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I am about to start a project in ABC methods and I need to first of all study ABC since I know nothing about it. I spent quite a bit looking for on-line tutorials and notes but I found nothing apart for some videos. Can you please suggest me some notes or even a textbook? Not to difficult since I do not really need to understand all the math behind it. I am looking for some material that could allow me to understand the reasons behind the use of ABC methods.

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There are many introductions to ABC, available, most of them on arXiv, so I am rather surprised the OP could only find videos! I agree with the earlier answer that the Wikipedia entry is very good; it actually stemmed from a PLoS Computational Biology article (that I refereed). In addition to Mark Beaumont's superb review, let me refer the OP to the depository of ABC related papers. Which was run by Erkan Buzbas from the University of Idaho till 2015. For more recent papers and extensions, you may check the 2015 and 2016 NIPS workshops on the topic, the ABC'ruise workshop, and our supremely recent ABC'ory workshop at BIRS, Banff, that contains videos for most talks.

Some chapters of The Handbook of Approximate Bayesian Computation edited by Sisson, Fan and Beaumont are available on arXiv as well, and the book should be out very soon.

The slides of several courses I gave on ABC are to be found on slideshare and range from one hour to ten hours presentations.

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Actually a great starting point is the very detailed Wikipedia article on the Approximate Bayesian Computation plus the enormous number of references below it (many of them are available online).

Check also the nice introductory blog entry by Rasmus Bååth and accompanying video and slides from UseR! 2015 conference.

The Handbook of Approximate Bayesian Computation by Sisson, Fan and Beaumont is not yet available but it is ought to be published Chapman & Hall/CRC this year.

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When I looked over the topic a few years ago I found the best review to be Approximate Bayesian Computation in Evolution and Ecology by Mark A. Beaumont.

Another good review (which focuses on MCMC methods for ABC) can be found in the Handbook of Markov Chain Monte Carlo chapter 12 (Likelihood-free Markov chain Monte Carlo by Scott A. Sisson and Yanan Fan.)

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