2 - Uncertainty

Concepts:

• Conditional Probability: P(a|b) =\frac{P(a\land b)}{P(b)}=\alpha P(a\land b)
• Random Variables
• Independence: P(a\land b)=P(a)P(b)
• Bayes’ Rule: P(b|a)=\frac{P(b)P(a|b)}{P(a)}
• Joint Probability
• Probability Rules:
  • Negation: P(\neg a)=1-P(a)
  • Inclusion-Exclution: P(a\lor b)=P(a)+P(b)-P(a\land b)
  • Marginalization:
    • P(a)=P(a,b)+P(a,\neg b)
    • P(X=x_{i})=\sum_{j} P(X=x_{i}, Y=y_{j})
  • Conditioning:
    • P(a) =P(b)P(a|b) + P(\neg b)P(a|\neg b)
    • P(X=x_{i})=\sum_{j}P(Y=y_{j})P(X=x_{i}|Y=y_{j})

Bayesian Network

A Bayesian network is a data structure(directed graphs) that represents the dependencies among random variables.

Concepts:

• Inference
  • Query, Evidence variables, Hidden variables, The goal
• Inference by Enumeration:
  P(X|e)=\alpha P(X,e)=\alpha \sum_{y}P(X,e,y)
• Approximate Inference (Sampling):
  • Each variable is sampled for a value according to its probability distribution.
  • Likelihood Weighting:
    • Fixing the evidence variables.
    • Sample the non-evidence variables.
    • Weight each sample by its likelihood: the probability of all the evidence occurring.

Markov Models

Concepts:

• Markov Assumption: the current state depends on only a finite fixed number of previous states.
• Markov Chain: a sequence of random variables where the distribution of each variable follows the Markov assumption.
• Observation and Hidden State
  • Hidden Markov Models: Hidden states generate observed event (Sensor model).
  • Sensor Markov Assumption: The evidence variable depends only on the corresponding state.
• Tasks:

Tasks	Definition
Filtering	Given observations from start until now, calculate the probability distribution for the current state.
Prediction	Given observations from start until now, calculate the probability distribution for a future state.
Smoothing	Given observations from start until now, calculate the probability distribution for a past state.
Most likely explanation	Given observations from start until now, calculate most likely sequence of events.