It's more likely to see 40% heads with a coin that lands heads 40% of the time than 50% or 10%. 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Bayes' Rule unpacks it: p(θ|X) = p(X|θ) p(θ) / p(X). Dynamoo writes "Bayesian filtering for spam is awfully clever stuff, touched on by Slashdot several times before.There's a very accessible article at BBC News explaining in fairly simple terms the drawbacks of current keyword-based filtering. Plugged into a more readable formula (from Wikipedia): Bayesian filtering allows us to predict the chance a message is really spam given the “test results” (the presence of certain words). Bayes first proposed his theorem in his 1763 work (published two years after his death in 1761), An Essay Towards Solving a Problem in the Doctrine of Chances . Interesting — a positive mammogram only means you have a 7.8% chance of cancer, rather than 80% (the supposed accuracy of the test). If we know nothing about θ, then all values of θ are equally likely, before. Knowing nothing else, the best guess is that 40% of future flips will land heads. This makes it easy to compute p(θ|X), which is called the posterior distribution, without any complex math. Spam filtering based on a blacklist is flawed — it’s too restrictive and false positives are too great. As the filter gets trained with more and more messages, it updates the probabilities that certain words lead to spam messages. Saying “100 in 10,000″ rather than “1%” helps people work through the numbers with fewer errors, especially with multiple percentages (“Of those 100, 80 will test positive” rather than “80% of the 1% will test positive”). Of the 99 remaining people, about 10% will test positive, so we’ll get roughly 10 false positives. p=0.4 actually is the best answer in a certain sense. And if you're not, then it could enhance the power of your analysis. Instead of saying that the rows/columns of U and V are normally distributed with zero mean and some precision matrix, we place hyperpriors on the mean vector and … That's why the MLE and MAP estimates were the same; the MAP estimate implicitly assumed a flat prior, or, no prior knowledge about the parameters. type of Probabilistic Graphical Model that can be used to build models from data and/or expert opinion x9 751.2793 171.902 4.370 0.000 413.409 1089.150 By now you may have a taste for Bayesian techniques and what they can do for you, from a few simple examples. Bayesian. Thank you, normalizing constant, for setting us straight! It turns out that it follows a beta distribution, and after seeing 2 heads and 3 tails, it's common to analyze this distribution as Beta(3,4): import numpy as np import scipy.stats as stats from matplotlib import pyplot as plt If a message has a 99.9% chance of being spam, it probably is. 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