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Lecture 4-Probability Slides

Lecture 4-Probability Slides - Cogsci 109 Virginia de Sa...

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1 Cogsci 109 Virginia de Sa desa at cogsci
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2 Data Modeling Suppose you are given some data and you would like to have a model that preserves properties of the data (e.g. interpoint distances, mean, variance, higher-order statistics) that you care about but can be easily operated on (e.g. compared to another dataset). -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 A very common approach is to model the data with a probability distribution (fit a probability distribution to the data). So we will now take an aside to delve into the basics of probability distributions.
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3 Probability Probability theory is the natural way to deal with computations about uncertain events. Both brains and computers must deal with uncertain events. Many people argue that in many problems the brain is performing optimally given the uncertainties that it has to deal with. “Performing optimally” usually means following the rules of probability.
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4 Motivation Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3 Motivation Motivation Our world is full of regularities, structure. It is useful for brains to learn about these regularities: brains construct models of the world. Models allow to correctly interpret ambiguous sensory inputs or even allow to predict future events: Model Parameters The World The Learner World State Projection Visible State Model State Inverse
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5 Random Variables Loosely, a random variable is a variable that has many values with different probabilities The outcome of a roll of a die is a discrete random variable The height of a randomly chosen person is a continuous random variable An event is a set of outcomes (e.g. die roll is odd)
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6 Probability Notation P ( a ) means probability that event a is true shorthand for Pr(a=true) usually called “probability of a” more detailed info e.g the probability of rolling a 6 on a die is 1/6 P(rolling 6) = 1/6
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7 Probability Axioms probability of event A, is between zero and one 0 P ( A ) 1 probability of some event occuing from the entire sample space S is one P ( S ) = 1 if events A and B are mutually exclusive , then P ( A or B ) = P ( A ) + P ( B )
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8 Example: Rolling two dice What is the probability of the outcome “getting doubles”? What is the probability of the event “the sum is less than 5”?
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9 Interpretation of Probability frequentist view: relative frequency P ( A ) = lim n - > N A N N number of experiments N A number of experiments where A happened
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10 Example: Rolling two dice What is the probability of the outcome “getting doubles”?
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11 Example: Rolling two dice What is the probability of the outcome “getting doubles”? 6 Die 1 Die 2 1 2 3 4 5 6 1 2 3 4 5 P(“getting doubles”)= 6/36 = 1/6
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12 Example: Rolling two dice What is the probability of the event “the sum is less than 5”?
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13 Example: Rolling two dice What is the probability of the event “the sum is less than 5”?
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Lecture 4-Probability Slides - Cogsci 109 Virginia de Sa...

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