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Gaussian Processes for Survival Analysis
Tamara Fernández, Nicolás Rivera, Yee Whye Teh
TL;DR
Survival data bring covariates and censoring that complicate inference and prediction. The paper combines a parametric baseline hazard with Gaussian-process variation and develops inference and an approximation scheme. Experiments on synthetic and real data demonstrate the model, while synthetic estimates near time zero remain difficult when the baseline prior differs from the data.
Problem
Survival data commonly include many covariates and censored event times, complicating inference and prediction.
Source: §1 · p. 1Method
Results
Experiments on synthetic and real data demonstrate the model; synthetic estimates improve with more observations, and real-data fits detect treatment differences at Karnofsky score 90.
Source: §1 · p. 2, §5.1 · p. 6, §5.2 · p. 8Takeaways & Limitations
The baseline can encode expert hazard knowledge while the nonparametric component adjusts incomplete or incorrect prior knowledge and accommodates covariates.
Source: §1 · p. 2, §2 · p. 2When synthetic data do not follow the Weibull or exponential prior-centered hazards, estimates near time zero are difficult with few observations.
Source: §5.1 · pp. 6, 8
Abstract
from arXiv · showhide
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.
1 Introduction
Survival analysis must handle covariates and censored event times, while existing approaches can restrict hazard shapes or make covariates and expert knowledge difficult to incorporate. The paper proposes a semiparametric Bayesian model combining a parametric baseline hazard with Gaussian-process variation.
1 Introduction
- Survival data often include many covariates and censored times, where only an upper or lower bound on the event time is observed.These characteristics make inference and prediction hard.Source: §1 · p. 1
- The Cox proportional hazards model imposes proportional, non-crossing hazard curves, a restriction that may be implausible for some datasets.Kaplan–Meier handles censoring but does not naturally incorporate covariates or prior knowledge.Source: §1 · p. 1
- Bayesian nonparametric alternatives often lack a direct way to incorporate covariates; ANOVA-DDP incorporates them without strong constraints, but expert knowledge remains difficult to include.The cited Gaussian-process models also retain the proportional-hazards assumption.Source: §1 · pp. 1–2
- A parametric baseline hazard combined with nonparametric Gaussian-process variation provides a semiparametric model that handles covariates and censoring while accommodating expert knowledge.The nonparametric component can amend incomplete or incorrect prior knowledge.Source: §1 · p. 2
- The paper studies prior properties, adapts an existing sampler for inference, and introduces random Fourier features to approximate the Gaussian process for scaling.Experiments use synthetic and real data.Source: §1 · p. 2
2 Model
The model represents a hazard as a parametric baseline modulated by a nonnegative Gaussian-process component, then extends the process to depend on time and covariates. Its prior yields proper survival functions under stated conditions.
2 Model
- The hazard is modeled as a parametric baseline multiplied by nonparametric Gaussian-process noise, allowing the baseline to encode expert knowledge and the noise to amend incomplete or incorrect prior beliefs.For a centered Gaussian process, the expected hazard is λ0(t)/2; exponential- and Weibull-form baselines are examples.Source: §2 · pp. 2–3
- The model includes kernel hyperparameters for a length scale and an overall variance.Source: §2 · p. 3
- Under regularity conditions on the stationary continuous Gaussian process and a baseline hazard bounded below by Kt^(α−1), the prior defines survival functions that tend to zero.The stated conditions hold for exponential and Weibull baseline hazards.Source: §2 · p. 3
2.1 Adding covariates
- A Gaussian-process kernel over time and covariates can encode additive effects or interactions between covariate values and time.The experiments use the interaction kernel; more complex structures can include further covariate interactions.Source: §2.1 · p. 3
- Stationary kernels for the covariate-specific components ensure that survival tends to zero over time for each fixed covariate value.The model evaluates the process at a pair of time and covariates.Source: §2.1 · p. 3
3 Inference
Thinning a baseline Poisson process augments each observed survival time with rejected points, avoiding an intractable Gaussian-process integral. The resulting sampler updates these latent points and the Gaussian process, with separate imputation steps for censoring.
3.1 Data augmentation scheme
- Thinning accepts the first proposed baseline-Poisson point with probability σ(l), representing earlier rejected points as latent data and avoiding numerical integration of the Gaussian-process hazard.The Gaussian process need only be evaluated at proposed points, rather than over its whole path.Source: §3.1 · p. 4
3.2 Inference algorithm
- For each observation, the inference algorithm samples rejected points conditional on its observed time and covariates, then samples the Gaussian process given the augmented data.The latter step is treated as Gaussian-process binary classification, with rejected and accepted points as the two classes.Source: §3.2 · p. 4
3.3 Adding censoring
- Right-censored observations contribute rejected points before the censoring time; left-censored observations use an imputed time truncated to [0, Ti], and interval censoring uses a similar imputation.The left-censoring imputation is repeated at the beginning of each iteration.Source: §3.3 · p. 5
4 Approximation scheme
The approximation replaces costly Gaussian-process matrix operations with random-feature parameters. As the number of features grows, the approximate process kernel approaches the target stationary kernel.
4 Approximation scheme
- Random-feature approximation addresses matrix inversions that scale poorly when datasets or augmented rejected-point sets are large.The approximation represents the process using random features; inference updates feature coefficients instead of performing matrix inversion.Source: §4 · pp. 5–6
- For the experimental additive time-and-covariate kernel, the random-feature process approximates the component kernels, with the approximation improving as the feature count m grows.The construction applies to stationary kernels.Source: §4 · pp. 5–6
5 Experiments
Experiments examine synthetic survival curves and real Veteran data using Weibull- and Exponential-baseline models. On synthetic data, estimates improve with sample size; on the Veteran data, the models capture the score-survival pattern, while the proposed method distinguishes treatments at score 90.
5 Experiments
- The synthetic study uses 25, 50, 100, or 150 observations per group, while the Veteran study uses 137 records and five covariates with 10-fold C-index evaluation.Experiments use 50 random features; Weibull and Exponential baseline hazards have separate priors.Source: §5.1 · p. 6, §5 · p. 6, §5.2 · p. 8
5.1 Synthetic Data
- With clean synthetic data, estimates improve as sample size grows; with noisy covariates, 150 points per group center estimates on the corresponding true survival functions.Figures 1 and 2 show the Exponential and Weibull models, respectively.Source: §5.1 · p. 6, Figure 1 · p. 7, Figure 2 · p. 7
- Synthetic-data estimates are harder with few observations because the generating distributions differ from the models’ Weibull and Exponential prior centers, with errors near time zero also reported.The paper notes this issue for the synthetic experiment.Source: §5.1 · pp. 6, 8, §5 · p. 6
5.2 Real data experiments
- For this dataset, all studied models associate higher Karnofsky performance scores with improved survival time; the proposed method detects treatment differences at score 90, and ANOVA-DDP has the lowest C-index.The experiment uses 10-fold cross-validation, computing C-index per fold; Figure 3 presents C-index and survival-curve comparisons.Source: §5.2 · p. 8, Figure 3 · p. 8
6 Discussion
The model combines an expert-informed parametric baseline hazard with a flexible nonparametric component. Random Fourier features help it scale to large datasets, but not to massive numbers of covariates, especially with many interactions.
6 Discussion
- The baseline hazard can incorporate expert knowledge, while the nonparametric component adds flexibility.The discussion describes the latter as allowing the model to amend incorrect or incomplete prior knowledge.Source: §6 · p. 9
- Random Fourier features help the algorithm run on large datasets, but remain insufficient for massive covariate sets, especially with many interactions.The discussion also identifies sensitivity to initial parameters and interpretable kernels as areas for future work.Source: §6 · p. 9
- Future work includes choosing initial parameters to avoid sensitivity problems and constructing kernels that experts can interpret.Source: §6 · p. 9
A Proof of proposition 1
The proof studies the integrated sigmoid of a centered Gaussian process and its variability. It bounds deviations along a sequence of times and uses the Borel-Cantelli lemma in its argument.
A Proof of proposition 1
- The proof analyzes the survival function using a hazard bounded below by a baseline term multiplied by the sigmoid of the Gaussian process.It states that the survival function can be written in terms of the integrated hazard.Source: A Proof of proposition 1 · p. 11
- The proof studies the expectation and variance of an integrated sigmoid of the Gaussian process, splitting a covariance integral into nearby and separated regions.It uses the centered process to obtain an expectation involving the sigmoid and divides the integral according to whether |t − s| < 1.Source: A Proof of proposition 1 · pp. 11, 13
- For a selected increasing sequence of times, the proof bounds deviation probabilities and applies the Borel-Cantelli lemma.The sequence is chosen so the deviation probabilities are at most n^-2.Source: A Proof of proposition 1 · p. 12