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Probability and Odds
Probability: It quantifies the likelihood of an event occurring. It's a value between 0 and 1, where 0 represents impossibility, 1 represents certainty, and values in between indicate varying degrees of likelihood. Odds : Odds express the ratio of favorable outcomes to unfavorable outcomes. They can be expressed as "X to Y" or as a fraction (e.g., 2:1 or 2/3). Here's how they relate: - If the probability of an event is (P), the odds in favor of that event are (P/(1-P)). - Conversely, if the odds in favor of an event are (O), the probability of that event is (O/(1+O)). For example: - If the probability of rain tomorrow is 0.3 (30%), the odds in favor of rain are (0.3/(1-0.3) = 0.3/0.7 = 3:7). - If the odds of winning a game are 2:1, the probability of winning is (2/(1+2) = 2/3). Dealing with Uncertainty In the context of self-driving cars, handling sensor noise and uncertainty is crucial. Here are some strategies: 1. Sensor Fusion: Combine data from multiple sensors (like cameras, LIDAR, radar) to get a more accurate picture. Algorithms like Kalman filters or particle filters help fuse sensor data. 2. Probabilistic Models: Use probabilistic models (e.g., Bayesian filters) to estimate the car's position and update it as new data arrives. 3. Path Planning: Plan routes that account for uncertainty. For instance, consider alternative routes if traffic worsens unexpectedly. 4. Risk Assessment: Assess the risk associated with different actions. Sometimes it's better to proceed cautiously (e.g., slow down) rather than making abrupt decisions. Remember, even though we can't eliminate uncertainty entirely, we can manage it effectively using these techniques.
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Probabilistic Reasoning
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The history of dealing with Uncertainty
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Probability plays a crucial role in various aspects of everyday life.
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Odds and Probabilities.
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Rule Of Bayes
The Bayes rule, also known as Bayes' theorem, is a fundamental concept in probability theory and statistics. It provides a way to update our beliefs about an event based on new evidence. Let's dive into the details. Bayes' Theorem Given two events, A and B, Bayes' theorem states: $$ P(A|B) = frac{P(B|A) cdot P(A)}{P(B)} $$ Where: - (P(A|B)) represents the **posterior probability** of event A given evidence B. - (P(B|A)) is the **likelihood** of observing evidence B given that event A has occurred. - (P(A)) is the **prior probability** of event A (our initial belief before considering evidence B). - (P(B)) is the **marginal likelihood** of observing evidence B. Medical Diagnosis Example Let's illustrate Bayes' theorem with a medical diagnosis scenario. Suppose we have a patient with symptoms (e.g., fever, cough) and we want to determine whether they have a specific disease (let's call it D). We have the following information: 1. Prior probability: (P(D)) (our initial belief about the patient having the disease). 2. Likelihood: (P(text{symptoms}|text{D})) (probability of observing symptoms given the patient has the disease). 3. Marginal likelihood: (P(text{symptoms})) (overall probability of observing the symptoms). Using Bayes' theorem, we can calculate the posterior probability: $$ P(D|text{symptoms}) = frac{P(text{symptoms}|text{D}) cdot P(D)}{P(text{symptoms})} $$ AI and Bayes In AI, Bayes' theorem is widely used in various applications: - Naive Bayes classifiers: These models assume that features are conditionally independent given the class label, making them efficient for text classification, spam filtering, and recommendation systems. - Hidden Markov Models (HMMs): HMMs use Bayes' theorem to estimate hidden states based on observed emissions (e.g., speech recognition, part-of-speech tagging). - Kalman filters: These recursive Bayesian filters estimate the state of a dynamic system (e.g., tracking objects in video sequences). Remember, Bayes' theorem allows us to update our beliefs as new evidence emerges, making it a powerful tool for reasoning under uncertainty.
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The concept of Prior and Posterior Odds
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How Bayes’ Rule helps us
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The concept of Likelihood Ratio
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The World of Naive Bayes Classification
The naive Bayes classifier is a probabilistic model based on Bayes' theorem. It's particularly useful for text classification tasks, spam filtering, sentiment analysis, and more. Here are the key points: 1. Bayes' Theorem: - Bayes' theorem relates the posterior probability of an event given evidence to the prior probability of the event and the likelihood of the evidence. - Mathematically, it's expressed as: $$ P(C|X) = frac{P(X|C) cdot P(C)}{P(X)} $$ where: - (P(C|X)) is the posterior probability of class (C) given evidence (X). - (P(X|C)) is the likelihood of evidence (X) given class (C). - (P(C)) is the prior probability of class (C). - (P(X)) is the evidence probability (a normalization factor). 2. Naive Assumption: - The "naive" part of naive Bayes comes from assuming that the features (variables) are conditionally independent given the class label. - In other words, the presence of one feature doesn't affect the presence of another feature, given the class. - This simplification allows us to compute probabilities more efficiently. 3. Text Classification Example: - Suppose we want to classify emails as either spam or not spam (ham). - Features (words) are the terms present in the email. - Given an email with features (X = {x_1, x_2, ldots, x_n}), we compute: - (P(text{spam}|X)) and (P(text{ham}|X)). - The class with the higher probability becomes the predicted class. 4. Training the Naive Bayes Classifier: - We estimate the prior probabilities (P(text{spam})) and (P(text{ham})) from the training data. - For each feature, we estimate the likelihoods (P(x_i|text{spam})) and (P(x_i|text{ham})). - The naive assumption allows us to multiply these probabilities together: $$ P(text{spam}|X) propto P(text{spam}) cdot prod_{i=1}^{n} P(x_i|text{spam}) $$ $$ P(text{ham}|X) propto P(text{ham}) cdot prod_{i=1}^{n} P(x_i|text{ham}) $$ 5. Smoothing: - To handle unseen features, we use smoothing techniques (e.g., Laplace smoothing) to avoid zero probabilities. 6. Predictions: - Compare (P(text{spam}|X)) and (P(text{ham}|X)) to make the final prediction. Remember, while the naive Bayes assumption simplifies the model, it often performs surprisingly well in practice.
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Naive Bayes Classifier for Spam Filters
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Spam filtering using the Naïve Bayes Algorithm.
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Spam and Ham Classification using Bayes Rule.
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Real-World AI And Ability To Handle Uncertainty
About Lesson

  1. Prior Odds:

    • Prior odds represent our initial belief or assessment about the likelihood of an event before considering any new evidence.
    • Mathematically, if we denote the probability of an event as (P(A)), the odds in favor of the event are given by: [ text{Prior Odds} = frac{P(A)}{1 – P(A)} ]

  2. Bayes’ Rule and Posterior Odds:

    • Bayes’ rule allows us to update our beliefs based on new evidence. It relates the prior odds to the posterior odds after incorporating this evidence.
    • Suppose we obtain new information (evidence) represented by (B). We want to find the updated odds, which we’ll call posterior odds.
    • The posterior odds are given by: [ text{Posterior Odds} = frac{P(A|B)}{1 – P(A|B)} ] where (P(A|B)) is the probability of the event (A) given the evidence (B).

  3. Bayes’ Rule Formula:

    • Bayes’ rule connects the prior odds and the likelihood of the evidence: [ P(A|B) = frac{P(B|A) cdot P(A)}{P(B)} ]
      • (P(A|B)) is the posterior probability of event (A) given evidence (B).
      • (P(B|A)) is the likelihood of observing evidence (B) if event (A) is true.
      • (P(A)) is the prior probability of event (A).
      • (P(B)) is the overall probability of observing evidence (B).

  4. Interpreting Posterior Odds:

    • If the posterior odds are greater than the prior odds, the evidence supports the occurrence of event (A).
    • If the posterior odds are less than the prior odds, the evidence weakens the belief in event (A).

Remember that Bayes rule is a powerful tool for updating probabilities based on new information.

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