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In the following image, taken from paper on Variational Autoencoders by Dr. Kingma and Dr. Welling, it is mentioned that probability of the data points given the parameters factorizes into product of individual probabilities. However, it is denoted as P{subscript theta}(D) instead of P(D|theta) and it is also later referred to as likelihood. So, does P{subscript theta}(D) and P(D|theta) mean the same and are just two ways of denoting the same thing?

enter image description here

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    $\begingroup$ It is just a notation. If the approach is not Bayesian, conditioning upon parameters $\theta$ is not meaningful and it is more traditional to index the data density by $\theta$. $\endgroup$
    – Xi'an
    May 11 at 18:00
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    $\begingroup$ It is almost always best to accept a source's notation as that source defines it, regardless of any superficial similarities that notation might bear to other materials you have seen. This quotation is clear about the meaning of its notation. $\endgroup$
    – whuber
    May 13 at 18:51

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