoint methodologies and can be used to calculate market shares, simulate scenarios etc. Their most interesting feature is that it is possible to tell if offering an option makes a product too complicated. They can also tell if their simple presence makes the product more appealing (halo effect). As far as we know, this is the first study in this promising field. The methodologies we propose are tested on two different datasets arising from studies conducted by SKIM. They have been developed with tests on simulated datasets. The software of choice for the estimation procedure was Sawtooth’s implementation of CBC HB. For reproducibility of experiments we also wrote a package in the open source language R reproducing the same algorithm. This package and Matlab codes used in simulations are found in the Appendix.Show less
