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As presented in the first article of Google Wavenet (https://arxiv.org/pdf/1609.03499.pdf) the model can approximate the joint probability of the whole sequence (raw audio waveform) using the chain rule. They implement this using some stacks of dilated-causal convolution. However in 2017 they presented a new article for neural audio synthesis of musical notes (https://arxiv.org/pdf/1704.01279.pdf). In this article the formula of the joint probability for the sequence(using chain rule) is different from the previous article. I believe that the chain rule in the first article is correct but I can't understand the same rule in the second one. For completeness I report the formula from the article:

enter image description here

I believe it should be given x1...x(i-1).

Is it correct?

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  • $\begingroup$ Looks like likelihood. $\endgroup$ – Carl Feb 18 at 18:28
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I think we see the same typo: The formula should be

$$ p(\{x_1,\cdots,x_N\})=\prod_{i=1}^{N}p(x_i|x_1, \cdots, x_{i-1}) $$

Confirming: the 2017 article explicitly says (section 2.1, 2nd paragraph) that probability is based on prior measurements, so yes, this was a typo.

Note that this arises from "unrolling" a joint probability distribution using Bayes theorem: $P(a,b,c)=P(c|a,b)P(a,b)=P(c|a,b)P(b|a)P(a).$ No approximations or assumptions are used; this is an exact relation.

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