The name is absent

86

the distribution on partitions induced by ties corresponds to a PPM with cohesions
c(Sfc) = a × (#Sfc — 1)!∙ Here #A denotes the cardinality of a set A and a is the total
mass parameter of the DP prior.

A feature of the PPM induced by the DP is that it is a priori exchangeable with
respect to the experimental units, which makes it inadequate for our application.

We now define a non-exchangeable PPM (NEPPM). In particular, applied to the
clustering of disease sub-types {1,... ,n} we define a model that increases the prior
probability of any two subtypes i and i' with equal prognoses x_i = x_i> to cluster
together.

In general, we define a probability model for random partitions of experimental
units {1,..., n} with categorical covariates x_i ∈ {1,..., Q} such that clusters with
homogeneous covariates are encouraged a priori.

We define

к

Pr{p_n = (Si,... ,S_κ∖) oc JJc(S_fc), where c(S_fc) = c_D(Sfc)d(Sfc), (4.3)
fc=l

Cn(Sfc) = a(#Sk — 1)! is the cohesion induced by the DP and

(i—rQ t ∖

∏q=l ^l_fcg! ʌ                               ,   .

(#Sfc + Q-l)!y  ∙                      ⁽ ' ^}

Here Q is the number of different categories, m_fcg the number of experimental units of
category q in the cluster S_fc and 7 is a nonnegative constant, common to all cohesions,
that gives strength to the cohesion of homogeneous clusters. We refer to d(S_fc) as a
“similarity function.” It serves the purpose of increasing the probability of forming
clusters with more homogeneous covariate values Xi, i ∈ Sk- The higher the value
of 7, the stronger the prior emphasis on homogeneous clusters. More homogeneous
clusters Sk have larger d(Sk). In our specific application, we have Q = 3 prognoses,
and mk_q for q = —1,0,1, are, respectively, the numbers of sarcoma subtypes with
poor, intermediate and good prognosis in the cluster Sk- As desired, the resulting
prior probability model is non-exchangeable.

<<   95   96   97   98   99   100   101   >>

More intriguing information