Bill Nye and Ken Ham debating on creationism

While I think creationism is bonkers and the debate itself wasn't that great (talking past each other), I think this debate is good mental exercise. How would you argue the anti-creationist case?
It forces us to clarify and formalize how we come about inference and how we reason. We ought to be able to explain where creationism is commiting reasoning mistakes.

By the way, that does not mean being able to convince creationists that creationism is false (or very low confidence compared to other theories). This may not be possible because all belief (assignment of probabilities to propositions) are necessarily subjective in the sense that they depend on prior beliefs. Two reasonable people may not converge even as they are both presented with the same mountain of evidence, depending on their prior beliefs.
Here are some excellent pointers on logical inference (Bayesian reasoning) by E.T. Jaynes:
Bayesian methods: general background (overview)
How does the brain do plausible reasoning? (overview)
Probability Theory: The Logic of Science (first three chapters of his book, with more details on Cox Theorem which derives Bayes' laws of probabilistic inference)

Prior belief about Genesis/God:
If I try an formalize Ham's inference, it would start like this:
P(Theory | All evidence) = P(Theory | All evidence, Genesis is true) . P(Genesis is true | All evidence) + P(Theory | All evidence, Genesis is not true) . P(Genesis is not true | All evidence).
Then you can remove the second half (because he assigns a very high probability to the "Genesis is true") and you are left with P(Theory | All evidence, Genesis is true).
From there he can proceed with normal scientific inference, but he is stuck in the frame of the Genesis.

So the obvious question, which Nye didn't ask during the debate, is why would you believe the Genesis? How can you assign such a high prior to that proposition?
The general way of determining priors is to pick the least "loaded" assumptions. What that means is that the assumptions don't lock your beliefs more than information allows. If you know nothing about a loaded dice, the prior you should assign is 1/6 to each face, until you have more information. In that sense, the prior for "this specific book is true" would have to be 0.5 or even less (if I give you a random book and I tell you many people believe it and it is taught to children, then what probability would you assign to "that book is true"?).

Secondly, as far as assumptions go, we ought to be more suspicious of assumptions that offer more degrees of freedom. This is also known as Ockham's Razor principle.
The proposition that an omnipotent god exists is the ultimate hypothesis with infinite degrees of freedom. Anything can be explained by tuning the parameters (God's choices, which cannot be explained). If you encounter surprising evidence, then it can automatically be explained. This actually makes such theories weaker. In the case of the God hypothesis, the theory would have to be infinitely weaker. On the other hand, assuming that the world is regular (same laws of physics apply at different times and places) has no degrees of freedom and is most supported by the evidence we have.

Thirdly, such powerful assumptions are very sensitive to error. Let me offer another one: "God exists and created the world 200 years ago". I don't see how creationists could find any evidence contrary to this claim. The only difference between this theory and the creationist theory is that it assumes that God can spawn people mid-life with memories (why not?), whereas creationists can say that the historical record back to Adam and Eve was lost.

Inference about the past, the present and the future:
Now, let's consider one of Ham's specific methodological argument: He points out that you cannot confirm predictions about the past, because we weren't there. He says historical science is different from observational science. That is wrong, inference and reasoning apply just the same.
Nye uses an example to illustrate how predictions can be made about the past: you may predict a missing link species between fish and lizard, then discover fossils of such animals.

But there is a more fundamental flaw, it fails to understand how logical inference works. There are two main ways to use inference: P(theory | data) and P(data | theory), and both are equally valid.
One is to establish confidence in a proposition (objects fall according to Newton's law of gravity): P(law | evidence).
The other is to apply laws in which we have high confidence to draw inferences (the comet was at this position in space a month ago, it will be at that position a month from now): P(proposition | law, evidence). This works because P(law | evidence) is very high, and is rounded up to 1 (that's a simplifying assumption).

In practice, I am sure creationists use this method all the time to infer "what happened". Just like Sherlock Holmes, if he enters a room where an accident happened, he could use his knowledge of laws of physics to infer what happened. If he is able to establish a strong chain of inference, then he is reasonable to have high confidence in his conclusion even though nobody was there.

During the talk, Ham even provides an example of creationist scientists using this method to confirm that all dogs evolved from a single "dog kind". He calls that the "creation orchard" (multiple trees). But if he supports the method to figure out that two dogs indeed probably have a common ancestor, then why doesn't he support the same method to figure out that dogs and cats have a common ancestor?
His answer is probably because "Genesis is true" and Genesis said that god created different animal kinds. This takes us back to my first point.

All that said, Ham is correct to point out that there remains high degrees of uncertainty about the past. For instance, historical dating methods based on radioactivity of different elements often disagree. Most likely because our guess (prior) about the initial concentrations of the different elements is incorrect.
We should not ignore this uncertainty, or incorrectly confuse high confidence with certainty (when different methods do agree, we could still be wrong). But that does not invalidate the method of reasoning and drawing inferences to the best of our knowledge.

PS: Ham makes a point in passing about morality which is unrelated to the debate, namely that "no god means no morality". I'll provide pointers for two rebuttals, by David Friedman and Peter Boghossian (in a broader talk about religion and rationality).