Gaussian Processes for Survival Analysis
We introduce a semi-parametric Bayesian model for survival analysis. The model is centred on a parametric baseline hazard, and uses a Gaussian process to model variations away from it nonparametrically, as well as dependence on covariates. As opposed to many other methods in survival analysis, our framework does not impose unnecessary constraints in the hazard rate or in the survival function. Furthermore, our model handles left, right and interval censoring mechanisms common in survival analysis. We propose a MCMC algorithm to perform inference and an approximation scheme based on random Fourier features to make computations faster. We report experimental results on synthetic and real data, showing that our model performs better than competing models such as Cox proportional hazards, ANOVA-DDP and random survival forests.
In the paper 'Gaussian Processes for Survival Analysis', the authors present a semi-parametric approach to survival time modeling in the presents of covariates.  The proposed method combines expert-knowledge in form of the parametric component, and a Gaussian Process (GP) as a non-parametric part to model differences between the former and the data.  The model, dubbed SGP, is compared to other, standard approaches in this area on synthetic and real world data.  The paper thoroughly introduces the method in section 2 by walking the reader through the layers of the hierarchical model in Equation (1) which is later extended in Equation (3) to account for covariates. In essence, the survival times are modeled as the first jumps of a Poisson process with an intensity assumed to be the product of an hand-crafted base function times a transformed GP.  The non-linear transformation of the GP assures that the model yields a proper survival function.  Section 3 shows that the SGP is tractable by defining a generative process to sample a survival time needed to perform inference. Also the (approximate) treatment of censored data points is discussed. To be able to handle large data sets, Section 4 describes an approximation scheme to the kernel based on a random Fourier series. Section 5 presents the experiments on one synthetic benchmark and a real world example, namely the Veteran data set available in R. For the latter, SGP is compared to three other methods for survival analysis.
The general idea of the paper seems promising, but I have several concerns:  1. The kernel choices in Equation (2) are not motivated and quite surprising to me. While a base kernel in 't' only makes a lot of sense, I don't understand the additive part for the covariates. Especially for the one used in the experiments, the summation of the product the covariates X_j, Y_j, and a kernel K_j(t,s) seems counterintuitive. Why should the covariance between to points grow if one of the covariates increases? Why not use a ARD (automatic relevance detection) kernel for all covariates combined rather than summing over all of them independently?  2. There is a discussion about the noise-parameters and the length-scales of the kernel, but what I missed was the actual method to adjust them given the data. A maximum likelihood estimation is mentioned, but only for the initialization. From experience, I know that this is crucial for 'high quality' GPs.  3. The experiments section is rather week. There is no comparison to the other algorithms on the synthetic benchmark, and the size of the real world benchmark with 137 samples and 5 covariates seems to small for the effort spent on approximations earlier in the paper. This amount of data should be no problem for a proper GP treatment without any approximations to the kernel.  4. Using GPs together with a parametric function to incorporate expert knowledge is a good idea, but the presented method makes a few ad-hoc choices to combine these two, e.g. the positive link function, or the particular kernel choice. An alternative could be a non-zero mean GP, where the mean function incorporates the expert knowledge. By modeling, for example, log(lambda) with this method, one could potentially enforce positiveness of the rate while also ensuring that the Survival function is well defined.   Overall, I think the paper is trying to accomplish too much at the same time: the GP based semi-parametric model, the (approximate) handling of censored data, the fast inference algorithm, and the random fourier kernel approximation. To me it feels that maybe just the first two point above without a focus the scalability, but with a more thorough empirical evaluation would make a more concise paper. The field of approximate GPs offers several possibilities to extend to larger data sets and many covariates.  Besides the technical points, the presentation of the material requires improvement: there are a few typos; citations are frequently used as nouns; and sometimes certain paragraphs feel out of place. For example, in line 55, Gaussian Cox Processes are mentioned for the first (and I think last time) without any context, citation or further explanation.   At this point, I have to recommend rejecting the paper, but I would like to encourage the authors to improve the paper for a future submission.