J. Sethuraman
Partition Based Nonparametric Priors
Partition based priors to analyze data arising
from failure models and censoring models
Jayaram Sethuraman
Department of Statistics
Florida State University
and
University of South Carolina
[email protected]
October 21, 2008
J. Sethuraman
Partition Based Nonparametric Priors
Summary
Failure and repair models
J. Sethuraman
Partition Based Nonparametric Priors
Summary
Failure and repair models
Censoring models
J. Sethuraman
Partition Based Nonparametric Priors
Summary
Failure and repair models
Censoring models
Partition based (PB) priors
J. Sethuraman
Partition Based Nonparametric Priors
Summary
Failure and repair models
Censoring models
Partition based (PB) priors introduced by Sethuraman and
Hollander (2008) turn out to be the natural priors for analyzing
both these kinds of data.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P. For
any set A, let PA denote the restriction of P to A.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P. For
any set A, let PA denote the restriction of P to A.
The general repair model postulates that there some environmental
random variables Y1 , Y2 , . . . , and
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P. For
any set A, let PA denote the restriction of P to A.
The general repair model postulates that there some environmental
random variables Y1 , Y2 , . . . , and
X1 |P ∼ P
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P. For
any set A, let PA denote the restriction of P to A.
The general repair model postulates that there some environmental
random variables Y1 , Y2 , . . . , and
X1 |P ∼ P
X2 |(X1 , Y1 , P) ∼ PA1
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P. For
any set A, let PA denote the restriction of P to A.
The general repair model postulates that there some environmental
random variables Y1 , Y2 , . . . , and
X1 |P ∼ P
X2 |(X1 , Y1 , P) ∼ PA1
X3 |(X1 , X2 , Y1 , Y2 , P) ∼ PA2
..
.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - I
Here X1 , X2 , . . . are dependent random variables whose
distribution depends on a pm P and we want to estimate P. For
any set A, let PA denote the restriction of P to A.
The general repair model postulates that there some environmental
random variables Y1 , Y2 , . . . , and
X1 |P ∼ P
X2 |(X1 , Y1 , P) ∼ PA1
X3 |(X1 , X2 , Y1 , Y2 , P) ∼ PA2
..
.
where An is a set that depends on X1 , . . . , Xn , Y1 , Y2 , . . . , Yn for
n = 1, 2, . . . .
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞),
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞), distribution of X2
is P (new component).
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞), distribution of X2
is P (new component).
Fix a number p in (0, 1). Let the environmental variable Y1 ∼
Uniform [0, 1] and independent of X1 .
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞), distribution of X2
is P (new component).
Fix a number p in (0, 1). Let the environmental variable Y1 ∼
Uniform [0, 1] and independent of X1 .
Let
(0, ∞)
if Y ≤ p
AX1 =
(X1 , ∞) if Y > p.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞), distribution of X2
is P (new component).
Fix a number p in (0, 1). Let the environmental variable Y1 ∼
Uniform [0, 1] and independent of X1 .
Let
(0, ∞)
if Y ≤ p
AX1 =
(X1 , ∞) if Y > p.
This is the Brown-Proschan model of randomized minimal repair.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞), distribution of X2
is P (new component).
Fix a number p in (0, 1). Let the environmental variable Y1 ∼
Uniform [0, 1] and independent of X1 .
Let
(0, ∞)
if Y ≤ p
AX1 =
(X1 , ∞) if Y > p.
This is the Brown-Proschan model of randomized minimal repair.
Many other repair models fall under this general repair model.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - III
For example if AX1 = (X1 , ∞), the distribution of X2 is that
obtained by minimal repair and if AX1 = (0, ∞), distribution of X2
is P (new component).
Fix a number p in (0, 1). Let the environmental variable Y1 ∼
Uniform [0, 1] and independent of X1 .
Let
(0, ∞)
if Y ≤ p
AX1 =
(X1 , ∞) if Y > p.
This is the Brown-Proschan model of randomized minimal repair.
Many other repair models fall under this general repair model.
Repair models can also be viewed as search models.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - IV
A Bayesian will put a prior distribution for P.
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - IV
A Bayesian will put a prior distribution for P.
He will also assume that the distribution of Yn given
X1 , . . . , Xn , Y1 , . . . , Yn−1 is independent of P, n = 1, 2, . . .
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - IV
A Bayesian will put a prior distribution for P.
He will also assume that the distribution of Yn given
X1 , . . . , Xn , Y1 , . . . , Yn−1 is independent of P, n = 1, 2, . . .
This leads to a nice simplification;
J. Sethuraman
Partition Based Nonparametric Priors
Failure and repair models - IV
A Bayesian will put a prior distribution for P.
He will also assume that the distribution of Yn given
X1 , . . . , Xn , Y1 , . . . , Yn−1 is independent of P, n = 1, 2, . . .
This leads to a nice simplification; we can assume that Y1 , Y2 , . . .
are constants.
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - 1
Here X1 , X2 , . . . are dependent random variables (actual censored
and uncensored observations) whose distribution depends on a pm
P and we want to estimate P.
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - 1
Here X1 , X2 , . . . are dependent random variables (actual censored
and uncensored observations) whose distribution depends on a pm
P and we want to estimate P.
The general censoring model postulates that there are potential
observations X1∗ , X2∗ , . . . , which are i.i.d. P, and there are
censoring (environmental) variables Y1 , Y2 , . . . . However, only
Z1 = min(X1∗ , Y1 ), Z2 = min(X2∗ , Y2 ), . . . are observed.
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - 1
Here X1 , X2 , . . . are dependent random variables (actual censored
and uncensored observations) whose distribution depends on a pm
P and we want to estimate P.
The general censoring model postulates that there are potential
observations X1∗ , X2∗ , . . . , which are i.i.d. P, and there are
censoring (environmental) variables Y1 , Y2 , . . . . However, only
Z1 = min(X1∗ , Y1 ), Z2 = min(X2∗ , Y2 ), . . . are observed.
Also observed are
X1∗
A1
X2∗
X2 |(X1 , Y1 , Y2 , P) ∼
A2
..
.
X1 |(Y1 , P) ∼
where A1 = (Z1 , ∞), A2 = (Z2 , ∞), . . . .
if X1∗ ∈ Ac1
if X1∗ ∈ A1
if X2∗ ∈ Ac2
if X2∗ ∈ A2
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - II
For left censoring, we define Zi = max(Xi∗ , Yi ) and take
Ai = [0, Zi ) in the model above.
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - II
For left censoring, we define Zi = max(Xi∗ , Yi ) and take
Ai = [0, Zi ) in the model above.
Similarly by taking An to be a set depending on
X1 , . . . , Xn−1 , Y1 , Y2 , . . . , Yn for n = 1, 2, . . . , in the model above,
we can simultaneously allow for all kinds of censoring and also
allow for dependence on previous X ∗ observations. This general
censoring model can also be used in search models.
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - II
For left censoring, we define Zi = max(Xi∗ , Yi ) and take
Ai = [0, Zi ) in the model above.
Similarly by taking An to be a set depending on
X1 , . . . , Xn−1 , Y1 , Y2 , . . . , Yn for n = 1, 2, . . . , in the model above,
we can simultaneously allow for all kinds of censoring and also
allow for dependence on previous X ∗ observations. This general
censoring model can also be used in search models.
As before, a Bayesian will put a prior distribution for P.
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - II
For left censoring, we define Zi = max(Xi∗ , Yi ) and take
Ai = [0, Zi ) in the model above.
Similarly by taking An to be a set depending on
X1 , . . . , Xn−1 , Y1 , Y2 , . . . , Yn for n = 1, 2, . . . , in the model above,
we can simultaneously allow for all kinds of censoring and also
allow for dependence on previous X ∗ observations. This general
censoring model can also be used in search models.
As before, a Bayesian will put a prior distribution for P.
He will also assume that the distribution of Yn given
X1 , . . . , Xn , Y1 , . . . , Yn−1 is independent of P, n = 1, 2, . . .
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - II
For left censoring, we define Zi = max(Xi∗ , Yi ) and take
Ai = [0, Zi ) in the model above.
Similarly by taking An to be a set depending on
X1 , . . . , Xn−1 , Y1 , Y2 , . . . , Yn for n = 1, 2, . . . , in the model above,
we can simultaneously allow for all kinds of censoring and also
allow for dependence on previous X ∗ observations. This general
censoring model can also be used in search models.
As before, a Bayesian will put a prior distribution for P.
He will also assume that the distribution of Yn given
X1 , . . . , Xn , Y1 , . . . , Yn−1 is independent of P, n = 1, 2, . . .
This leads to the nice simplification;
J. Sethuraman
Partition Based Nonparametric Priors
Censoring Models - II
For left censoring, we define Zi = max(Xi∗ , Yi ) and take
Ai = [0, Zi ) in the model above.
Similarly by taking An to be a set depending on
X1 , . . . , Xn−1 , Y1 , Y2 , . . . , Yn for n = 1, 2, . . . , in the model above,
we can simultaneously allow for all kinds of censoring and also
allow for dependence on previous X ∗ observations. This general
censoring model can also be used in search models.
As before, a Bayesian will put a prior distribution for P.
He will also assume that the distribution of Yn given
X1 , . . . , Xn , Y1 , . . . , Yn−1 is independent of P, n = 1, 2, . . .
This leads to the nice simplification; we can assume that
Y1 , Y2 , . . . are constants.
J. Sethuraman
Partition Based Nonparametric Priors
Partition Based Prior Distributions
The models described earlier for dependent data X1 , X2 , . . . lead
us, in a natural fashion, to Partition based (PB) Priors. These
were introduced in Sethuraman and Hollander (2008).
J. Sethuraman
Partition Based Nonparametric Priors
Partition Based Prior Distributions
The models described earlier for dependent data X1 , X2 , . . . lead
us, in a natural fashion, to Partition based (PB) Priors. These
were introduced in Sethuraman and Hollander (2008).
Let B = (B1 , . . . , Bm ) be a finite partition of X . Let
P(B) = (P(B1 ), . . . , P(Bm )) be the partition probability vector,
and let (PB1 , . . . , PBm ) be the vector of restricted pm’s.
J. Sethuraman
Partition Based Nonparametric Priors
Partition Based Prior Distributions
The models described earlier for dependent data X1 , X2 , . . . lead
us, in a natural fashion, to Partition based (PB) Priors. These
were introduced in Sethuraman and Hollander (2008).
Let B = (B1 , . . . , Bm ) be a finite partition of X . Let
P(B) = (P(B1 ), . . . , P(Bm )) be the partition probability vector,
and let (PB1 , . . . , PBm ) be the vector of restricted pm’s.
There is a one-to-one correspondence between pm’s P and the pair
(P(B), (PB1 , . . . , PBm )) since
P(B) =
m
X
P(Bi )PAi (B).
1
The pair (P(B), (PB1 , . . . , PBm )) is just another way to understand
a pm P.
J. Sethuraman
Partition Based Nonparametric Priors
Partition based Priors
Suppose that P(B), PB1 , . . . , PBm are independent,
with P(B) having a pdf proportional to h(y),
and PBi having a distribution Gi , i = 1, . . . , m.
J. Sethuraman
Partition Based Nonparametric Priors
Partition based Priors
Suppose that P(B), PB1 , . . . , PBm are independent,
with P(B) having a pdf proportional to h(y),
and PBi having a distribution Gi , i = 1, . . . , m.
Then we say that P has a Partition based (PB) prior distribution
H(B, h, G)
where G = G1 × · · · × Gm .
J. Sethuraman
Partition Based Nonparametric Priors
Partition based Priors
Suppose that P(B), PB1 , . . . , PBm are independent,
with P(B) having a pdf proportional to h(y),
and PBi having a distribution Gi , i = 1, . . . , m.
Then we say that P has a Partition based (PB) prior distribution
H(B, h, G)
where G = G1 × · · · × Gm .
The Gi ’s are probability measures on the space of probability
measures and there are m of them.
J. Sethuraman
Partition Based Nonparametric Priors
Partition based Priors
Suppose that P(B), PB1 , . . . , PBm are independent,
with P(B) having a pdf proportional to h(y),
and PBi having a distribution Gi , i = 1, . . . , m.
Then we say that P has a Partition based (PB) prior distribution
H(B, h, G)
where G = G1 × · · · × Gm .
The Gi ’s are probability measures on the space of probability
measures and there are m of them. Hopefully, we will use standard
nonparametric priors for Gi , i = 1, . . . , m,
J. Sethuraman
Partition Based Nonparametric Priors
Partition based Priors
Suppose that P(B), PB1 , . . . , PBm are independent,
with P(B) having a pdf proportional to h(y),
and PBi having a distribution Gi , i = 1, . . . , m.
Then we say that P has a Partition based (PB) prior distribution
H(B, h, G)
where G = G1 × · · · × Gm .
The Gi ’s are probability measures on the space of probability
measures and there are m of them. Hopefully, we will use standard
nonparametric priors for Gi , i = 1, . . . , m, so that we also know the
corresponding posterior distributions GiX .
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors
Let α be a finite non-zero measure on X and take Gi to be DαAi ,
i = 1, . . . , m.
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors
Let α be a finite non-zero measure on X and take Gi to be DαAi ,
i = 1, . . . , m. In this case, we will call the PB prior H(B, h, G) as
a Partition Based (PB) Dirichlet prior and denote it by D(B, h, α).
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors
Let α be a finite non-zero measure on X and take Gi to be DαAi ,
i = 1, . . . , m. In this case, we will call the PB prior H(B, h, G) as
a Partition Based (PB) Dirichlet prior and denote it by D(B, h, α).
If B ∗ is a subpartition of B, then a PB Dirichlet prior on B is also a
PB Dirichlet prior on B ∗ .
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors
Let α be a finite non-zero measure on X and take Gi to be DαAi ,
i = 1, . . . , m. In this case, we will call the PB prior H(B, h, G) as
a Partition Based (PB) Dirichlet prior and denote it by D(B, h, α).
If B ∗ is a subpartition of B, then a PB Dirichlet prior on B is also a
PB Dirichlet prior on B ∗ .
A PB Dirichlet prior D(B, h, α) is the Dirichlet prior Dα if and only
of
α(A1 )−1
h(y) ∝ y1
α(Am )−1
· · · ym
.
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions in standard nonparametric Bayesian
problem
For any nonparametric prior G , let G X be the posterior distribution
in the standard nonparametric Bayesian problem.
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions in standard nonparametric Bayesian
problem
For any nonparametric prior G , let G X be the posterior distribution
in the standard nonparametric Bayesian problem.
That is the unknown pm P has prior distribution G , and the the
data X given P has distribution P. G X is simply the notation for
the the posterior distribution of P given X .
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 1
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 1
Consider the problem where P has the PB prior
H(B, h, G)
and the data X given P has distribution PA .
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 1
Consider the problem where P has the PB prior
H(B, h, G)
and the data X given P has distribution PA .
Suppose that A = ∪ri=1 Bi , and define yA =
Pr
i=1 yi .
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 1
Consider the problem where P has the PB prior
H(B, h, G)
and the data X given P has distribution PA .
Suppose that A = ∪ri=1 Bi , and define yA =
Pr
i=1 yi .
Let R be the random index such that X ∈ BR ; note that
R ∈ {1, . . . , r }.
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 2
Theorem
Then the posterior distribution of P given X is
H(B, hX , G X )
where
hX (y ) ∝ h(y) · yR /yA
and
G X = G1 × · · · GR−1 × GRX × GR+1 × · · · Gm .
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 3
Suppose that P has the PB Dirichlet distribution D(B, h, α).
Given P, let the data X have distribution PA where, as before,
A = ∪ri=1 Bi .
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 3
Suppose that P has the PB Dirichlet distribution D(B, h, α).
Given P, let the data X have distribution PA where, as before,
A = ∪ri=1 Bi .
Then the posterior distribution of P given X is
D(B, hX , α + δX ).
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 3
Suppose that P has the PB Dirichlet distribution D(B, h, α).
Given P, let the data X have distribution PA where, as before,
A = ∪ri=1 Bi .
Then the posterior distribution of P given X is
D(B, hX , α + δX ).
The condition A = ∪ri=1 Bi is not required when the prior is a PB
Dirichlet prior; the finite partition B can be enlarged to a new
partition B ∗ by including A;
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 3
Suppose that P has the PB Dirichlet distribution D(B, h, α).
Given P, let the data X have distribution PA where, as before,
A = ∪ri=1 Bi .
Then the posterior distribution of P given X is
D(B, hX , α + δX ).
The condition A = ∪ri=1 Bi is not required when the prior is a PB
Dirichlet prior; the finite partition B can be enlarged to a new
partition B ∗ by including A;
the posterior distribution will still be a PB Dirichlet distribution on
the partition on B ∗ .
J. Sethuraman
Partition Based Nonparametric Priors
Posterior distributions of PB priors with repair data - 3
Suppose that P has the PB Dirichlet distribution D(B, h, α).
Given P, let the data X have distribution PA where, as before,
A = ∪ri=1 Bi .
Then the posterior distribution of P given X is
D(B, hX , α + δX ).
The condition A = ∪ri=1 Bi is not required when the prior is a PB
Dirichlet prior; the finite partition B can be enlarged to a new
partition B ∗ by including A;
the posterior distribution will still be a PB Dirichlet distribution on
the partition on B ∗ .
Even more; The partition B can be enlarged with the random
restriction sets based on all of data, and still a blind application of
the method above will give the correct answer. See Sethuraman
and Hollander (2008).
J. Sethuraman
Partition Based Nonparametric Priors
Illustration a Dirichlet Prior as a PB prior
As an illustration, we will write a Dirichlet prior as a partition
based prior on a partition determined by the data and compute the
posterior; we will see that we still obtain the correct answer.
J. Sethuraman
Partition Based Nonparametric Priors
Illustration a Dirichlet Prior as a PB prior
As an illustration, we will write a Dirichlet prior as a partition
based prior on a partition determined by the data and compute the
posterior; we will see that we still obtain the correct answer.
Suppose that P ∼ Dα , X |P ∼ P.
J. Sethuraman
Partition Based Nonparametric Priors
Illustration a Dirichlet Prior as a PB prior
As an illustration, we will write a Dirichlet prior as a partition
based prior on a partition determined by the data and compute the
posterior; we will see that we still obtain the correct answer.
Suppose that P ∼ Dα , X |P ∼ P. Let A = (X − , X + ) and let
α(A) > 0.
J. Sethuraman
Partition Based Nonparametric Priors
Illustration a Dirichlet Prior as a PB prior
As an illustration, we will write a Dirichlet prior as a partition
based prior on a partition determined by the data and compute the
posterior; we will see that we still obtain the correct answer.
Suppose that P ∼ Dα , X |P ∼ P. Let A = (X − , X + ) and let
α(A) > 0. Let B = {A, Ac }.
J. Sethuraman
Partition Based Nonparametric Priors
Illustration a Dirichlet Prior as a PB prior
As an illustration, we will write a Dirichlet prior as a partition
based prior on a partition determined by the data and compute the
posterior; we will see that we still obtain the correct answer.
Suppose that P ∼ Dα , X |P ∼ P. Let A = (X − , X + ) and let
α(A) > 0. Let B = {A, Ac }.
α(A)−1 α(Ac )−1
y1
Then rewrite Dα = H(B, h, Dα ) with h ∝ y1
and
J. Sethuraman
Partition Based Nonparametric Priors
Illustration a Dirichlet Prior as a PB prior
As an illustration, we will write a Dirichlet prior as a partition
based prior on a partition determined by the data and compute the
posterior; we will see that we still obtain the correct answer.
Suppose that P ∼ Dα , X |P ∼ P. Let A = (X − , X + ) and let
α(A) > 0. Let B = {A, Ac }.
α(A)−1 α(Ac )−1
Then rewrite Dα = H(B, h, Dα ) with h ∝ y1
y1
and a
blind application of the main theorem will show that the posterior
α(A) α(Ac )−1
is H(B, hX , Dα+δX ), with hX ∝ y1 y1
, which luckily
reduces to the correct answer, namely Dα+δX .
J. Sethuraman
Partition Based Nonparametric Priors
Bayes and Frequentist Estimates of the Survival Function
Bayes and W−S Estimates from AC Data Set (small) of Whittaker −Samaniego
1
0.9
0.8
WS Estimate
Survival Probability
0.7
0.6
0.5
0.4
0.3
0.2
Bayes Estimate
0.1
0
0
50
100
150
200
Failure Age (Hours)
250
300
350
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
2. The potential data may be censored,
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
2. The potential data may be censored, i.e. there are sets
A1 , A3 . . . . and we know the value of Xi if Xi ∈ Aci ,
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
2. The potential data may be censored, i.e. there are sets
A1 , A3 . . . . and we know the value of Xi if Xi ∈ Aci ,
otherwise we say the Xi is censored and we only know that Xi ∈ Ai .
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
2. The potential data may be censored, i.e. there are sets
A1 , A3 . . . . and we know the value of Xi if Xi ∈ Aci ,
otherwise we say the Xi is censored and we only know that Xi ∈ Ai .
3. Here the set An can depend on (X1 , Y1 , . . . , Xn−1 , Yn−1 , Yn ),
where Y1 , Y2 , . . . are the censoring variables.
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
2. The potential data may be censored, i.e. there are sets
A1 , A3 . . . . and we know the value of Xi if Xi ∈ Aci ,
otherwise we say the Xi is censored and we only know that Xi ∈ Ai .
3. Here the set An can depend on (X1 , Y1 , . . . , Xn−1 , Yn−1 , Yn ),
where Y1 , Y2 , . . . are the censoring variables.
4. The distribution of Yn given X1 , . . . , Xn , Y1 , . . . , Yn−1 is
independent of P, n = 1, 2, . . . (usual assumption in Bayesian
methods giving nice simplifications;
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 1
1. Suppose that, given P, the potential data X1 , X2 , . . . are i.i.d.
P.
2. The potential data may be censored, i.e. there are sets
A1 , A3 . . . . and we know the value of Xi if Xi ∈ Aci ,
otherwise we say the Xi is censored and we only know that Xi ∈ Ai .
3. Here the set An can depend on (X1 , Y1 , . . . , Xn−1 , Yn−1 , Yn ),
where Y1 , Y2 , . . . are the censoring variables.
4. The distribution of Yn given X1 , . . . , Xn , Y1 , . . . , Yn−1 is
independent of P, n = 1, 2, . . . (usual assumption in Bayesian
methods giving nice simplifications; can assume that Y1 , Y2 , . . .
are constants.)
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 2
Let P have prior H(B, h, G) and suppose that the censoring set A
satisfies A = ∪ri=1 Bi ; define yA = ∪ri=1 yi .
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 2
Let P have prior H(B, h, G) and suppose that the censoring set A
satisfies A = ∪ri=1 Bi ; define yA = ∪ri=1 yi .
Let us consider just only one potential observation X and also
assume that it is censored by A. (The uncensored case is the
standard nonparametric Bayesian problem.)
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 3
Theorem
Then the posterior distribution of P given that the potential data
X is censored,
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 3
Theorem
Then the posterior distribution of P given that the potential data
X is censored, is
H(B, h(y) · yA , G).
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 3
Theorem
Then the posterior distribution of P given that the potential data
X is censored, is
H(B, h(y) · yA , G).
(Do not need independence among PB1 , . . . , PBm ).
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 3
Theorem
Then the posterior distribution of P given that the potential data
X is censored, is
H(B, h(y) · yA , G).
(Do not need independence among PB1 , . . . , PBm ).
Theorem
If there are censoring sets A1 , . . . , An , each of which is a sum of
sets in the partition B,
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 3
Theorem
Then the posterior distribution of P given that the potential data
X is censored, is
H(B, h(y) · yA , G).
(Do not need independence among PB1 , . . . , PBm ).
Theorem
If there are censoring sets A1 , . . . , An , each of which is a sum of
sets in the partition B,then the posterior distribution of P given
that all the data X1 , . . . , Xn are censored
J. Sethuraman
Partition Based Nonparametric Priors
PB Priors and Censored Data - 3
Theorem
Then the posterior distribution of P given that the potential data
X is censored, is
H(B, h(y) · yA , G).
(Do not need independence among PB1 , . . . , PBm ).
Theorem
If there are censoring sets A1 , . . . , An , each of which is a sum of
sets in the partition B,then the posterior distribution of P given
that all the data X1 , . . . , Xn are censored is
H(B, h(y) · yA1 · · · yAn , G).
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors and Censored Data
Simpler results can be stated for PB Dirichlet priors and Dirichlet
priors.
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors and Censored Data
Simpler results can be stated for PB Dirichlet priors and Dirichlet
priors. We do not need the condition that the censoring sets are
sums of sets in the partition B.
In particular, suppose that P has a Dirichlet prior Dα . Let
A1 , . . . , An be the censoring sets for the potential data X1 , . . . , Xn ,
which are i.i.d. P. Let P
B = (B1 , . . . , Bm ) be the partition based on
A1 , . . . , An . Let YAi = j:Bj ⊂Ai yj , i = 1, . . . , n.
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors and Censored Data
Simpler results can be stated for PB Dirichlet priors and Dirichlet
priors. We do not need the condition that the censoring sets are
sums of sets in the partition B.
In particular, suppose that P has a Dirichlet prior Dα . Let
A1 , . . . , An be the censoring sets for the potential data X1 , . . . , Xn ,
which are i.i.d. P. Let P
B = (B1 , . . . , Bm ) be the partition based on
A1 , . . . , An . Let YAi = j:Bj ⊂Ai yj , i = 1, . . . , n.
Theorem
Then the posterior distribution of P given that all the data
X1 , . . . , Xn are censored is
D(B, h(y) · yA1 · · · yAn , α)
α(B1 )−1
where h(y) ∝ y1
α(Bm )−1
· · · ym
,
J. Sethuraman
Partition Based Nonparametric Priors
PB Dirichlet Priors and Censored Data
Simpler results can be stated for PB Dirichlet priors and Dirichlet
priors. We do not need the condition that the censoring sets are
sums of sets in the partition B.
In particular, suppose that P has a Dirichlet prior Dα . Let
A1 , . . . , An be the censoring sets for the potential data X1 , . . . , Xn ,
which are i.i.d. P. Let P
B = (B1 , . . . , Bm ) be the partition based on
A1 , . . . , An . Let YAi = j:Bj ⊂Ai yj , i = 1, . . . , n.
Theorem
Then the posterior distribution of P given that all the data
X1 , . . . , Xn are censored is
D(B, h(y) · yA1 · · · yAn , α)
α(B1 )−1
where h(y) ∝ y1
α(Bm )−1
· · · ym
,
and this is a PB Dirichlet distribution.
J. Sethuraman
Partition Based Nonparametric Priors
Dirichlet Priors
The moral of this talk is that if you start even from a
Dirichlet prior, and the data consists of general repair or
general censored data, then the posterior will be a PB
Dirichlet distribution.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -1
Susarla and Van Ruzin (1976) considered right censoring
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -1
Susarla and Van Ruzin (1976) considered right censoring (i.e. the
censoring set Ai = [ai , ∞) among random variables on [0, ∞).).
They also used a Dirichlet prior for P and obtained the expectation
of the df F (x) under the posterior distribution.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -1
Susarla and Van Ruzin (1976) considered right censoring (i.e. the
censoring set Ai = [ai , ∞) among random variables on [0, ∞).).
They also used a Dirichlet prior for P and obtained the expectation
of the df F (x) under the posterior distribution.
Their example has been revisited by many authors.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -1
Susarla and Van Ruzin (1976) considered right censoring (i.e. the
censoring set Ai = [ai , ∞) among random variables on [0, ∞).).
They also used a Dirichlet prior for P and obtained the expectation
of the df F (x) under the posterior distribution.
Their example has been revisited by many authors.
We will show how their example works with the use of PB priors.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -2
Their data set is
0.8, 1.0+, 2.7+, 3.1, 5.4, 7.0+, 9.2, 12.1+,
where a+ denotes that the censoring set was [a, ∞) and that
potential observation was censored.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -2
Their data set is
0.8, 1.0+, 2.7+, 3.1, 5.4, 7.0+, 9.2, 12.1+,
where a+ denotes that the censoring set was [a, ∞) and that
potential observation was censored.
They used a Dirichlet prior for P with parameter α, which was
8 times the exponential distribution with failure rate 0.12.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -2
Their data set is
0.8, 1.0+, 2.7+, 3.1, 5.4, 7.0+, 9.2, 12.1+,
where a+ denotes that the censoring set was [a, ∞) and that
potential observation was censored.
They used a Dirichlet prior for P with parameter α, which was
8 times the exponential distribution with failure rate 0.12.
The posterior distribution given the uncensored observations is
Dirichlet with parameter α∗ = α + δ0.8 + δ3.1 + δ5.4 + δ9.2 . We can
take this to be the prior distribution and say that the remaining
data 1.0+, 2.7+, 7.0+, 12.1+,
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example -2
Their data set is
0.8, 1.0+, 2.7+, 3.1, 5.4, 7.0+, 9.2, 12.1+,
where a+ denotes that the censoring set was [a, ∞) and that
potential observation was censored.
They used a Dirichlet prior for P with parameter α, which was
8 times the exponential distribution with failure rate 0.12.
The posterior distribution given the uncensored observations is
Dirichlet with parameter α∗ = α + δ0.8 + δ3.1 + δ5.4 + δ9.2 . We can
take this to be the prior distribution and say that the remaining
data 1.0+, 2.7+, 7.0+, 12.1+, are all censored.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example - 3
The partition formed by the censoring sets is
B = (B1 , . . . , B5 ) = ([0, a1 ), [a2 , a3 ), . . . , [a4 , ∞))
= ([0, 1.0), [1.0, 2.7), [2, 7, 7.0), [7.0, 12.1), [12.1, ∞)).
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example - 3
The partition formed by the censoring sets is
B = (B1 , . . . , B5 ) = ([0, a1 ), [a2 , a3 ), . . . , [a4 , ∞))
= ([0, 1.0), [1.0, 2.7), [2, 7, 7.0), [7.0, 12.1), [12.1, ∞)).
The posterior distribution given all the data is therefore
D(B, h(y)Y2 Y3 Y4 Y5 , α∗ )
where
Yj = yj + · · · + y5 , j = 2, . . . , 5 are the tail sums of the y ’s and
α∗ (Bj )−1
h(y) ∝ ×51 yj
.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example - 4
The expectation of P(Bj ) under this posterior is the expectation of
yj under the finite dimensional distribution with pdf proportional to
h(y)Y2 Y3 Y4 Y5 .
Q
The transformation yj = zj 1j−1 (1 − zr ), j = 1, . . . , 4 makes
z1 , . . . , z4 into independent Beta variables. This makes the
calculation of the expected value of P(Bj ) very easy.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example - 4
The expectation of P(Bj ) under this posterior is the expectation of
yj under the finite dimensional distribution with pdf proportional to
h(y)Y2 Y3 Y4 Y5 .
Q
The transformation yj = zj 1j−1 (1 − zr ), j = 1, . . . , 4 makes
z1 , . . . , z4 into independent Beta variables. This makes the
calculation of the expected value of P(Bj ) very easy.
The expected value of P([a, ∞)) can be computed just as easily
for any a ∈ Bi since
E (P([a, ∞))) = E (yi )E (PBi ([a, ai ))) + E (Yi+1 )
(Explain independence)
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example - 4
The expectation of P(Bj ) under this posterior is the expectation of
yj under the finite dimensional distribution with pdf proportional to
h(y)Y2 Y3 Y4 Y5 .
Q
The transformation yj = zj 1j−1 (1 − zr ), j = 1, . . . , 4 makes
z1 , . . . , z4 into independent Beta variables. This makes the
calculation of the expected value of P(Bj ) very easy.
The expected value of P([a, ∞)) can be computed just as easily
for any a ∈ Bi since
E (P([a, ∞))) = E (yi )E (PBi ([a, ai ))) + E (Yi+1 )
(Explain independence) and
E (PBi ([a, ai ))) = α∗ ([a, ai ))/α∗ ([ai−1 , ai )).
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -1
Two way censoring
Suppose that we write the data from the example of Susarla-Van
Ryzin as
0.8−, 1.0+, 2.7+, 3.1−, 5.4−, 7.0+, 9.2−, 12.1+.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -1
Two way censoring
Suppose that we write the data from the example of Susarla-Van
Ryzin as
0.8−, 1.0+, 2.7+, 3.1−, 5.4−, 7.0+, 9.2−, 12.1+.
where a− indicates that a potential data was censored to the
interval [0, a) (i.e. right censored).
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -1
Two way censoring
Suppose that we write the data from the example of Susarla-Van
Ryzin as
0.8−, 1.0+, 2.7+, 3.1−, 5.4−, 7.0+, 9.2−, 12.1+.
where a− indicates that a potential data was censored to the
interval [0, a) (i.e. right censored).
The data generates a partition B = (B1 , . . . , B9 ) of 9 intervals.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended - 2
If we use a Dirichlet prior Dα , it can also be viewed as PB Dirichlet
on this partition. The prior distribution of
P(B) = (P(B1 ), . . . , P(B9 )) is the finite dimensional Dirichlet with
pdf proportional to
α(B )−1
h(y) = ×91 yi i .
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended - 2
If we use a Dirichlet prior Dα , it can also be viewed as PB Dirichlet
on this partition. The prior distribution of
P(B) = (P(B1 ), . . . , P(B9 )) is the finite dimensional Dirichlet with
pdf proportional to
α(B )−1
h(y) = ×91 yi i .
The posterior distribution of P is a PB Dirichlet distribution
D(B ∗ , h(y)Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 , α)
whereP
P
Zj = j1 yi , Yj = 9j yi .
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended - 2
If we use a Dirichlet prior Dα , it can also be viewed as PB Dirichlet
on this partition. The prior distribution of
P(B) = (P(B1 ), . . . , P(B9 )) is the finite dimensional Dirichlet with
pdf proportional to
α(B )−1
h(y) = ×91 yi i .
The posterior distribution of P is a PB Dirichlet distribution
D(B ∗ , h(y)Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 , α)
whereP
P
Zj = j1 yi , Yj = 9j yi .
This also illustrates how to handle any kind of censoring in an
arbitrary space.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -3
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -3
A closed form expression for E (yj ) under the posterior distribution
is not available. However, it is just
Eh (yj Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 )
.
Eh (Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 )
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -3
A closed form expression for E (yj ) under the posterior distribution
is not available. However, it is just
Eh (yj Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 )
.
Eh (Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 )
where Eh denotes expectation under the finite dimensional
Dirichlet distribution D(α(B1 ), . . . , α(B9 )).
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -3
A closed form expression for E (yj ) under the posterior distribution
is not available. However, it is just
Eh (yj Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 )
.
Eh (Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9 )
where Eh denotes expectation under the finite dimensional
Dirichlet distribution D(α(B1 ), . . . , α(B9 )).
We can generate samples (y1r , . . . , y9r ), r = 1, . . . , N from h(y) by
using independent Gamma random variables and approximate the
above as
PN
r r r r r r r r r
r =1 yj Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9
r r r r r r r r
r =1 Z1 Y3 Y4 Z5 Z5 Y7 Z8 Y9
PN
.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -4
This method is justified by the Law of Large Numbers and good
rates of convergence are well known.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -4
This method is justified by the Law of Large Numbers and good
rates of convergence are well known.
It does not use imputation and MCMC methods, which add
another level of randomness in the final answer.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -4
This method is justified by the Law of Large Numbers and good
rates of convergence are well known.
It does not use imputation and MCMC methods, which add
another level of randomness in the final answer.
This method will handle any kind of censoring and is applicable to
distributions in multidimensional spaces.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -4
This method is justified by the Law of Large Numbers and good
rates of convergence are well known.
It does not use imputation and MCMC methods, which add
another level of randomness in the final answer.
This method will handle any kind of censoring and is applicable to
distributions in multidimensional spaces.
Of course, we can estimate many features other than just the
distribution function, i.e median, quantiles, etc.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -5
We use our method to recalculate the estimates of F (t) based on
Susarla-Van Ryzin data. When the transformation to independent
Beta variables is used we get the same results as Susarla and Van
Ryzin. However, this depends on the fact that all the censoring
were right censoring.results.
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -5
We use our method to recalculate the estimates of F (t) based on
Susarla-Van Ryzin data. When the transformation to independent
Beta variables is used we get the same results as Susarla and Van
Ryzin. However, this depends on the fact that all the censoring
were right censoring.results.
We also use on LLN method to compute estimates of F (t). A
comparison of the results is given below:
F̂ (t)
t
0.80
1.00+
2.70+
3.10
5.40
7.00+
9.20
12.10+
Exact
0.1083
0.1190
0.2071
0.3006
0.4719
0.5256
0.6823
0.7501
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -5
We use our method to recalculate the estimates of F (t) based on
Susarla-Van Ryzin data. When the transformation to independent
Beta variables is used we get the same results as Susarla and Van
Ryzin. However, this depends on the fact that all the censoring
were right censoring.results.
We also use on LLN method to compute estimates of F (t). A
comparison of the results is given below:
F̂ (t)
t
0.80
1.00+
2.70+
3.10
5.40
7.00+
9.20
12.10+
Exact
0.1083
0.1190
0.2071
0.3006
0.4719
0.5256
0.6823
0.7501
LLN
0.1082
0.1189
0.2068
0.3000
0.4708
0.5244
0.6810
0.7487
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -5
We use our method to recalculate the estimates of F (t) based on
Susarla-Van Ryzin data. When the transformation to independent
Beta variables is used we get the same results as Susarla and Van
Ryzin. However, this depends on the fact that all the censoring
were right censoring.results.
We also use on LLN method to compute estimates of F (t). A
comparison of the results is given below:
F̂ (t)
t
0.80
1.00+
2.70+
3.10
5.40
7.00+
9.20
12.10+
Exact
0.1083
0.1190
0.2071
0.3006
0.4719
0.5256
0.6823
0.7501
LLN
0.1082
0.1189
0.2068
0.3000
0.4708
0.5244
0.6810
0.7487
SSS
0.1083
0.1190
0.2084
0.3011
0.4706
0.5261
0.6802
0.7476
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example, Extended -5
We use our method to recalculate the estimates of F (t) based on
Susarla-Van Ryzin data. When the transformation to independent
Beta variables is used we get the same results as Susarla and Van
Ryzin. However, this depends on the fact that all the censoring
were right censoring.results.
We also use on LLN method to compute estimates of F (t). A
comparison of the results is given below:
F̂ (t)
t
0.80
1.00+
2.70+
3.10
5.40
7.00+
9.20
12.10+
Exact
0.1083
0.1190
0.2071
0.3006
0.4719
0.5256
0.6823
0.7501
LLN
0.1082
0.1189
0.2068
0.3000
0.4708
0.5244
0.6810
0.7487
SSS
0.1083
0.1190
0.2084
0.3011
0.4706
0.5261
0.6802
0.7476
KME
0.1250
0.1450
0.1250
0.3000
0.4750
0.4750
0.7375
0.7375
J. Sethuraman
Partition Based Nonparametric Priors
Susarla-Van Ryzin Example with Two-way Censoring
We take the data from Susarla-Van Ryzin and change the
uncensored observations to be left censored. We use the same prior
distribution for P and obtain estimates of F (t) using the LLN
approximation.
F̂ (t)
t
0.80-
1.00+
2.70+
3.10-
5.40-
7.00+
9.20-
12.10+
LLN
0.1737
0.1937
0.3625
0.3968
0.5274
0.5990
0.6796
0.7721
J. Sethuraman
Partition Based Nonparametric Priors
Graphical Comparisons
Estimates of the df from censored (all types) data, based on PB priors
0.9
0.8
Two−way censoring
0.7
F(t)
0.6
Prior df
0.5
0.4
0.3
Right censoring
0.2
0.1
0
0
5
t
10
15
J. Sethuraman
Partition Based Nonparametric Priors
Some References
Ferguson, T. (1973) A Bayesian Analysis of Some Nonparametric
Problems. Ann. Statist. 1, 209–230.
Ferguson, T. and Phadia, E. (1979) Bayesian Nonparametric
Estimation Based on Censored Data Ann. Statist. 7, 163–186.
Doksum, K. (1974) Tailfree and Neutral and Random Probabilities
and their Posterior Distributions. Ann. Probab. 2, 183–201.
Doss, Hani (1994) Bayesian Nonparametric Estimation for
Incomplete Data via Successive Substitution Sampling. Ann.
Statist. 22, 1763–1786.
Susarla, V. and Van Ryzin, J. (1976) Nonparametric Bayesian
Estimation of Survival Curves from Incomplete Observations. Jour.
Amer. Statist. Assoc. 71 897–902.
Sethuraman, J. and Hollander, M. (2008) Nonparametric Bayes
Estimation in Repair Models Jour. Statist. Plan. Inf. – To appear .