Mathematics / advanced
Information, coding, and statistical limits
Quantify information through entropy, coding theorems, channel capacity, lossy compression, and statistical inequalities.
Your route through mathematics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 advanced
Information, coding, and statistical limits
Quantify information through entropy, coding theorems, channel capacity, lossy compression, and statistical inequalities.
- 01
Entropy, chain rules and mutual information
Measure finite-alphabet uncertainty in bits and distinguish information shared with a whole variable from information shared with its individual components.
33 min - 02
Relative entropy and data processing
Compare probability laws through divergence and explain why applying the same observation channel cannot increase distinguishability.
35 min - 03
Typical sequences and asymptotic equipartition
Derive iid typical-set probability and cardinality from the law of large numbers without treating a finite type class as the whole typical set.
36 min - 04
Source coding and Kraft's inequality
Use binary code trees to connect integer description lengths with entropy and distinguish lossless variable-length coding from almost-lossless block compression.
36 min - 05
Channel capacity and random coding
Derive finite-alphabet memoryless capacity as an asymptotic rate and understand how a random ensemble proves the existence of deterministic codes.
37 min - 06
Converse theorems and Fano's inequality
Turn residual message uncertainty into finite-block error bounds and identify the precise strength of a weak converse.
36 min - 07
Gaussian channels and water filling
Fix real-use and power conventions before allocating signal power across independent Gaussian noise modes.
39 min - 08
Rate–distortion theory and test channels
Trade block description rate against average distortion and distinguish an optimizing joint distribution from an implemented symbol-by-symbol codec.
40 min - 09
Hypothesis testing and error exponents
Separate fixed-type-I-error and Bayesian testing limits, with explicit hypothesis orientation and base-two exponents.
40 min - 10
Information bottlenecks and variational bounds
Preserve information relevant to a target while checking the direction, units and population scope of tractable variational objectives.
41 min