Assume that we are interested is classifying cases into two classes based on some covariate information. We then have data that consist of a binary outcome \(y\) nad a \(p\)-dimensional covariate vector \(\mathbf{x}\). The classic way of doing the classification is then to specify a discriminative model, i.e. a model for \(Y|\mathbf{X}=\mathbf{x}\), i.e. conditioning on the covariate values in order to avoid specifying a distribution for those. The logistic regression model is an example of such a model. The model parameters are
then typically estimated using the conditional likelihood of \(Y|\mathbf{X}=\mathbf{x}\).
An alternative is to use generative models for discrimination. The model
for \(Y|\mathbf{X}=\mathbf{x}\) is then specified indirectly via the conditional distributions of \(\mathbf{X}|Y=y\) for \(y=0,1\)and the marginal probability \(\pi_Y = P(Y = 1)\), as
\(P(Y = 1 \vert \mathbf{x}) = \frac{\pi_Yp(\mathbf{x}\vert Y=1)}{(1-\pi_Y)p(\mathbf{x}\vert Y=0) + \pi_Yp(\mathbf{x}\vert Y=1)}\). In this setting, the parameters are typically estimated via the likelihood functions of \(\mathbf{X}|Y=y\) separately for the two classes.
One possible project is to investigate this type of approach in the setting of fraud detection. The aim is then to construct a model for predicting whether a new incoming case is fraudulent, based on a collection of previous cases, with an associated indicator of whether the case is fraudulent or not. One of the characteristics of such data is that the marginal disribution of the covariates are be quite different, and include both continous and discrete variables. One option in such a case is to model the joint distribution of the covariates, conditioning on the class, with a copula (see for instance [1] for an introduction). Another important characteristic is that they typically are class imbalanced, meaning that the fraction of fraudulent cases in the data is small. This could for instance mean that it is best to have a complex model for the non-fraudulent class, for which data are abundant, and a much simpler model for the fraudulent class, for which data are scarce. The aim of this project would be to investigate how well a generative approach works for fraud detection, and more specifically if the above mentioned strategy of more deteiled modelling of the non-fraudulent class is advantageous or not.
Another possible project is to investigate this type of approach in the setting of immune receptor binding. The covariate information is then a short (typically length 15) sequence of amino acids whereas the response is an indicator of whether the immune receptor in question binds to a given pathogen. This has already been attempted with Naive Bayes type models, assuming that the covariates are conditionally independent given information on whether they are binding or not. However, the belief is that such a model is too simplistic, and that dependence between the covariates is important to capture. As all the covariates are categorical, this is not trivial. The typical way of modelling such dependence is via a regression model, but that not suitable within the given setting. Other possibilities include graphical models and also copulas. The aim of this project is to investigate different ways of modelling the dependence between the sequence positions in the binder and non-binder classes, and if including such dependence in the model is beneficial for predicting binding.