This document discusses uncertainty quantification for Bayesian survival analysis using nonparametric priors on the hazard function. It presents a theorem establishing the minimax convergence rate for the piecewise exponential model with a Gaussian Haar-histogram prior on the log-hazard. The proof uses a maximal inequality and shows the posterior contracts around the true hazard function at the optimal rate.

1. Uncertainty quantification for
Bayesian survival analysis
Isma¨el Castillo1 St´ephanie van der Pas2
1Laboratoire de Probabilit´es, Statistique et Mod´elisation, Sorbonne Universit´e
2Mathematical Institute, Leiden University
BFF Conference, April 29, 2019

2. Survival model
T : event time; C : censoring time.
Y = min{T, C}; δ = 1{T ≤ C}.
Independent right censoring.
We observe n independent pairs (Y1, δ1), . . . , (Yn, δn).
Survival function: P(T > t)
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3. Survival objects
Hazard function Cumulative hazard Survival
λ(t) = lim
∆t→0
P(t ≤ T < t + ∆t | T ≥ t)
∆t
.
Λ(t) =
t
0
λ(u)du.
S(t) = e−Λ(t).
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4. Goal
Pointwise intervals Credible band
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5. The piecewise exponential model
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Hazard
E.g. Gamerman (1991, 1994), Ibrahim, Chen and Sinha
(2013), Kalbfleisch (1978), Arjas and Gasbarra (1994),
Nieto-Barajas and Walker (2002).
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6. Bayesian survival analysis
Kim and Lee (2004), Kim (2006), BvM for survival objects
using neutral to the right processes.
De Blasi, Peccati, Pr¨unster (2009), CLTs for linear and
quadratic functionals of the hazard, using kernel mixtures
with respect to a completely random measure.
De Blasi and Hjort (2009), BvM in competing risks setting,
using a beta process prior.
Castillo (2012), semiparametric BvM for the Cox model,
using a Riemann-Liouville type process.
Donnet, Rivoirard, Rousseau, Scricciolo (2017),
L1-concentration for priors where the normalized hazard
and its integral are specified independently.
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7. Assumptions
Hazard function Cumulative hazard Survival
There exists τ > 0 such that S0(τ) > 0 and
Pλ0
(C ≥ τ) > 0. We take τ = 1.
There exists ρ > 0 such that λ0(t) ≥ ρ for all t ∈ [0, τ].
C admits a continuous density g with respect to the
Lebesgue measure on [0, τ).
There exists ρ > 0 such that g(t) ≥ ρ for all t ∈ [0, τ).
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8. Our approach
Our setting is non-conjugate and the techniques potentially
apply to many priors.
Key result: establish convergence at the minimax rate in
. ∞-norm.
Using tools from Castillo (2012), Castillo (2014), Castillo &
Rousseau (2015).
Next, obtain nonparametric BvM in appropriately weighted
multiscale space, and use continuous mapping and the
functional delta method.
As in Castillo & Nickl (2014), Gill (1989).
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9. Haar wavelets
ψ00
ψ10 ψ11
ψ20 ψ21 ψ22 ψ23
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10. Smoothness
For β > 0, M > 0 and a set ψ = {ψlk }:
Cψ
β,M
= λ ∈ C([0, 1]) : sup
l≥0
max
0≤k<2l
2l(β+1/2)
| ψlk , λ 2| ≤ M .
For the standard H¨older class H(β, M) and any B > 0:
H(β, M) ⊂ Cψ
β,M for β ≤ 1 when {ψlk } are the Haar
wavelets.
H(β, M) ⊂ Cψ
β,M for all β ≤ B if {ψlk } is smooth enough.
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11. Haar-histogram priors
Defined on r = log λ, with given cut-off Ln such that for some
δ > 0, 2Ln n
1
1+2δ :
r =
Ln
l=−1
2l −1
k=0
Zlk ψlk , Zlk ∼ N(0, σ2
l ).
Prior on log of hazard Prior on log of hazard Prior on log of hazard
Prior on hazard Prior on hazard Prior on hazard
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12. Dyadic histogram priors
With IJ
k = [k2−J, (k + 1)2−J), k = 1, . . . , 2J:
r =
2J
k=1
Zk IJ
k .
Prior on log of hazard Prior on log of hazard Prior on log of hazard
Prior on hazard Prior on hazard Prior on hazard
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13. Autoregressive histogram priors
Building on the autoregressive idea in Arjas and Gasbarra
(1994), we construct dependent histograms such that, with λk
the value on interval IJ
k on the hazard-scale:
E[λk | λk−1, . . . , λ1] = λk−1.
Var(λk | λk−1, . . . , λ1) = σ2
(λk−1)2
.
Prior on hazard Prior on hazard Prior on hazard
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14. . ∞-rate result for Haar-histogram priors
Theorem. Let X(n) = (Y1, δ1), . . . , (Yn, δn) be the observations.
Suppose λ0 ∈ Cβ,M
ψ with β > 1/2, some M > 0, and ψ the Haar
wavelets. Set α = min{β, 1}. Let the prior on the hazard be the
Gaussian Haar-histogram:
r =
Ln
l=−1
2l −1
k=0
Zlk ψlk , Zlk ∼ N(0, σ2
l ),
where σl = 2−l(1/2+α) and Ln = log2{(n/ log n)1/(2α+1)} .
Then, with ε∗
n,α = (log n
n )α/(2α+1), there exists M > 0 such that:
En
λ0
λ − λ0 ∞dΠ(λ | X(n)
) ≤ Mε∗
n,α.
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15. Proof idea
λ − λ0 ∞ ≤ λ − λ∗
Ln ∞ + λ∗
Ln
− λ0,Ln ∞ + λ0,Ln
− λ0 ∞,
where
λ∗
Ln
is a ’frequentist centering’;
λ0,Ln
is the L2-projection of λ0 on Vect{ψlk }.
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16. Maximal inequality
λ − λ∗
Ln ∞ ≤
1
√
n
Ln
l=0
2l/2
√
n max
0≤k<2l
| λ − λ∗
Ln
, ψlk 2|.
For any t > 0:
Eλ0
EΠn
max
0≤k<2l
t
√
n| λ − λ∗
Ln
, ψlk 2|
≤ log


2l −1
k=0
Eλ0
EΠn
et
√
n λ−λ∗
Ln
,ψlk 2
+ e−t
√
n λ−λ∗
Ln
,ψlk 2


On sets An = {λ : λ − λ0 1 ≤ εn}, there exists a C > 0
independent of l, k such that:
E et
√
n λ−λ∗
Ln
,ψlk 2
| Xn
, An ≤ e
t2
2
C
(1 + op(1)).
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17. References
Arjas E and Gasbarra D (1994). Nonparametric Bayesiana inference from right censored survival data, using the
Gibbs sampler. Statistica Sinica, 505–24.
Castillo I (2012). A semiparametric Bernstein-von Mises theorem for Gaussian process priors. Probability Theory
and Related Fields 152, 53–99.
Castillo I (2014). On Bayesian supremum norm contraction rates. The Annals of Statistics 42, 2058–91.
Castillo I and Nickl R (2014). On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures.
The Annals of Statistics 42, 1941–69.
Castillo I and Rousseau J (2015). A Bernstein-von Mises theorem for smooth functionals in semiparametric
models. The Annals of Statistics 43, 2353–83.
De Blasi P and Hjort N (2009). The Bernstein-von Mises theorem in semiparametric competing risks models.
Journal of Statistical Planning and Inference 139, 2316–28.
De Blasi P, Peccati G and Pr¨unster I (2009). Asymptotics for posterior hazards. The Annals of Statistics 37,
1906–45.
Donnet S, Rivoirard V, Rousseau J and Scricciolo C (2017). Posterior concentration rates for counting processes
with Aalen multiplicative intensities. Bayesian Analysis 12, 53–87.
Gamerman D (1991). Dynamic Bayesian models for survival data. Applied Statistics, 63–79.
Gamerman D (1994). Bayes estimation of the piece-wise exponential distribution. IEEE Transactions on
Reliability 43, 128–31.
Gill R (1989). Non- and semi-parametric maximum likelihood estimators and the Von Mises method, Part 1.
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Ibrahim JG, Chen MH, Sinha D (2013). Bayesian survival analysis. Springer Science & Business Media.
Kalbfleisch JD (1978). Non-parametric Bayesian analysis of survival time data. Journal of the Royal Statistical
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Kim Y (2006). The Bernstein-von Mises theorem for the proportional hazard model. The Annals of Statistics 34,
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