Trade-tape entropy versus a coin-flip no-skill baseline

Entropy as an uncertainty score

Information entropy is a bit-scale score of uncertainty in a sequence of symbols. It does not use what those symbols mean, so the same formula can treat the bits as coin tosses or as wins and losses.

A constant one-symbol sequence has entropy of zero. A fair two-outcome sequence matches a coin toss and has entropy of one bit.

When the sequence is a trade tape, the evaluation target is information entropy below the coin-flip no-skill baseline. That reading means there is less uncertainty about the next outcome.

Rank-1 entropy on a short string

On a 16-symbol two-outcome string, rank-1 entropy is 1 bit when the two symbols each appear eight times, and 0.99 bits when the counts are nine and seven.

The 1-bit reading is the coin-flip no-skill baseline for that alphabet. The 0.99-bit reading sits only a short step under that equal-probability ceiling.

Entropy of rank and percent below random

Pairing symbols raises the no-skill ceiling to 2 bits. The more patterned of the two example strings sat 4.64% below that ceiling versus 2.04% for the flatter string, and triple grouping widened the gap to 14.78% versus 1.66%.

Those distances are percent-below-random scores after entropy of rank is computed on consecutive pairs or triples. Grouping can surface pairwise or longer dependencies that single-symbol counts miss.

How a closed tape is encoded

A closed trade list can be encoded as win versus loss, or as win, loss, and a near-zero band. That win-loss-breakeven alphabet fills the rank-1 table, and consecutive pairs of those labels become rank-2 elements.

Sign-only entropy ignores profit and loss size. Magnitude enters the same test only after returns are placed in quantized return bins such as larger than 2% gain, 1 to 2% gain, near zero, 1 to 2% loss, and worse than 2% loss.

Expected value beside the ranks

Mathematical expectation is kept beside the entropy ranks because a tape can still have usable expected value when label frequencies sit near 50% for two symbols or 33.3% for three, provided the reward-to-risk ratio is greater than 1.

Expected value is a separate dollar filter. It asks whether reward-to-risk still bounds the next loss when the sequence looks random.