A Bayes method of a Monotone Hazard Rate via Spaths

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A Bayes method of a Monotone Hazard Rate via
S-paths

• Man-Wai Ho
• National University of Singapore

Cambridge, 9th August 2007
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Agenda

• Overview of Estimation of Monotone Hazard Rates
• What is an S-path?
• A class of random monotone (non-decreasing or
  non-increasing) hazard rates
• Posterior analysis via S-paths
• A Markov chain Monte Carlo (MCMC) method
• Numerical Examples

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Hazard Rate / Hazard Function
Lifetime

• The hazard rate at time ,
• is interpreted as the instantaneous probability
  of failure of an object.
• A wide variety of shapes
• The simplest case of a constant hazard rate
  corresponds to an exponential lifetime
  distribution.
• Cases of increasing or decreasing hazard rate
  lifetime distributions that are of a heavier or
  lighter tail, respectively, compared with an
  exponential distribution.

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Estimation of Monotone Hazard Rates Bayesian
Approach

• A non-increasing (or decreasing) hazard rate on
  the positive line may be written in the
  mixture form
• where is the indicator function of a
  set .
• The unknown measure is modeled as a random
  measure / random process.
• for a non-decreasing hazard rate.

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Motivation of this Work

• Bayes estimate (posterior mean) of an increasing
  hazard rate as a sum over e-vectors Dykstra
  Laud (1981) or m-vectors Lo Weng (1989)
  based on extended / weighted Gamma processes.
• Lo Weng (1989) characterized the posterior
  distribution of any random hazard rate
  
• with being a weighted gamma process in
  terms of random partitions p.

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Motivation of this Work

• Recently, James (2002, 2005) generalized the
  result of Lo Weng (1989) about general hazard
  rates
• by modeling as a completely random measure
• Analogously obtained a characterization of the
  posterior distribution in terms of partitions p.
• A completely random measure Kingman (1967,
  1993) includes Gamma process, weighted Gamma
  process, generalized gamma process Brix (1999),
  stable process and many other random measures as
  special cases.

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Motivation of this Work
General Hazard Rates
?
Partitions structure
S-paths structure
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What is an S-path?

• An integer-valued vector
  satisfying
• Denote

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What is an S-path?

• Served as an alternative for clustering
  integers provided that only (i) number of
  elements, and (ii) the maximum index, in each
  cluster are concerned.
• Suppose The path
  conveys
• One cluster of 2 integers with 3 as the maximum
  index
• Another cluster of 2 integers with 4 as the
  maximum index
• A combinatorial reduction of p, e.g., if

correspondence
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What is an S-path?

• One of the advantages of using S-paths over
  partitions for a fixed , the space of all S
  is much less than the space of all p ( )

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A Class of Random Decreasing Hazard Rate

• Consider a class of random decreasing hazard
  rates on defined by
• where is a completely random measure on
  .
• This class contains the decreasing counterpart of
  the models considered by Dykstra Laud (1981)
  based on extended/weighted Gamma process

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The Completely Random Measure

• An Independent increment process uniquely
  characterized by the Laplace functional
• Alternatively, can be represented in a
  distributional sense as
• where is a Poisson random
  measure on with intensity measure

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The Data

• The data is observed upon a right-censorship
  scheme.
• Suppose we collect observations, denoted by
  ,
  from items with monotone hazard rates until
  time .
• are
  completely observed failure times, and
• 
  are the right-censored times.

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Posterior Analysis

• The likelihood is given by Aalen (1975, 1978)
• where
• The posterior distribution of is
  proportional to the product of the likelihood and
  the prior

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Posterior Analysis

• A streamline proof
• Note
• Augment the latent variables
  and work with the joint
  distribution of
• Apply Proposition 2.3 in James (2005) to get a
  nice product form
• recognize from the posterior distribution that
  the information carried by a partition about the
  remaining members other than the maximal element
  in any cell is irrelevant when

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Posterior Analysis

• Theorem 1 The posterior law of given the
  data can be described by a three-step experiment
• An S-path
  has a distribution
• where for any integer
• and

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Posterior Analysis

• Given S, there exists
  independent pairs of , denoted
  by
• where is distributed as
• Given has a distribution
  as
• where is a completely random measure
  characterized by

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The Bayes Estimate

• The Bayes estimate (posterior mean) of a
  decreasing hazard rate given is given by
• where
• if otherwise 0.

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The Bayes Estimate

• The posterior mean is a sum over all S-paths with
  coordinates. The computation is
  formidable even when even though
  the total number of S is smaller than that of p
• NO exact simulation method for S!
• Possible strategies
• Develop Markov chain Monte Carlo (MCMC)
  algorithms or sequential importance sampling
  (SIS) methods for sampling S (not
  straightforward!)
• Use algorithms for sampling p or latent
  variables?

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The Bayes Estimate

• Consistent estimate due to the weak consistency
  of the posterior by Dragichi Ramamoorthi (2003)
• Always less variable than as a
  specialization of James (2002, 2005)
• according to a Rao-Blackwellization argument
• by a discrete uniform conditional distribution
  of pS,T and constancy of for all

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An MCMC method

• To draw a Markov chain of S-paths, which has a
  unique stationary distribution given by
• Generalize an accelerated path (AP) sampler Ho
  (2002) -- an efficient MCMC method for sampling
  S-paths in Bayesian nonparametric models based on
  Dirichlet process and Gamma process.
• An improvement over a naïve Gibbs sampler Ho
  (2002) for S-paths.

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The AP Sampler

• A transition cycle contains steps.
• At step
• Obtain
  from the current path

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The AP Sampler

• Note that the current S is given by
• where
• Then, the new S will be
• for
  , with (transition)
  probability proportional to
• Repeat for steps
  to finish one cycle

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The AP Sampler the Transition Probabilities

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Evaluation of the Bayes Estimate

• Start with an arbitrary path
• Repeat cycles to yield a Markov chain
• Compute the ergodic average
• to approximate the sum
• in the posterior mean of the monotone hazard rate

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Rationale behind the AP Sampler

• Markov chain is defined by a transition kernel
• A Markov chain has a unique stationary
  distribution
• Irreducible transition kernel all states in the
  space communicate with each other within one
  cycle
• Construct reducible kernels such
  that the stationary distribution of the resulted
  chain is the target distribution and the product
  of them is irreducible Hastings (1970 Tierney
  (1994)

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Numerical Examples

• Gamma process for (i.e.,
  ) with shape measure
• Lifetime data of an item with a hazard rate
• Data are generated subject to the
  censoring rate is about 15
• Monte Carlo size is
• Initial path

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Comparisons between Different Methods

• Many commonly-used partition-based methods Lo,
  Brunner Chan (1996) Ishwaran James (2003)
  James (2005)
• Replicate 1000 independent hazard estimates by
  each of the three available methods (i) the AP
  sampler, (ii) the naïve Gibbs path (gP) sampler,
  and (iii) the gWCR sampler in Lo, Brunner and
  Chan (1996).

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Comparisons between Different Methods

• At different time points, the three averages are
  close to each other, yet the standard errors vary
  substantially.
• The standard error of hazard rate estimates
  produced by the AP sampler is the smallest among
  all the three methods.
• The AP sampler definitely outweighs the naive
  Gibbs path sampler and beats the closest
  competitor, the gWCR sampler, by a comfortable
  margin.

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Conclusions

• Tractable posterior distribution and Bayes
  estimate in terms of S-paths
• A Rao-Blackwellization result for S over p
• An efficient numerical method for sampling
  S-paths
• Accentuate the importance of study and usage of
  S-paths in models with monotonicity constraints
  (e.g., symmetrical unimodal density, monotone
  density, unimodal density, bathtub-shaped hazard
  rate,)

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THANK YOU!