This Perl program computes the entropy $$H=-\sum_{n=1}^K p_n log_2(p_n)$$ of some discrete probability distributions. The things computed are the entropy of a single symbol alphabet, a four-symbol alphabet with probabilities $1/3$, $1/4$, $1/4$, $1/6$, an alphabet with four, six, and eight equally-likely symbols, a two-symbol alphabet with probabilities $1/x$ and $1 - 1/x$, and an n-symbol alphabet with probabilities $(1/2, 1/4, ..., 1/2^n, 1/2^n)$.

This is the output of the program:

Entropy of a single constant: 0 Entropy of alphabet with probabilities 1/3/, 1/4, 1/4, 1/6: 1.95914791702724 Entropy of four equally-likely symbols: 2 Entropy of six equally-likely symbols: 2.58496250072116 (log_2 (6) = 2.58496250072116) Entropy of eight equally-likely symbols: 3 Entropy of a two-symbol alphabet with probabilities 1/n, 1 - 1/n: 1 0 2 1 3 0.91829583405449 4 0.811278124459133 5 0.721928094887362 6 0.650022421648354 7 0.591672778582327 8 0.543564443199596 9 0.503258334775646 10 0.468995593589281 Entropy of an n-symbol alphabet, probs (1/2, 1/4, ... 1/(2^n), 1/(2^n)): n = 2: entropy = 1.5 n = 3: entropy = 1.75 n = 4: entropy = 1.875 n = 5: entropy = 1.9375 n = 6: entropy = 1.96875 n = 7: entropy = 1.98438 n = 8: entropy = 1.99219 n = 9: entropy = 1.99609 n = 10: entropy = 1.99805 n = 11: entropy = 1.99902

This is the program itself:

#!/home/ben/software/install/bin/perl use warnings; use strict; # A small error factor. use constant { eps => 1e-10, }; print "Entropy of a single constant:\n"; print &entropy (1), "\n\n"; print "Entropy of alphabet with probabilities 1/3/, 1/4, 1/4, 1/6:\n"; my @p = (1/3, 1/4, 1/4, 1/6); print &entropy (@p), "\n\n"; print "Entropy of four equally-likely symbols:\n"; my @p3 = (1/4) x 4; print &entropy (@p3), "\n\n"; print "Entropy of six equally-likely symbols:\n"; my @p2 = (1/6) x 6; print &entropy (@p2), " (log_2 (6) = ", log2 (6), ")\n\n"; print "Entropy of eight equally-likely symbols:\n"; my @p4 = (1/8) x 8; print &entropy (@p4), "\n\n"; print "Entropy of a two-symbol alphabet with probabilities 1/n, 1 - 1/n:\n"; for my $factor (1..10) { my @p5 = (1 - 1/$factor, 1/$factor); print $factor, " ", entropy (@p5), "\n"; } print "\n"; print "Entropy of an n-symbol alphabet, probs (1/2, 1/4, ... 1/(2^n), 1/(2^n)):\n"; for my $factor (1..10) { my @q; for my $j (0..$factor) { $q[$j] = 1/(2**($j+1)); } $q[$factor + 1] = 1/(2**($factor+1)); printf "n = %d: entropy = %g\n", $factor + 1, entropy (@q); } print "\n"; exit; # Log to the base 2, so log2 (2) = 1, log2 (4) = 2, etc. sub log2 { my ($n) = @_; return log ($n) / log (2); } # Given a set of discrete probabilities, "@p", compute the entropy. sub entropy { my (@p) = @_; # Do some checks on the inputs before proceeding. my $sum = 0; for (@p) { if ($_ < 0 || $_ > 1) { warn "$_ is not a probability.\n"; return undef; } $sum += $_; } if (abs ($sum - 1) > eps) { warn "@p does not sum to 1.\n"; return undef; } # The return value. my $h = 0; for (@p) { if ($_ == 0) { next; } $h += - $_ * log2 ($_); } return $h; }

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