... system.^1.1

It is not essential that the input be perceptual. However, perceptual modalities are rich, expressive, intuitive and non-obtrusive. One could take other measurements if they help infer behaviour, internal state or intentionality.

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... game)^2.1

Of course, the individuals need not be in the same physical space and could be interacting through a virtual environment.

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... integral^5.1

Either analytically or by sampling techniques

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... space^5.2

This is the typical task of unsupervised learning.

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... output^5.3

This is the typical task of supervised learning.

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... density.^5.4

A proof by example can be found in the appendix.

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... optimization.^6.1

We shall focus on maximization in this document but most arguments should apply to minimization problems as well.

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...$\Sigma$^6.2

In this sense, $\Sigma$ acts like a variational parameter.

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...>0^6.3

Here, a is also only an integer traditionally but this is not necessary for the following derivation. Also note that a does not truly have to be positive since $\sin(-x) = sin(x+\pi)$ can be used to factor out the negativity.

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... covariances^7.1

In fact, step 1) and step 2) can be performed simultaneously since they only involves taking derivatives and setting to 0.

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...[5] ^7.2

A classic example of this is that of dystal learning to recover the angle at which to throw a ball when trying shoot it a pre-specified distance. Since throwing a ball at 90 or 0 degrees is equivalent (and any other (90-x) or (x) combination), the expected value is to always throw the ball at 45 degrees which is incorrect and would have probability $\approx 0$.

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... constraints^11.1

In fact, in assuming a joint density, the model is a parent of the ${\cal X}$ data instead of being a co-parent with it.

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