Not to rain on your parade, but doesn't it make sense if you are talking about a specific bear that you would also be talking about a specific forest? Otherwise you would see a bear and just know that (s)he lives in a forest, big or small. For me, "the" makes more sense. Grammatically may be a different matter, but isn't language about meaning? Shouldn't the sentences make a little bit of sense? Well, I am no Finn, or native English speaking, and as I have seen repeatedly, Finnish doesn't exactly fit English in all respects. Add to that our general confusion of how to use English (UK, American, Dutch, Germans, French, Polish, etc) and making a translation that is "ok" becomes a lot harder. I like reporting errors as much as the next guy, but I am not sure this is an error.
doesn't it make sense if you are talking about a specific bear that you would also be talking about a specific forest?
It does, but that doesn't mean that the definite article is the only option. "A bear lives in the big forest" is a perfectly sensible sentence if that bear wasn't mentioned before. And "The bear lives in a big forest" is also a sensible sentence if the bear was mentioned before, but the forest wasn't. Using definite articles for both would make sense if both the forest and the bear were mentioned before.