A Brief Tutorial on Bayesian Thinking. A Brief Tutorial on Bayesian Thinking Example: learning about a proportion We first discuss the similarities and differences between classical and Bayesian methods for a problem of learning about a population proportion. For a particular Big Ten university, we are interested in estimating the proportion p of athletes who graduate within six years. For a particular year, forty-five of seventy-four athletes admitted to the university graduate. Assuming that this sample is representative of athletes admitted during other years, what have we learned about the proportion of all athletes who will graduate within six years?
Specifically, we will consider two types of inferences. We wish to construct an interval that we are pretty confident contains the unknown value of p. The classical approach The classical 95% interval estimate for the proportion $p$ for a large sample is given by where denotes the sample proportion of athletes who graduate within six years, and n is the size of the sample.
Causal Markov condition. The Markov condition (sometimes called Markov assumption) for a Bayesian network states that any node in a Bayesian network is conditionally independent of its nondescendents, given its parents. A node is conditionally independent of the entire network, given its Markov blanket. The related causal Markov condition is that a phenomenon is independent of its noneffects, given its direct causes.[1] In the event that the structure of a Bayesian network accurately depicts causality, the two conditions are equivalent. However, a network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition. Recursive Bayesian estimation. Recursive Bayesian estimation, also known as a Bayes filter, is a general probabilistic approach for estimating an unknown probability density function recursively over time using incoming measurements and a mathematical process model.
In robotics[edit] A Bayes filter is an algorithm used in computer science for calculating the probabilities of multiple beliefs to allow a robot to infer its position and orientation. Essentially, Bayes filters allow robots to continuously update their most likely position within a coordinate system, based on the most recently acquired sensor data. This is a recursive algorithm. It consists of two parts: prediction and innovation. If the variables are linear and normally distributed the Bayes filter becomes equal to the Kalman filter. In a simple example, a robot moving throughout a grid may have several different sensors that provide it with information about its surroundings. Model[edit] The true state The denominator is constant relative to filtering. Recursive Bayesian estimation. Markov property. In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process.
It is named after the Russian mathematician Andrey Markov.[1] A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state, not on the sequence of events that preceded it. A process with this property is called a Markov process. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time.
Both the terms "Markov property" and "strong Markov property" have been used in connection with a particular "memoryless" property of the exponential distribution.[2] The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. Introduction[edit] History[edit] Let with . Naive Bayes classifier. A naive Bayes classifier is a simple probabilistic classifier based on applying Bayes' theorem with strong (naive) independence assumptions. A more descriptive term for the underlying probability model would be "independent feature model". An overview of statistical classifiers is given in the article on pattern recognition. Introduction[edit] In simple terms, a naive Bayes classifier assumes that the value of a particular feature is unrelated to the presence or absence of any other feature, given the class variable.
For example, a fruit may be considered to be an apple if it is red, round, and about 3" in diameter. For some types of probability models, naive Bayes classifiers can be trained very efficiently in a supervised learning setting. Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers have worked quite well in many complex real-world situations. Probabilistic model[edit] over a dependent class variable through . For given the category . . Is: ML_Class2.pdf.
Bayes' Rule. By Kevin Murphy. Intuition Here is a simple introduction to Bayes' rule from an article in the Economist (9/30/00). "The essence of the Bayesian approach is to provide a mathematical rule explaining how you should change your existing beliefs in the light of new evidence. In other words, it allows scientists to combine new data with their existing knowledge or expertise. In symbols Mathematically, Bayes' rule states likelihood * prior posterior = ------------------------------ marginal likelihood or, in symbols, P(e | R=r) P(R=r) P(R=r | e) = ----------------- P(e) where P(R=r|e) denotes the probability that random variable R has value r given evidence e.
P(e) = P(R=0, e) + P(R=1, e) + ... = sum_r P(e | R=r) P(R=r) This is called the marginal likelihood (since we marginalize out over R), and gives the prior probability of the evidence. Example of Bayes' rule Here is a simple example, based on Mike Shor's Java applet. This, of course, is just the true positive rate of the test. Entry-level Books.