Information Theory Overview

Questions and Answers

Match the following terms with their definitions in Information Theory:

Self Information = The amount of information produced by a single event Probability P(E) = The likelihood of an event occurring Logarithmic function = Function that satisfies axioms of information content Axioms = Basic rules that govern the measurement of information

Match the following measurement units with their logarithm base:

Bits = Base 2 Nats = Base e Hartlys = Base 10 Logarithm base = Function used to measure information content

Match the following events with their information content:

Drawing a king of hearts from a pack of cards = 5 bits Drawing any card from a pack of 32 cards = Log2(32) bits Certain event occurring = 0 bits Drawing an event with probability 0.5 = -1 bit

Match the following statements with their corresponding information conversion relationships:

<p>1 hartlys = 1/log10(2) bits 1 nat = 1/log_e(2) bits 1 bit = 1/log2(10) hartlys Log10(1/P(x)) = Y log10(2) bits</p> Signup and view all the answers

Match the following concepts with their corresponding conditions:

<p>I(E) is 0 = If P(E) = 1 I(E∩F) = If E and F are independent events Decreasing function = I(E) must decrease with the increase of P(E) Information content = Expressed in different units based on logarithm base</p> Signup and view all the answers

Study Notes

Information Theory Overview

Information Theory studies the quantification, storage, and communication of information.
The fundamental concept in Information Theory includes self-information and uncertainty surrounding events.

Measurement of Information

Shannon defines the information content ( I(E) ) of an event ( E ) as a function of its probability ( P(E) ).
Axioms for information measurement:
- ( I(E) ) is a decreasing function of ( P(E) ).
- ( I(E) = 0 ) when ( P(E) = 1 ), indicating no new information from a certain event.
- If ( E ) and ( F ) are independent events, then ( I(E \cap F) = I(E) + I(F) ).

Logarithmic Function

The logarithmic function is the only function satisfying the axiom conditions for measuring information.
Information can be expressed as:
- ( I(E) = \log(P(E)) ) or ( I(E) = -\log(P(E)) )
Units of measurement depend on the logarithm base:
- Bits for base 2
- Nats for base ( e )
- Hartlys for base 10

Example Calculation

For a standard deck of 32 playing cards:
- Probability of drawing the king of hearts ( P(E) ) is ( 1/32 ).
- The amount of information ( I(E) ) is calculated as:
  - ( I(E) = \log_2 (1/P(E)) = \log_2 (32) = 5 ) bits.

Conversion of Measures

Standard conversions include:
- ( \log_2(1/P(x)) = y ) bits translates to ( 1/P(x) = 2^y ).
- For different logarithm bases, rearrangements help convert measures:
  - ( y = \frac{\log_{10}(1/P(x))}{\log_{10}(2)} ) bits
  - ( 1 ) hartlys = ( 1/\log_{10}(2) ) bits
  - ( 1 ) nat = ( 1/\log_{e}(2) ) bits
  - ( 1 ) bit = ( 1/\log_{2}(e) ) nats.

Additional Conversion Insights

Further conversion relations:
- ( 1 ) hartlys = ( 1/\log_{10}(e) ) nats
- ( 1 ) nat = ( 1/\log_{e}(10) ) hartlys
- ( 1 ) bit = ( 1/\log_{2}(10) ) hartlys.

Description

Explore the fundamentals of Information Theory in this quiz. Learn about concepts such as self-information, uncertainty, and the measurement of information as defined by Shannon's axioms. Test your understanding of how information content relates to probability.