Is It All the Info or Just Some of It?

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In conventional probability, we would imagine all the events that could happen and assign a number to them. So like we'd imagined, like it's just a list of numbers. And now you're saying, let's promote that from the start to a matrix,. To instead of n numbers and nd by n array of numbers. You have both the diagonal part of the matrix, which is your old fashioned set of probabilities, and more in fo is lurking there off diagonal and this. This lets you both compute the marginal probability, but then also keep some info about what was going on in the thing you are now ignoring.

Mathematics is often thought of as the pinnacle of crisp precision: the square of the hypotenuse of a right triangle isn’t “roughly” the sum of the squares of the other two sides, it’s exactly that. But we live in a world of messy imprecision, and increasingly we need sophisticated techniques to quantify and deal with approximate statistical relations rather than perfect ones. Modern mathematicians have noticed, and are taking up the challenge. Tai-Danae Bradley is a mathematician who employs very high-level ideas — category theory, topology, quantum probability theory — to analyze real-world phenomena like the structure of natural-language speech. We explore a number of cool ideas and what kinds of places they are leading us to.

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Tai-Danae Bradley received her Ph.D. in mathematics from the CUNY Graduate Center. She is currently a research mathematician at Alphabet, visiting research professor of mathematics at The Master’s University, and executive director of the Math3ma Institute. She hosts an explanatory mathematics blog, Math3ma. She is the co-author of the graduate-level textbook Topology: A Categorical Approach.

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