Lotfi Zadeh wrote:
> X and Y are random variables taking values in the set (1, 2, ...,n).
> The entries in the joint probability matrix, P, are of the form
> "approximately aij," where the aij take values in the unit interval and add
> up to unity. What is the marginal probability distribution of X? Four special
> cases: (a) "approximately aij," is interpreted as an interval centering on
> aij; (b) "approximately aij," is interpreted as a triangular fuzzy number
> centering on aij; (c) "approximately aij," is interpreted as a uniform
> probability distribution over an interval centering on aij; and (d)
> "approximatelt aij," is interpreted as a triangular probability density
> function centering on aij.
Let me take the Bayesian stance and consider P_ij (corresponding to the
probability that X=i and Y=j) a random variable (probability). P is
defined through the approximate (imprecise) matrix A. In a Bayesian
context, A is specified through the prior, the likelihood function and
the data. In a fuzzy context, A is given directly.
Notice that the variable P_ij is _not independent_ of other variables
P_kl in the matrix P. Namely, the sum P_** should be equal to 1. We
should therefore reject all combinations that may be consistent with
A_**, but that result in a non-normalized P. It is a responsibility of A
to allow at least one valid combination.
I will not deal with problems a-d. Instead, I will assume that the n*n
matrix P is a normalized random vector. Furthermore, I will assume that
the Bayesian posterior of P has the Dirichlet distribution (also present
in Walley's approach). The Dirichlet distribution is specified in terms
of counts. The higher the counts, the "crisper" the P. The lower the
counts, the "fuzzier" the P.
Therefore:
p ~ Dirichlet(a_11,a_12,a_21,a_22,...,a_nn)
It is then easy to prove that
p_.* ~ Dirichlet(a_11+a_12+...+a_1n, ... , a_n1+a_n2+...+a_nn)
p_*. ~ Dirichlet(a_1i+a_2i+...+a_ni, ... , a_1i+a_2i+...+a_ni)
Best regards,
Aleks
--
dr. Aleks Jakulin
http://kt.ijs.si/aleks/
Department of Knowledge Technologies,
Jozef Stefan Institute, Ljubljana, Slovenia.
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