# Wavenet joint probability

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:

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

Is it correct?

• Looks like likelihood. – Carl Feb 18 at 18:28

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