Kaosium used an old post of yours where you estimated probabilities of TOD based on a bell curve of GE. At the time you were arguing that the 11:30 TOD was impossible which btw I agree with as stated many times.
You had a 5% chance of TOD being at 9:20.
The current discussion is different because some here are claiming that the Naruto computer interaction at 9:26 provides proof positive that kids couldn't be involved.
It has come to light that the time of eating is not certain. Not only when the girls brought out the pizza but when Meredith started eating. Combining that with the vagaries of GE I don't believe a proof of innocence results. While it is clear that a defendant need not prove innocence if one claims this does it I disagree.
The ILE did not prove guilt beyond a reasonable doubt but the digestion scenario doesn't prove innocence for many reasons.
Ah, but for our purposes (and the court's purposes), the statistical analysis that matters is Bayesian conditional analysis.
That's to say, we must pose the following question:
given that we know that the victim was still alive at 9.00pm, what are the probabilities of her dying at 9-9,30, 9.30-10, 10-10.30, 10.30+?
So even though it's statistically unlikely that Meredith died later than 9pm - the important thing is that
we know this to be the case. The correct evaluation in the light of this conditional knowledge is to look at the shape of the probability distribution curve to the right of the time we know to be the effective "start" time on death (corresponding to 9pm in our case). If the "curve" were a horizontal flat line beyond this point, then effectively one could say that there's an equal probability of death occurring at any time later than 9pm (regardless of whether death later than 9pm is improbable in itself).
But the curve is
not a horizontal flat line in this case. It's the end of a bell curve that slopes down to zero by around 10.30pm. And this effectively means that given we know that Meredith was still alive at 9pm, it's possible to state that it's still far more probable that she died between 9 and 9.30 than between 9.30 and 10, and that it's impossible that she died later than 10.30.
In fact, we actually have nested conditional probabilities to look at in this case, since (as you point out) there is also some ambiguity as to when the meal started. But that's OK. What we can do to deal with this ambiguity is run a series of analyses based on various different meal start times within the reasonable range.
So we could do:
Analysis 1: If the meal started at 5.30pm, then given that the victim was still alive at 9.00pm, what are the probabilities of her dying at 9-9,30, 9.30-10, 10-10.30, 10.30+?
Analysis 2: If the meal started at 6.00pm, then given that the victim was still alive at 9.00pm, what are the probabilities of her dying at 9-9,30, 9.30-10, 10-10.30, 10.30+?
Analysis 3: If the meal started at 6.30pm, then given that the victim was still alive at 9.00pm, what are the probabilities of her dying at 9-9,30, 9.30-10, 10-10.30, 10.30+?
Analysis 4: If the meal started at 7.00pm, then given that the victim was still alive at 9.00pm, what are the probabilities of her dying at 9-9,30, 9.30-10, 10-10.30, 10.30+?
Ironically (and perhaps counter-intuitively) a later meal start time actually makes it
more statistically likely that Meredith died shortly after 9pm. The reason for this is the shape of the bell curve at and after (say) 2 hours after ingestion (corresponding in our case to a 7pm meal start time), compared to the shape of the bell curve at and after 3 hours after ingestion (corresponding to a 6pm meal start time). If this requires further explanation and clarification, I'll be happy to do so but will probably need to construct a number of graphs to illustrate the principles.