Perhaps is used when one does not wish to be too definite or assertive in the expression of an opinion.
"perhaps not surprisingly, he was cautious about committing himself"
But, what is the meaning of the phrase perhaps not surprisingly in the above sentence? I hope you will give another usage example.
I take it that you understand that surprisingly, set in front of the main clause and separated from it with a comma, modifies the entire main clause:
Surprisingly, he was cautious about committing himself is equivalent to
It is surprising that he was cautious about committing himself.
Perhaps and not modify surprisingly:
Perhaps not surprisingly, ... is equivalent to
Perhaps it is not surprising that...
This might be said, for instance, if an ordinarily bold man were unusually cautious in some particular circumstance. At first thought this would surprise you; but thinking more deeply might suggest a possible reason for his caution, and you would realize that perhaps you should not be surprised.
