is a function defined on
. To find
, there are several ways. In heuristic search, an estimation function
of
and a tree structure of X are introduced, then using
as a guidance to find the optimal solution
on the tree. The same problem can be solved by the quotient space method. First, we transform the problem of finding the optimal solution on space
into a set of its quotient spaces
. Then, by letting the grain-size of the quotient space [X] approaching to zero, then the optimal solution on [X] will approach to the optimal solution on X. If an estimation function of [
f ] on
is introduced, then we may solve the problem on the quotient spaces. From the above statement we can see the connection between heuristic search and quotient space problem solving methods.
On the other hand, in statistical inference some statistic is used to estimate function
. Thus, the statistical inference method can be used to judging which the solution is.
These mean that we can integrate heuristic search, quotient space method and statistical inference to form a new statistical heuristic search model – a quotient space model of statistical heuristic search.
Assume that
is a set of random variables in a basic probability space
,
is a sequence of hierarchical quotient spaces of
, and
are their corresponding equivalence relations, where
denotes that
is a quotient space of
, and
is a finite set.
, let
be a statistic of
. Based on
implementing a statistical inference S, if
is accepted and
, then
is a goal and success; otherwise fail.
If
, then
is partitioned into
by equivalence relation
. From
extracting statistic
implement statistic inference S based on
,…. Where, statistic
is extracted from a subset of
, so it represents the global information of the subset.
This is a quotient space model of statistical heuristic search.