Week 2 Uncertainty

• The sum probability of a world "small omega", in the set of all possible worlds "big omega", should equal 1.

  
• Unconditional Probability: refers to degree of belief in a proposition, without any other data.

  Example: rolling a fair die with a 1/6 probability.
• Conditional Probability: refers to degree of belief in a proposition, given that other evidence is provided.

  Example: rolling two dice with the sum of 10, knowing one dice already rolled a 4.
• Random Variable: probability variable with a domain of possible values.

  Example: weather = {sun, rain, snow, dry}
• Probability Distribution

  
• Independence: one event occurring doesn't affect the probability of the other

  Example: rolling a 2 on one dice, does not make it more/less likely another die will roll a 5.
• Bayes Rule

  probability of b given a, equals the probability of b times the probability of a given b, divided by the probability of a

  
• Probability Rule #1

  Negation

  probability of not a = 1 - probability a

  
• Probability Rule #2

  Inclusion-exclusion

  probability of a or b = probability of a add probability of b, minus probability a and b

  
• Probability Rule #4

  Marginalization of a random variable

  
• Probability Rule #3

  Marginalization

  probability a = probability of a and b, add the probability of a and not b

  allows for turning a joint probability -> into a single

  
• Probability Rule #5

  Conditioning

  probability of a = probability of a given b times probability b, add probability of a given not b times probability of not b

  turns a conditional probability -> into a single

  
• Probability Rule #5b

  Conditioning of a random variable

  
• Bayesian Network: data structure representing dependency between random variables

  directed graph connects nodes (representing random variables) with probability

  P (x | Parents (x))

  
• Markov Assumption:

  assumes a current state is only dependent on a finite number of previous states
• Markov Chain:

  sequence of random variables

  
• Computers can assume information in the hidden state, from observations

  
• Sensor Markov Assumption:

  assumes evidence variable depends only on the corresponding state
• Hidden Markov Model