I came across many sentences in empirical economics papers using the term:
prima facie evidence
Is there any difference between "prima facie" evidence and "preliminary evidence"?
Prima facie (at first face) means evidence or results that appear to conclude a certain point or argument.
Legally this evidence would be presented at a preliminary hearing (as preliminary evidence) to establish the existence of a prima facie case. Therefore prima facie evidence is not actually preliminary evidence in law until it's presented as part of a preliminary hearing.
In other contexts they can be presumed to be the same thing.
"prima facie" is the immediately apparent evidence without digging deeper. "preliminary" implies a process of conclusion, so it's more about an exhaustion of readily available evidence.
If a candidate leaves the court house tearing his hairs, that could be considered prima facie evidence of him having lost an important vote. It does not reach the level of "preliminary evidence". "Preliminary evidence" would be what you have after counting all mail ballots and/or precincts with finishied tallies.