30 秒でわかる
Grant Sanderson explains why an event with probability p carries −log₂(p) bits of self-information, using successive reductions of equally likely possibilities.
- 1Halving possibilitiesHalving equally likely possibilities yields one bit.
- 2ProbabilityAn event of probability p has self-information −log₂(p) bits.
- 3The logarithmEntropy averages self-information across possible events.
ポイント
- Halving equally likely possibilities yields one bit.
- An event of probability p has self-information −log₂(p) bits.
- Entropy averages self-information across possible events.
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Information in bits
From halving possibilities to a logarithmic formula.
I introduce the bit as a unit for the information associated with an observed event. This short comes from a longer lesson that applies information theory to Wordle.
If an observation reduces equally likely possibilities by a factor of two, it provides one bit of information. Reducing them by a factor of four gives two bits, and by a factor of eight gives three.
For an event with probability p, the number of bits I satisfies p = 2^(−I). Equivalently, 2^I = 1/p.
Taking the base-two logarithm gives I = log₂(1/p) = −log₂(p). Less probable events carry more self-information under this definition.
This is the self-information of one observed event. Entropy is the expected self-information over a probability distribution, a related but distinct quantity.
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