Uncertainty & Probabilistic Reasoning Course

Uncertainty & Probabilistic Reasoning

• This course covers Probabilistic Reasoning, focusing on its intuitive and mathematical concepts.
• It includes Bayes' Theorem, its applications in decision-making under uncertainty, and its use in probabilistic models for data analysis.
• The course also discusses Markov Chains and Hidden Markov Models for time series analysis and pattern recognition, evaluating their model performance.

Objectives

• Students will be able to explain probabilistic reasoning, including Bayes' Theorem and its use in decision-making.
• Students will be able to use probabilistic models based on Bayes' Theorem to represent and analyze uncertainty in various situations (e.g., classification, diagnosis, prediction).
• Students will be able to apply Markov Chains and Hidden Markov Models to analyze time series and patterns, assessing model performance.

Reasoning Under Uncertainty

• Uncertainty plays a crucial role in various situations, including travel delays, insurance claims, object recognition, game playing, and medical diagnoses.
• Systems capable of understanding uncertainty and its effects perform better than those that do not.
• Uncertainty should be well-represented for effective outcomes.

Two (Toy) Examples

• Uncertainty is inherent in many events, prompting the investigation of their possible causes.
• When presented with new evidence, the likelihood associated with each hypothesised cause is subject to change.
• Probabilistic reasoning quantifies the likelihood of a hypothesised cause. A numerical representation captures the likelihood of the cause.

Probability Theory: Variables and Events

• A random variable represents an observation, outcome, or event with an uncertain value.
• The set of all possible outcomes of a random variable is its domain.
• Boolean random variables have two outcomes.

Probability Theory: Atomic Events

• Atomic events are complete specifications of random variable values.
• The set of all possible atomic events is exhaustive and mutually exclusive.

Probability Theory: Probabilities

• Probabilities of random variable outcomes are assigned values between 0 and 1.
• Probabilities of necessarily true events equal 1. Probabilities of impossible events equal 0.
• Probability of a disjunction of two events is (P(A)+P(B)-P(A and B).

Probability Theory: Relation to Set Theory

• Probability theory principles can be intuitively grasped through set theory. Probability of a union of two sets can be determined using the relation of their set intersections.

Probability Theory: Conditional Probability

• Conditional probability expresses the likelihood of an event given another event.
• Independence implies that one event doesn't affect the likelihood of another. The conditional probability of event a, given event b, is equal to the probability of event a if they are independent.

Combining Probabilities: The Product Rule

• The Product Rule combines basis and conditional probabilities to determine the likelihood of multiple events occurring together.

Bayes' Rule

• Bayes' rule provides a way to calculate the likelihood of a hypothesis given the evidence.
• It's fundamental for modern probabilistic AI, enabling effective inference.

Bayes' Rule: Combining Evidence

• Applying Bayes' Rule allows combining multiple pieces of evidence.
• Complex problems involving multiple effects lead to more intricate causal models.

Bayes' Rule + Conditional Independence

• Conditional independence of evidence makes models computationally more manageable when dealing with multiple evidence parameters. A cause can lead to multiple effects. However, the effects alone cannot lead to the cause. This principle helps create more compact causal models.

Bayes' Nets

• Bayes' nets visually represent conditional independence relationships, compressing complex models with many parameters.

The Bayesian Brain

• The human brain's decision-making processes effectively implement Bayes' rule.
• The brain combines sensory data to determine the probability of possible outcomes.

The Non-Bayesian Brain

• Some inferencing processes are not easily explained by probabilistic reasoning alone.
• A system where one wheel's rotation affects another may not have a probabilistic basis.

Probabilistic Reasoning Over Time

• Methods for analyzing time-dependent or sequential data, like Markov Chains (MC) and Hidden Markov Models (HMM), are discussed.

Markov Chain

• A Markov Chain is a method for representing and analyzing temporal or sequential data.
• The likelihood of the next state depends solely on the current state, not earlier states.

Markov Chain: Weather Prediction Example

• A Markov Chain can model the probability of future weather conditions based on past sequences of weather.
• This example demonstrates how to use Markov Chains to predict future events based on past sequences.

Markov Chain: Example Exercise

• Analyzing probability of a sequence of weather events. Several calculations are needed to determine the probability of a future sequence of weather events based on knowledge of earlier sequences.

Markov Chains

• A Markov Chain is used to predict future outcomes due to preceding events.

Markov Assumption

• The Markov Model Assumption states that the probability of the next state depends solely on the current state, not past states.

Markov Model

• A Markov Model describes dependencies of current information on previous information, focusing on sequential data.
• It is composed of several elements including states, transition schemes, and the emission of outcomes, or outputs.

Hidden Markov Model (HMM)

• HMM describes systems where state transitions are unobservable. Observed sequences of outputs are derived from these unseen states.

HMM Mathematical Model

• Bayes' Theorem calculations are used to determine the probability of hidden states given observed outputs.
• Markov Property and Independent Observations assumptions are applied to simplify computations in HMM models.

HMM Parameters

• HMM models use parameters to determine probabilities related to state transitions, outputs, and starting states.
• Probability calculations use Matrix multiplication.

Three Basic Problems of HMM

• Probability Evaluation: calculating the probability of observed output sequences given an HMM model.
• Optimal State Sequence: identifying optimal hidden state sequences corresponding to observed output sequences.
• Parameter Estimation: adjusting model parameters to best match observed output sequences.

HMM Example: Coins & Dice

• A fictional example demonstrating a way to model sequential data processes with an HMM based on dice throwing.

Types of HMMs

• Categorizes different HMM structures based on how states interact.

Case Study

• Real-world applications of HMM models illustrate various use cases.

Summary

• Uncertainty and probabilistic reasoning are integral aspects of AI.
• Statistical approaches like Bayes' Rule and HMMs are used to model probabilistic reasoning and uncertainty in a more systematic way than purely logical approaches.
• HMMs are useful in modeling sequential or temporally ordered data.