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Stock Trading : Given the current and past price movements for a stock, determine whether the stock should be bought, held or sold. A model of this decision problem could provide decision support to financial analysts.
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Customer Segmentation : Given the pattern of behaviour by a user during a trial period and the past behaviours of all users, identify those users that will convert to the paid version of the product and those that will not. A model of this decision problem would allow a program to trigger customer interventions to persuade the customer to covert early or better engage in the trial.
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Shape Detection : Given a user hand drawing a shape on a touch screen and a database of known shapes, determine which shape the user was trying to draw. A model of this decision would allow a program to show the platonic version of that shape the user drew to make crisp diagrams. The Instaviz iPhone app does this.
These examples give a good sense of what a machine learning problem looks like. There is a corpus of historic examples, there is a decision that needs to be modelled and a business or domain benefit to having that decision modelled and efficaciously made automatically.
Some of these problems are some of the hardest problems in Artificial Intelligence, such as Natural Language Processing and Machine Vision (doing things that humans do easily). Others are still difficult, but are classic examples of machine learning such as spam detection and credit card fraud detection.