Please consider the following sentences:
- Den Wolken nach zu urteilen, wird es offensichtlich regnen.
- Dem Wetterbericht nach zu urteilen wird es Morgen schön.
Can I say instead:
- Den Wolken nach wird es offensichtlich regnen.
- Dem Wetterbericht nach wird es Morgen schön.
Yes, the meaning of the sentences is equivalent and both versions are fine and common.
etwas nach zu urteilen
You can only use this when referring to a thing or concept which can in the widest possible sense be judged or interpreted. So you would not use it when simply referring to or quoting a statement either, but one can use it, if it is a more convoluted one which needs possibly interpretation. You cannot use it when referring to a person themselves. Concerning cloud observations or weather forecasts: both is perfectly fine. So if one wants to make a distinction, it could be along these lines:
Max nach sind sie gestern ins Kino gegangen
According to Max... / Going by Max' words...
Nach Max' Erzählung zu Vampiren zu urteilen, wird er Dr. Frankenstein schon gesehen haben. (but Max did not explicitly talk about that movie)
Judging Max' story telling about vampires... / If Max' talking about vampires is anything to go by... / Going by Max' words...
