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Questions and Answers
What is a characteristic of a Hidden Markov Model?
What is a characteristic of a Hidden Markov Model?
In a Hidden Markov Model, what does the observation sequence represent?
In a Hidden Markov Model, what does the observation sequence represent?
Which of the following best describes the relationship between the observations and hidden states in a Hidden Markov Model?
Which of the following best describes the relationship between the observations and hidden states in a Hidden Markov Model?
What type of model is a Hidden Markov Model categorized as?
What type of model is a Hidden Markov Model categorized as?
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What can NOT be observed in the scenario presented for the Hidden Markov Model?
What can NOT be observed in the scenario presented for the Hidden Markov Model?
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What is the probability of transitioning from a Sunny day to a Rainy day?
What is the probability of transitioning from a Sunny day to a Rainy day?
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If today is Rainy, what is the likelihood that tomorrow will be Cloudy?
If today is Rainy, what is the likelihood that tomorrow will be Cloudy?
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Which of these statements correctly describes first-order Markov models in the context of weather?
Which of these statements correctly describes first-order Markov models in the context of weather?
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What can be inferred about the transition from Cloudy to Sunny based on the given probabilities?
What can be inferred about the transition from Cloudy to Sunny based on the given probabilities?
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If today is Foggy, what is the probability that tomorrow will be Sunny?
If today is Foggy, what is the probability that tomorrow will be Sunny?
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What is the sum of the probabilities of transitioning to any weather condition from a Cloudy day?
What is the sum of the probabilities of transitioning to any weather condition from a Cloudy day?
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Given today's weather is Sunny, what is the probability that tomorrow is both Sunny and then Rainy the day after?
Given today's weather is Sunny, what is the probability that tomorrow is both Sunny and then Rainy the day after?
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What does the Bayesian brain theory suggest about how the brain processes sensory information?
What does the Bayesian brain theory suggest about how the brain processes sensory information?
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Which of the following statements accurately reflects a limitation of Bayes' rule in brain function?
Which of the following statements accurately reflects a limitation of Bayes' rule in brain function?
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In a Bayesian framework, which of the following best describes the nature of the information utilized?
In a Bayesian framework, which of the following best describes the nature of the information utilized?
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What is a possible application of a Markov Chain as mentioned in the content?
What is a possible application of a Markov Chain as mentioned in the content?
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Which of the following concepts is described as being utilized alongside Bayes’ rule?
Which of the following concepts is described as being utilized alongside Bayes’ rule?
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What does the statement 'P is next of 0.6' likely indicate?
What does the statement 'P is next of 0.6' likely indicate?
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How does the brain reportedly combine incoming sensory information according to the Bayesian model?
How does the brain reportedly combine incoming sensory information according to the Bayesian model?
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What does the content imply about the use of Bayes' rule in the brain's inference process?
What does the content imply about the use of Bayes' rule in the brain's inference process?
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What is the correct calculation for the probability of it being Rainy two days from now given Cloudy today?
What is the correct calculation for the probability of it being Rainy two days from now given Cloudy today?
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Which statement accurately describes a Markov Model?
Which statement accurately describes a Markov Model?
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What is one of the primary goals of using Markov models?
What is one of the primary goals of using Markov models?
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How many sequences were identified to arrive at Rainy two days from Cloudy today?
How many sequences were identified to arrive at Rainy two days from Cloudy today?
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If today's weather is Sunny, what transitions could lead to Rainy two days later based on the provided model?
If today's weather is Sunny, what transitions could lead to Rainy two days later based on the provided model?
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Which of the following is NOT a component of a Markov Model?
Which of the following is NOT a component of a Markov Model?
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What probability does $P(w3=R|w2=C)$ imply?
What probability does $P(w3=R|w2=C)$ imply?
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What are the properties of the set of all possible atomic events in the context of two Boolean random variables?
What are the properties of the set of all possible atomic events in the context of two Boolean random variables?
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What is the total computed probability of it being Rainy two days from now if today is Cloudy?
What is the total computed probability of it being Rainy two days from now if today is Cloudy?
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Which of the following probabilities correctly represents a conditional probability?
Which of the following probabilities correctly represents a conditional probability?
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Which statement is true regarding independent events?
Which statement is true regarding independent events?
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What is the probability that an event that is necessarily false occurs?
What is the probability that an event that is necessarily false occurs?
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Which parameter in a Hidden Markov Model (HMM) represents the state-transition probabilities?
Which parameter in a Hidden Markov Model (HMM) represents the state-transition probabilities?
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What formula represents the probability of the disjunction of two events?
What formula represents the probability of the disjunction of two events?
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What is the role of the emission probability in an HMM?
What is the role of the emission probability in an HMM?
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If $P(Mary ext{_}Calls = true) = 0.1$, what does this imply about the probability of Mary not calling?
If $P(Mary ext{_}Calls = true) = 0.1$, what does this imply about the probability of Mary not calling?
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What is the total number of possible atomic events when considering two Boolean random variables?
What is the total number of possible atomic events when considering two Boolean random variables?
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How do discrete outputs in an HMM typically get modeled?
How do discrete outputs in an HMM typically get modeled?
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Which of the following statements about probabilities is incorrect?
Which of the following statements about probabilities is incorrect?
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What constitutes the 'initial state probabilities' (𝜋) in an HMM?
What constitutes the 'initial state probabilities' (𝜋) in an HMM?
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Which of the following defines the fixed number of states in a Hidden Markov Model?
Which of the following defines the fixed number of states in a Hidden Markov Model?
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Which statement describes a key component of the probability evaluation in an HMM?
Which statement describes a key component of the probability evaluation in an HMM?
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In modeling outputs, what distinguishes between a pmf and a pdf?
In modeling outputs, what distinguishes between a pmf and a pdf?
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What do the parameters of an HMM collectively represent?
What do the parameters of an HMM collectively represent?
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Study Notes
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.
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Description
Explore the fundamentals of probabilistic reasoning in this course. Delve into Bayes' Theorem, its applications in decision-making, and learn how to utilize Markov Chains for time series analysis. Gain skills in using these models to evaluate uncertainty and improve data analysis.
