Basic probability theory and statistics
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The document outlines fundamental concepts in probability theory and statistics relevant to machine learning and AI, including definitions of random experiments, sample spaces, and random variables. It distinguishes between discrete and continuous random variables, describes probability and conditional probability, and discusses variance and different types of probability distributions. Additionally, it introduces operations such as conditioning, reduction, and marginalization applied to probability distributions.
Basic probability theory and statistics
- 1. Swipe Basic Probability Theory and Statistics
- 2. These are some very fundamental terms/concepts related to probability and statistics that often come across any literature related to Machine Learning and AI. Random Experiment Sample Space Random Variables Probability Conditional Probability Variance Probability Distribution Joint Probability Distribution Conditional Probability Distribution (CPD) Factor Basic Probability Theory and Statistics
- 3. A random experiment is a physical situation whose outcome cannot be predicted until it is observed. Random Experiment
- 4. A sample space, is a set of all possible outcomes of a random experiment. Sample Space
- 5. A random variable, is a variable whose possible values are numerical outcomes of a random experiment. There are two types of random variables. Discrete Random Variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,…….. Discrete random variables are usually (but not necessarily) counts. Continuous Random Variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Random Variables
- 6. Probability is the measure of the likelihood that an event will occur in a Random Experiment. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. Probability
- 7. Conditional Probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, is usually written as P(A|B). Conditional Probability
- 8. The variance of a random variable X is a measure of how concentrated the distribution of a random variable X is around its mean. Variance
- 9. Is a mathematical function that maps the all possible outcomes of an random experiment with it’s associated probability. It depends on the Random Variable X , whether it’s discrete or continues. Discrete Probability Distribution: The mathematical definition of a discrete probability function, p(x), is a function that satisfies the following properties. This is referred as Probability Mass Function. Continuous Probability Distribution: The mathematical definition of a continuous probability function, f(x), is a function that satisfies the following properties. This is referred as Probability Density Function. Probability Distribution
- 10. If X and Y are two random variables, the probability distribution that defines their simultaneous behaviour during outcomes of a random experiment is called a joint probability distribution. Joint Probability Distribution
- 11. If Z is random variable who is dependent on other variables X and Y, then the distribution of P(Z|X,Y) is called CPD of Z w.r.t X and Y. It means for every possible combination of random variables X, Y we represent a probability distribution over Z. There are a number of operations that one can perform over any probability distribution to get interesting results. Some of the important operations are :- Conditioning/Reduction Marginalisation Conditional Probability Distribution (CPD)
- 12. If we have a probability distribution of n random variables X1, X2 … Xn and we make an observation about k variables that they acquired certain values a1, a2, …, ak. It means we already know their assignment. Then the rows in the JD which are not consistent with the observation is simply can removed and that leave us with lesser number of rows. This operation is known as Reduction. Conditioning/Reduction
- 13. This operation takes a probability distribution over a large set random variables and produces a probability distribution over a smaller subset of the variables. This operation is known as marginalising a subset of random variables. This operation is very useful when we have large set of random variables as features and we are interested in a smaller set of variables, and how it affects output. Marginalisation
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