Differential Model Inference with Imperfect Information

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Density Ratio estimation (DRE) is the practice of estimating the ratio between two probability density functions (PDFs). DRE's ability to characterise the relationship between two PDFs naturally lends itself to many applications such as outlier detection, Generative Adversarial Networks (GANs) , and general binary classification. Furthermore, DRE is a technique which can be applied to the EPSRC research area of Natural Language Processing. In our research we aim to adapt various DRE methods, and downstream applications of these methods, to be robust when working with imperfect data. Imperfect data itself can take many forms such as missing data, corrupted data, and even adversarial data each of which need to be taken into careful consideration when trying to adapt DRE approaches. Imperfect data is a key issue within DRE as "few key points" can have a large impact on estimates making DRE very sensitive to any irregularities within the data. While there is a vast number of DRE approaches, very few of them explicitly account for the any case of imperfect data. Some work has been done regarding the impact of missing data on DRE, however this work exclusively focuses on the case of uniform missing patterns. There are many applications in which such an assumption is unrealistic and the probability of an observation being missing depends in some way in the value of the observation itself. For example, many measuring instruments are more likely to err when when attempting to measure more extreme values, while in questionnaires, participants are less likely to answer a question if they deem their answer to be embarrassing or unfavourable. Both of these examples lead to non-uniform missing patterns within the data. In such a case, naive implementation of a complete case approach with any DRE procedure can lead to estimating a different density ratio to our true density ratio and thus give inconsistent estimations. Our initial aim is therefore to adapt DRE procedures to this scenario of non-uniform missing data. When doing so there are multiple considerations An additional aim of the PhD will be to adapt downstream applications of DRE to the case of imperfect data. One of these applications we are looking to adapt is Neyman-Pearson (NP) classification. Neyman-Pearson (NP) classification is an application of DRE in which one wants to create a classification procedure which strictly controls miss-classification for one class. A potential application of NP classification is for use in disease diagnosis. Within this setting, falsely classifying a diseased individual as healthy could be far more damaging than classifying an individual who is healthy as diseased. As such we would like to construct a procedure for classifying individuals which has a strict control on the probability of miss-classifying a healthy individual as diseased. While major NP classification procedures leverage DRE, there is still opportunity for imperfect data to impact the procedure outside of the DRE. Therefore, we aim to address this and make the entirety of the NP classification procedure robust to non-uniform missing data. Again we will look to expand this to multi-dimensional settings. Another way we intend to extend this work is to look into cases where the missingness structure is in some way unknown. In this case we will explore how this missingness structure can be first learned before we perform our adapted DRE procedure or any downstream applications.