Information-Theoretic Bounds

Origin. Shannon (1948); Soviet contributions applying information-theoretic bounds to statistical inference (Kolmogorov), control (Glushkov), and complexity theory.

Mechanism. Fundamental limits on what can be achieved with information. Shannon's channel capacity bounds reliable communication. Kolmogorov complexity bounds description length. Entropy bounds prediction accuracy. These bounds are independent of algorithm — no method can beat them. Knowing the bound tells you whether to keep searching for better methods or accept the fundamental limit.

Procedure. Identify the information-theoretic problem: communication (channel capacity), compression (entropy), learning (sample complexity), control (Ashby's law). Compute or estimate the relevant bound. Compare current performance to the bound. If close to the bound, further improvement is marginal; effort is better spent elsewhere. If far from the bound, better algorithms exist in principle.

Applies to. System design, algorithm evaluation, feasibility analysis, knowing when to stop optimizing.

Limitations. Bounds are often asymptotic or existential — they guarantee something is possible without showing how to achieve it. The gap between the bound and what practical algorithms achieve can be large. Also: bounds depend on assumptions (memoryless channel, i.i.d. data); violated assumptions invalidate the bounds.