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I believe that statistical AI uses inductive thought processes. For example, deducing a trend from a pattern, after training.

What are some examples of successfully applied statistical AI to real-world problems?

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There are several examples. For example, one instance of using Statistical AI from my workplace is:

  1. Analyzing the behavior of the customer and their food-ordering trends, and then trying to upsell by recommending them the dishes which they might like to order/eat. This can be done through the apriori and FP-growth algorithms. We then, automated the algorithm, and then the algorithm improves itself through an Ordered/Not-Ordered metric.
  2. Self-driving cars. They use reinforcement and supervised learning algorithms for learning the route and the gradient/texture of the road.
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There are many online services that use statistical neural networks for recommendations. For example, we have a well known service here in Russia that could give it's users recommendations for movies and shows to watch and books to read. Its recommendation core is based on many things known about a user: what movies/books he or she loves and what not, analyses his or her friends like and so on. While you have only a few items rayed it will give you very strange recommendations but then it becomes more accurate and really could give you some true gems.

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Not strictly examples of AI, but related to the greater AI project: But us in the psychology / cognitive science side of things sure love our bayesian modelling!

In fact there are people who believe that a theory grounded in such analysis would ultimately bring us to a unified theory of the brain and cognition!

Unfortunately to my knowledge, these theories are not yet complete or testable in interesting ways as they are grounded more in the philosophy end of things. More so the claims that the psychologists make are rather weak: that hypothesis updating and inference is Bayesian-like (which isn't super exciting to be honest) (but my knowledge in this area is not super complete)

Alas, more work needs to be done but at least there is psychological support for the claim that cognition is Bayesian-like.

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Statistical AI is widely used in finance for asset management (particularly hedge funds) and trade execution looking at high-speed small data sets, lots of HMMs and SSMs, but nobody talks about it because it provides proprietary riches.

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