  For pattern recognition like image recognition, it has become clear that each
machine-learning dictionary data actually became data in probability space
belonging to Euclidean space. However, the distances in the Euclidean space and
the distances in the probability space are separated and ununified when machine
learning is introduced in the pattern recognition. There is still a problem
that it is impossible to directly calculate an accurate matching relation
between the sampling data of the read image and the learned dictionary data. In
this research, we focused on the reason why the distance is changed and the
extent of change when passing through the probability space from the original
Euclidean distance among data belonging to multiple probability spaces
containing Euclidean space. By finding the reason of the cause of the distance
error and finding the formula expressing the error quantitatively, a possible
distance formula to unify Euclidean space and probability space is found. Based
on the results of this research, the relationship between machine-learning
dictionary data and sampling data was clearly understood for pattern
recognition. As a result, the calculation of collation among data and
machine-learning to compete mutually between data are cleared, and complicated
calculations became unnecessary. Finally, using actual pattern recognition
data, experimental demonstration of a possible distance formula to unify
Euclidean space and probability space discovered by this research was carried
out, and the effectiveness of the result was confirmed.
