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Fisher Kernel Martin Sewell Department of Computer Science University College London April 2007 (last updated September 2008) 1 Introduction In common with all kernel methods, the support vector machine technique in- volves two stages: first non-linearly map the input space into a very high di- mensional feature space, then apply a learning algorithm designed to discover linear patterns in that space. This paper concerns the first stage. The basic idea is to train a (generative) hidden Markov model on market data to derive a Fisher kernel for a (discriminative) support vector machine. If each data point is a (possibly varying length) sequence, each may be used to train a probabil- ity model (HMM), with the average of the models in the training set used to construct a global HMM. It is then possible to calculate how much a new data point would ‘stretch’ the parameters of the global model. 2 Literature Review Jaakkola and Haussler (1999a) introduced the Fisher kernel (named in honour of Sir Ronald Fisher), thus creating a generic mechanism for incorporating gener- ative probability models into discriminative classifiers such as SVMs. Jaakkola and Haussler (1999b) introduced a generic class of probabilistic regression mod- els and a parameter estimation technique that can make use of arbitrary kernel functions. Jaakkola, Diekhans and Haussler (1999) applied the Fisher kernel method to detecting remote protein homologies which performed well in classi- fying protein domains by SCOP superfamily. Jaakkola, Diekhans and Haussler (2000) found that using the Fisher kernel significantly improved on previous methods for the classification of protein domains based on remote homologies. Moreno and Rifkin (2000) used the Fisher