When dealing with right-censored data, where some outcomes are missing due to a limited observation period, survival analysis focuses on predicting the time until an event of interest occurs. Multiple classes of outcomes lead to a classification variant: predicting the most likely event, an area known as competing risks.

1. 12/06/2025
Survival Models:
Proper Scoring Rule and Stochastic
Optimization for Competing Risks
Julie Alberge
PhD student under the supervision of Judith Abécassis and Gaël
Varoquaux
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2. Collaborators
Vincent Maladière Olivier Grisel Judith Abécassis Gaël Varoquaux
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3. What is survival analysis?
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4. Right-censored time-to-event data
Censored patients: patients who
didn’t experience the event
during the observation study.
Goal: Predict the CDF of their
death for each patient in time.
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5. Survival Analysis
Notations:
: Time of event of interest.
: Censoring time.
: Observed time.
The event that occurred.
Hyp: Non-informative censoring:
Cumulative Incidence Function:
Survival Function:
T* ∈ ℝ+
C
T = min(T*, C)
Δ ∈ {0,1}
T* C|X
F*(ζ|x) = ℙ(T* ≤ ζ|X = x)
S*(ζ|x) = ℙ(T* > ζ|X = x)
Any given time
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6. How to deal with the censoring distribution?
Inverse Propensity Censoring Weighting (IPCW)
• Compute the probability of being censored
• Weight samples by inverse probability
Time horizon
Less observed
outcomes
More weight
per person
(increasing censoring)
Many observed
outcomes
Less weight
per person
-> Learn and evaluate
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7. Usual evaluation metrics
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8. Integrated Brier Score
• Proper Scoring Rule
• “How close to the real probabilities are
our estimates?”
Lower is better
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BS(t) =
1
n
n
∑
i=1
𝕀
(yi ≤ t ∧ δi = 1)
(0 − ̂
S(t|xi))2
̂
G(yi)
+
𝕀
(yi > t)
(1 − ̂
S(t|xi))2
̂
G(t)

9. C-index
Assess the ranking power of the models.
Commonly used by practitioners.
T*
i
≤ T*
j
⟶ S(τ|xi) < S(τ|xj)
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10. Competing Risks
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11. Competing risks setting
When there is only one event of interest (predict the death, recovery, etc.)
—> Survival Analysis
When there are multiple events (such as the cause of death, etc.).
—> Competing Risks Setting
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12. Competing Risks Setting
Notations:
: Time of the
fi
rst event of interest.
: Censoring time.
: Observed time.
K: Number of events of interest.
The event that occurred.
Hyp: Non-informative censoring:
kth Cumulative Incidence Function:
T* ∈ ℝ+
C
T = min(T*, C)
Δ ∈ {0,1,...,K}
T* C|X
F*
k
(τ|x) = ℙ(T* ≤ τ ∩ Δ* = k|X = x)
Any given time
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Buckets
Buckets
Perfect
calibratio
Model's CR
D-calibratio
Event 1
Aggregate results to obta
the D-CR calibration
Computation of the
probability of the event
at the true time event
Other event
observed
Event K
observed
CIFs
probability
CIFs
probability
Event K
Event 1
...
...

13. Motivation of our work
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14. What do we want to achieve?
• Handles competing risks (limited results available).
• Predict the
fi
rst event for each patient correctly.
• Good trade-o
ff
between predictions and
fi
tting time.
• A model that handles tabular data well.
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15. Survival Models:
Proper Scoring Rule and Stochastic
Optimization for Competing Risks
Published in AISTATS 2025
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16. Proper scoring rule
• A scoring rule evaluates a distribution over an observation and gives a
corresponding score ( , ).
• The better the score, the better the model
fi
ts the observation.
Intuition:
Optimal proper scoring rule <-> Oracle distribution
ℓ
𝒫
Y
ℓ
𝒫
Y
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17. Our proper Scoring rule
This can be givan to any estimator with stochastic optimization
Teaching a model how to survive 17

18. SurvivalBoost
• Implementation of the previous loss using Gradient Boosting Trees.
• Using a feedback loop to learn the distribution of .
C
SurvivalBoost Algorithm: one iteration
(Time is
sampled
uniformly.)
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19. Our Benchmark
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20. Metrics
• Mean of Integrated Brier Score
• C-index
• Introduction of the accuracy in time:
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δi1ti≤τ
Highest probability
Number of non
censored individuals
Non
censored individuals

21. Competing Risks:
Benchmark
Trade-o
ff
prediction/training time Performances on the
mean IBS compared to
fi
tting time for each model on the
SEER dataset (300k datapoints)
Accuracy at time SEER dataset (300k datapoints)
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22. Survival Analysis:
Benchmark
Trade-o
ff
prediction/training time in survival usage Performances on the IBS compared to
fi
tting time for each model.
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23. Survival Analysis:
Scalability
Trade-o
ff
prediction/training time in survival usage Performances on the IBS compared to
fi
tting time for each model.
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24. Python Library
The model is implemented in a Python
library named hazardous, which
includes documentation, examples, and
several useful metrics.
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25. Conclusion
• A plug-in Proper Scoring Rule for competing risks and survival analysis.
• An implementation with Gradient Boosting Trees due to the presence of
categorical data.
• Outstanding performances in benchmarks in the competing risks and the survival
analysis setting
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26. Future work
Submitted - Under review
We try to answer:
• What is the calibration in the competing risks setting?
• How to measure it?
• How to recalibrate any given method?
Tristan Haugomat
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