Absence of evidence IS evidence of absence. There is literally a formula used to determine how much the absence of evidence is indeed evidence of absence:
P(H | E) = P(H) x P(E | H) / P(E).
This shows how much the evidence supports the explanation, and it has a perfectly valid formulation for absence of evidence:
P(H | ~E) = P(H) x P(~E | H) / P(~E).
Example using some made up stats: If dangerous fires are rare (1%) but smoke from barbecues is fairly common (10%), and 90% of dangerous fires make smoke then:
P(Dangerous Fire | Smoke) = P(Dangerous Fire) * P(Smoke | Dangerous Fire) / P(Smoke)
=1% x 90% / 10%
Absence of evidence version is derived from each term’s compliments:
P(Smoke | Dangerous Fire) + P(No Smoke | Dangerous Fire) = 100%
P(Dangerous Fire) + P(No Dangerous Fire) = 100%
P(Smoke) + P(No Smoke) = 100%
P(Dangerous Fire | No Smoke) = P(Dangerous Fire) x P(No Smoke | Dangerous Fire) / P(No Smoke)
=1% x 10% / 90%
Absence of evidence lowers the posterior probability, therefore absence of evidence (smoke) is evidence of absence (dangerous fire).