lecture2 - CSE 6740 Lecture 2 How Do I Learn a Simple...

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Unformatted text preview: CSE 6740 Lecture 2 How Do I Learn a Simple Model? (Probability and inference) Alexander Gray [email protected]c.gatech.edu Georgia Institute of Technology CSE 6740 Lecture 2 – p. 1/3 2 Today 1. Probability and random variables (What is “data”? What is a “model”?) 2. Asymptotics and point estimation (What is “estimation/learning”?) 3. About the course CSE 6740 Lecture 2 – p. 2/3 2 Probability and random variables What is “data”? What is a “model”? CSE 6740 Lecture 2 – p. 3/3 2 Mathematical Notation Pay attention to my comments about the notation for various things. Statistical notation is highly variable and ambiguous and can make things confusing, all on its own. CSE 6740 Lecture 2 – p. 4/3 2 Samples Spaces and Events If we toss a coin twice then the sample space , or set of all possible outcomes or realizations ω , is Ω = { HH,HT,TH,TT } . An event is a subset of this set; for example the event that the first toss is heads is A = { HH,HT } . CSE 6740 Lecture 2 – p. 5/3 2 Probability We’ll assign a real number P ( A ) to each event A , called the probability of A . To qualify as a probability, P must satisfy three axioms: 1. P ( A ) ≥ for every A 2. P (Ω) = 1 3. If A 1 ,A 2 ,... are disjoint then P parenleftBigg ∞ uniondisplay i =1 A i parenrightBigg = ∞ summationdisplay i =1 P ( A i ) . (1) Note that frequentists and Bayesians agree on these. CSE 6740 Lecture 2 – p. 6/3 2 Random Variables A random variable is a mapping, or function X : Ω → R (2) that assigns a real number X ( ω ) to each outcome ω . For example, if Ω = braceleftbig ( x,y ) : x 2 + y 2 ≤ 1 bracerightbig and our outcomes are samples ( x,y ) from the unit disk, then these are some random variables: X ( ω ) = x , Y ( ω ) = y , Z ( ω ) = x + y . CSE 6740 Lecture 2 – p. 7/3 2 Data and Statistics The data are specific values of random variables. A statistic is just any function of the data/random variables. Any function of a random variable is itself a random variable. CSE 6740 Lecture 2 – p. 8/3 2 Distribution Functions Suppose X is a random variable, x a specific value of it (data). Cumulative distribution function (CDF): the function F : R → [0 , 1] (sometimes F X ) defined by F ( x ) = P ( X ≤ x ) ....
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lecture2 - CSE 6740 Lecture 2 How Do I Learn a Simple...

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