We've found a large and continuously growing number of "birdlike theropods" after Archaeopteryx, but none before. OK, we can't expect every genuine Archae-ancestor to be fossilised and found, and some descendants of the original stock might be expected to survive beyond Archae, but why do they ALL appear after instead of before? Just how many paradoxical instances will we need before we call it a piece of evidence?

Jurassic fossils are rarer than Cretaceous fossils; it is also hypothetically possible that if we assume maniraptorans gave rise to birds, it may have been early in maniraptoran existence (though the more this is true, the weaker the case - see below).

(By chance, only a small fraction of the evidence appears between Archaeopteryx and the end of the J (assuming the China stuff is K) so the principle needs only slight re-phrasing to refer to either 'Pre-Archae' or 'Pre-K', though at the moment they amount to the same thing).

Say the ratio of maniraptorans living in the K to those living in the J was:

A:B;

Say the ratio of the chances of an indiv living in the K ever being found to that of one in the J is:

C:D;

The ratio of the chances of a maniraptoran we have found coming from the K to from the J is:

(A/B)*(C/D).

The probability of any maniraptoran we have found coming from the K is:

1/(1 + 1/ ((A/B) * (C/D)) ).

In row 1, if about 12 times as many mani's existed post-Archae as pre, and a mani-type creature was three times as likely to be fossilised and found in the K as the J, then 171 mani's found post Archae (& as always in the table, none pre Archae) has a probability of 1%.

In other words, being slightly generous with the search effort and geological luck, we can assume (at 1% confidence level) that with the mani's we've actually found, there were at least 12.2 times as many mani's existing post-Archae as pre.

What if it is claimed the maniraptorans had only just started prior to Archae? Row 2 shows that if 99.1% of all the mani's that ever lived were post-Archae, then with the mani's we've found, the K would have to be the less productive (0.3663 of the J, assuming Archae was at the J/K border).

This is more than BAMM believers would allow, so let's adjust the findability ratio back to 3 for Row 3: Now we would have to find 1383 post-Archae for the result to be significant at the 1% level.

However, if we were content with a 5% confidence level, we would only have to wait for 900 post-Archae mani's, as shown in row 4; in row 5 we see we would need 2076 to be confident at the 0.1% level.

A few more variations with very high certainty levels are shown in rows 6-10; notice though, that in order to make the game 'interesting' we have had to assume only about 1% of the mani's that ever lived came before Archae. But this is so perilously close as to be almost indistinguishable from the 2F principle we are trying to show - that all mani's followed Archaeopteryx!

And enough of all this 1% and 0.1% confidence level stuff - who's being tried for murder here? Who says we need to do more than decide the balance between two ideas? There's certainly no way these figures will ever be able to prove anything against secondary flightlessness! A fair-minded person who has to make a choice between two possible theories will not be looking for a 99% confidence level in either direction.

Row 11 shows a scenario where 82/83rds (that's 98.795%) of the manis lived post Archae, with reasonable findability ratio, and the current number of finds. This gives a 50% probability, showing that the idea that maniraptorans had just started about the time of Archaeopteryx is perfectly consistent with the evidence (as opposed to being undeniably proven).

Row 12 shows that when we have 206 identified mani's all from the K, this will be fully consistent with the idea that at least 99% of all mani's occured in K (or post Archae), and Row 13 shows that with a slightly more generous findability the evidence is consistent with less than one in 62 (that's 1.6%) mani's living pre Archae.

As shown in row 14, by the time we have 638 mani's all post Archae, it will be most likely that no more than 1% of K-type mani's, if that, lived pre Archae.

Proof of 2F's absolute truth:

Still some way from undeniable proof at 1% confidence level and above, using this evidence alone.

Balance of probability:

Definitely on the side of 2F.

Adequate evidence that 2F is a viable theory that ought to be included in any consideration of this issue:

Definite. 100%.

Some people admit they're not experts in the area but still say this analysis is invalid. I'm not very happy about that, but I will admit myself that you don't have to be an expert to hold an opinion on certain probabilistic issues. I didn't. However, let them just answer this: if 171 manis before and none after are not sufficient evidence, how about 371? How about 1000? Could even 2500 still be put down to preservational bias? At what point would they say the evidence becomes meaningful?

Many people (some of whom may have considerable investment in the BAMM theory) will claim (wrongly) that it is wise to cling on to an old theory until there is no longer any possibility of it being true. "You'll have to prove..." they typically ask of any new theory, conveniently forgetting that their own theory is entirely unproven. In fact the wisest position for the disinterested is to consider all possible theories as possible, [according to their apparent likelihood##]. What the above analysis shows is that 2F is at least as likely as BAMM, and getting likelier. It seems reasonable to conclude that anyone who refuses even to acknowledge the existence of 2F as a genuine, possible hypothesis, with at least the worth of BAMM, is showing...shall we say...less than adequate scientific integrity!

Cell A15:   1/(B15*((1/(POWER(D15,(1/C15)))) -1))
Cell B16:   1/(A16*((1/(POWER(D16,(1/C16)))) -1))
Cell C17:   LOG(D17,(1/(1+(1/(A17*B17)))))
Cell D18:   POWER((1/(1+(1/(A18*B18)))),C18)