I would sooner hypothesize that Meredith’s last meal actually took place closer to 19:00 than 18:00, given the vagueness of the testimony on the matter. This puts her within 2 standard deviations, perhaps even 1.5.
If we model the meal start-time as a normal distribution, then it’ll be simple to add it to the model and combine it with the other sources of uncertainty, since two normal distributions sum to a new normal distribution with a variance equal to the sum of the variances. Though now that I mention it, a lot of the other bits of uncertainty might be somewhat log-normal because they might multiply the time rather than add to it.
But, granting a non-normal distribution, it’s really difficult for me to see how it could significantly work against Raffaele, given where the 25th and 75th percentiles are. Probability mass would have to be transferred to the extreme right tail from somewhere else; how do you propose to do this in a way that isn’t specifically tailored to yield the desired bottom line?
(1a) What does your gastric lag-time model look like, such that you don’t get significantly more surprised by going out to 22:00 than 21:00?
Good question. Let me look here at some more papers. One source of uncertainty is that I don’t know if we care in this case about 2% or 10% or something else.
The gated study you cited shows 81.5 minutes using unknown-to-me methods, perhaps the meal was larger or different from the other studies.
So I guess I would reluctantly discard the concept of attempting solely normal distributions, since this already is looking too right-tailed. So this is too complex for me to easily model, I can only say that intuitively even if we use 19:00, then if a genie tells me it’s at least 120 minutes, then I wouldn’t be much more surprised by 150 minutes or 180 minutes. The first three studies above looked like they were behaving at 10, 25, and 23, and then your example jumped to more than 3x the highest figure so far. So jumping again to even 3x of your number wouldn’t be more than a one-in-ten surprise, especially given the numerous factors I’ve itemized.
(2) What is your probability of guilt, conditioned on death having occurred (a) before 21:30? (b) before 22:00?
If we’re not taking systemic uncertainty into account, then it’s still going to be quite a large probability of guilt. However, I would say that, compared with 23:00, (a) would shift me by about 15:1 on the grounds that the computer evidence would have to be mis-analyzed, or (more likely) Raffaele would have had to manufacture the computer alibi (recall Raffaele is a computer engineer), and (b) by 5:1 on the grounds that the timetable gets a bit tighter than in the 23:00 case. Keep in mind that I’m currently not yet bothering to weigh the eyewitness testimony at all in my assessment of guilt.
Slippage is a priori unlikely, especially with the ligatures applied (professional opinion), and hence given a level of gastric contents consistent with the meal in question, there’s no reason to believe any significant slippage occurred.
I believe the independent court expert more than hearsay that an unknown FRCPath claimed that, even without ligatures, complete slippage is “well-nigh impossible”.
And note this: “The lag phases after 4 and 10% (v/v) ethanol, beer, and red wine were not significantly different from that of water… the inhibitory effect of ethanol and alcoholic beverages is mainly induced by a prolongation of the gastric emptying phase (without affecting the lag phase)...”
That’s a good point, so I hereby drop the alcohol point altogether for the non-slippage case.
Here is another source characterizing any lag time over 150 minutes as “extremely delayed”. By comparison, “normal” is 50-100 min and “delayed” is 100-150. For half-emptying time, over 200 minutes is “extremely delayed”.
This seems to be for small easily-digested test meals, as far as I can tell. No hospital is going to serve a patient a pizza to determine how well their diabetes is under control. ;-)
Just how large do you think the standard deviation is? If you believe in the Massei theory, you have to come up with a lag time of four hours at minimum. I can’t find any evidence that that is anywhere close to being within normal human parameters. Can you?
I see that large, fried, and/or starchy meals have much larger T(50) times than other meals, and I don’t have any lag times for those. Since T(50) times are frequently unexpectedly large, and since lag times correlate in some large but unknown way with T(50) times, I infer a significant probability that lag times are frequently unexpectedly large as well.
Let me float one scenario. I’d presume that starch increases the T(50) time so much because it can take a long time for large amounts of starch to convert to sugar in the stomach. Does almost the entire portion of starch need to get converted to sugar before any starch can go to the duodenum? If so, then the lag time for a large starch meal would be close to the T(50) time.
On the other hand, if you want to believe the time of death was earlier, you run into other problems...
Sounds like a whole other discussion.
So what is your probability distribution for time of death?
Based on just stomach evidence, and ignoring expert testimony, I’d have to say it most likely happened around 19:00. So that’s not very useful.
If we take a leap of faith and use the 317 minutes T(50) for 700 kcal fried pasta but don’t believe the starch needs to convert first, then I’d revert to a 1⁄4 guess for lag time on the basis the ratio decreases as T(50) grows, resulting in 80 +/- 6 minutes, so that model fails for me as well, dang it.
Factoring in that it wasn’t before 21:00, but still ignoring expert testimony, I’ll have to take an “inside view” and try to generate hypotheses as to why it took so long. I’ll currently guess that for to get us out to 21:00, either the starch needs to convert to sugar first (40%), or else there was slippage after the body was discovered (5%), or that there was slippage when the body was moved by one or more perps before being discovered by the police and “ligatured” (55%). I’m open to other suggestions. Unfortunately the gated 81-minute median study isn’t currently helpful in this regard, because I have to ask myself, why was this study 81 minutes, instead of the others that were 25 or 10 or 40 minutes? But if we can find out whatever X factor increased it to 81 minutes, then might be able to judge how much of that X factor we had in our case, and whether we had more or less X factor than in the study. Anyway, overall I’ll guess 30% for 21:00-21:30, 20% for 21:30-22:00, 25% for 22:00-23:00, 7% for 23:00-23:30.
Now let’s factor in expert testimony. Since none of our models are working very well, and since the literature that I’ve seen doesn’t converge on a single simple model anyway, I think in the end I’ll go with the independent expert testimony. The experts have access to gated medical journals and even some kind of summary chart of different times under different situations in the literature, as well as forensic experience, which I don’t have. They also get to factor in the body temperature, which I’ve been ignoring.
I’m not following this discussion in detail, but I’m glad you guys are having it—I think it’s a worthwhile case study, and the flavour of the way it’s discussed is informative without getting into the detail of the subject matter.
(2) What is your probability of guilt, conditioned on death having occurred (a) before 21:30? (b) before 22:00?
If we’re not taking systemic uncertainty into account, then it’s still going to be quite a large probability of guilt. However, I would say that, compared with 23:00, (a) would shift me by about 15:1 on the grounds that the computer evidence would have to be mis-analyzed, or (more likely) Raffaele would have had to manufacture the computer alibi (recall Raffaele is a computer engineer), and (b) by 5:1 on the grounds that the timetable gets a bit tighter than in the 23:00 case.
If we don’t take systemic uncertainty into account, then 15:1 isn’t much of a shift in the face of numbers like 2500:1 or 200000:1 that you were giving for the knife. On the other hand, if we do take systemic uncertainty into account (as we ultimately must), a shift of 15:1 or even 5:1 would be significant, given your estimate of .95 probability of guilt, or 19:1 odds. Crudely approximating this as 20:1, it would take you down to 4:3 (p = 4⁄7 = 0.57) or 4:1 (p= 4⁄5 = 0.8) respectively. I imagine that if you were to believe the 21:26 computer interaction, the pre-22:00 odds could potentially go down to something like 4:3 as well. This indicates that it may be worthwhile to keep pursuing the time-of-death issue.
Keep in mind that I’m currently not yet bothering to weigh the eyewitness testimony at all in my assessment of guilt.
Well, I was already assuming that you didn’t believe Curatolo (for one thing, he gives Amanda and Raffaele an alibi for 21:30-23:00!). If you do, that will have to be dealt with separately. But in general, not only is eyewitness testimony among the weakest forms of evidence, but it cuts both ways in this case, as the broken-down car illustrates.
Slippage is a priori unlikely, especially with the ligatures applied (professional opinion), and hence given a level of gastric contents consistent with the meal in question, there’s no reason to believe any significant slippage occurred.
I believe the independent court expert more than hearsay that an unknown FRCPath claimed that, even without ligatures, complete slippage is “well-nigh impossible”.
Keep in mind, however, that Ronchi’s speculation about slippage was based on the mistaken assumption that ligatures had not been applied. So there may not be as much contradiction here as it appears (especially since it isn’t clear to me that the FRCPath’s opinion necessarily pertained to the non-ligature case, depending on how standard ligatures are).
How much slippage do you think may have occurred?
Here is another source characterizing any lag time over 150 minutes as “extremely delayed”. By comparison, “normal” is 50-100 min and “delayed” is 100-150. For half-emptying time, over 200 minutes is “extremely delayed”.
This seems to be for small easily-digested test meals, as far as I can tell. No hospital is going to serve a patient a pizza to determine how well their diabetes is under control. ;-)
If you can find a reference to support the idea that a lag time in excess of four or even three hours would not be highly unusual for a small-to-moderate pizza meal eaten by a healthy adult human, I will update appropriately. Conversely, if you can’t (and I haven’t been able to so far), I don’t see how you can derive the level of uncertainty you need to make the Massei theory plausible in the face of all the other data. Even acknowledging the wide variation in lag times depending on the type of meal used in the studies, they are all on the short end; there is no indication, anywhere (that I have come across), of the kind of extremes that we would need at the long end. Not so much as a passing remark. Isn’t this a bit suspicious?
I see that large, fried, and/or starchy meals have much larger T(50) times than other meals, and I don’t have any lag times for those. Since T(50) times are frequently unexpectedly large, and since lag times correlate in some large but unknown way with T(50) times, I infer a significant probability that lag times are frequently unexpectedly large as well.
I agree, of course; I just don’t agree that “frequently unexpectedly large” translates to anything like “regularly in excess of four hours”.
Let me float one scenario. I’d presume that starch increases the T(50) time so much because it can take a long time for large amounts of starch to convert to sugar in the stomach. Does almost the entire portion of starch need to get converted to sugar before any starch can go to the duodenum? If so, then the lag time for a large starch meal would be close to the T(50) time...
If we take a leap of faith and use the 317 minutes T(50) for 700 kcal fried pasta but don’t believe the starch needs to convert first, then I’d revert to a 1⁄4 guess for lag time on the basis the ratio decreases as T(50) grows, resulting in 80 +/- 6 minutes, so that model fails for me as well, dang it.
I’ll ask Yvain (LW’s medical student extraordinaire) if he knows anything about the mechanisms involved and the plausibility of your proposed scenario. At the very least I expect he’ll know someone who knows something. (Update: Yvain says he doesn’t know any more than we do.)
I’m confused about the notation T(50): does this refer to half-time, or total emptying time? Because the 317 minutes for fried pasta was total emptying time.
I’m open to other suggestions. Unfortunately the gated 81-minute median study isn’t currently helpful in this regard, because I have to ask myself, why was this study 81 minutes, instead of the others that were 25 or 10 or 40 minutes? But if we can find out whatever X factor increased it to 81 minutes, then might be able to judge how much of that X factor we had in our case, and whether we had more or less X factor than in the study.
I’ll try to find this out from people who have read the study.
Anyway, overall I’ll guess 30% for 21:00-21:30, 20% for 21:30-22:00, 25% for 22:00-23:00, 7% for 23:00-23:30.
Not that it’s surprising, given the roughness of all these estimates, but there seems to be an inconsistency with your other probabilities: if you believe p = 95% for guilt overall, p = 80% conditioned on before 22:00, and p = 57% conditioned on before 21:30, then you have 40 percentage points’ worth of guilt to distribute among the 50% before 22:00, and only 17 of those are taken from the 30% before 21:30; leaving you with 23% to be distributed among the 20% between 21:30 and 22:00, which is impossible.
(EDIT: And also, 55% worth of guilt-probability to be distributed among the 50% post-22:00 probability, likewise impossible.)
Now let’s factor in expert testimony. Since none of our models are working very well, and since the literature that I’ve seen doesn’t converge on a single simple model anyway, I think in the end I’ll go with the independent expert testimony.
Which testimony? Ronchi didn’t give any testimony about lag time, as far as I know.
They also get to factor in the body temperature, which I’ve been ignoring.
Unfortunately, one of the many scandals of this case is that the body temperature measurement was delayed until 12 hours after the discovery of the body, limiting its usefulness.
Thanks for another well-researched reply, let’s have a couple more posts on this, and then turn to the DNA for a bit.
On the other hand, if we do take systemic uncertainty into account (as we ultimately must), a shift of 15:1 or even 5:1 would be significant, given your estimate of .95 probability of guilt, or 19:1 odds.
The problem is that systemic uncertainty works both ways. If I see there being, say, 10 times as much evidence for guilt then there is for innocence, I’ll still cap the probability of guilt at .95 anyway, due to systemic uncertainty. If I change my mind and decided there was 5, or 20, times as much evidence for guilt, the basic conclusion won’t change.
To look at it another way, I expect that if we examine ten pieces of evidence as to whether the Earth is flat, on average one of the pieces can easily point to the Earth being flat at a 10:1 ratio by chance. You would need to either have a much stronger piece of evidence among the first ten pieces, or else have more than one of the pieces point to the Earth being flat, to show that something is true given the first ten pieces of evidence.
How much slippage do you think may have occurred?
There’s a ton of factors here, I’ll guess that if there’s slippage, it’s about 50% that the entire contents would slip; probably our digestion process is evolutionary designed to make the food pass through easily by that stage. Probably another 50% that a suspiciously large amount of food would be found in the small intenstine. I could narrow it down more if I knew how large the volume of the first part of the small intestine to the first bend is, whether the first part of the small intestine was searched, whether the rest of the small intestine was searched, how fast food is evacuated from the duodenum, whether food keeps getting evacuated from the duodenum after the stress of being threatened with a knife occurs, whether peristalsis continues to push food through the small intestine after stress, whether diffusion of food through the small intestine walls continues after stress or even death, and how fast peristalsis and diffusion work.
I’ll add that a search for ‘”empty duodenum” forensics’ suggests to me that, as far as I can tell, almost nobody except for Amanda Knox’s defense has ever cared whether a duodenum was empty or not. That probably puts an upper-bound on how useful this evidence is; if it were reliable, I would expect it come up more often in online appeals-courts decisions, and in trial reporting. I also can’t find any literature on this, which is odd if it’s a useful way to narrow down time of death. So based on the “evidence of absence”, let me propose the following hypothesis:
Vacant Duodenum Hypothesis: “An empty duodenum is not, by itself, definitive proof for or against any time-of-death. The main reason to search the duodenum is in hopes of actually finding food there; no matter what the time-of-death scenario, there is always at least a 1⁄10 chance that the duodenum will be empty when examined.”
If you can find a reference to support the idea that a lag time in excess of four or even three hours would not be highly unusual for a small-to-moderate pizza meal eaten by a healthy adult human, I will update appropriately.
So far, we don’t have data either way about lag times (not median) for a pizza, nor how a follow-up snack affects it. BTW do you know something I don’t about the size of the pizza?
Conversely, if you can’t (and I haven’t been able to so far), I don’t see how you can derive the level of uncertainty you need to make the Massei theory plausible in the face of all the other data. Even acknowledging the wide variation in lag times depending on the type of meal used in the studies, they are all on the short end; there is no indication, anywhere (that I have come across), of the kind of extremes that we would need at the long end.
So far there’s no indication of >180 or even >120 either, right? Is the main point of disagreement that if you see the numbers:
10, 25, 23, 82, 48
and if a genie tells you the next number is above 150, then you’re saying “it’s almost certainly between 150 and 180!” and I’m saying “these numbers are all over the place, it’s more likely to be near 150 than near 300, but there’s a signficant chance it’s a lot bigger than 150.”
I’m confused about the notation T(50): does this refer to half-time, or total emptying time? Because the 317 minutes for fried pasta was total emptying time.
My bad, I misread the abstract. Doesn’t significantly change the scenarios though.
Unfortunately, one of the many scandals of this case is that the body temperature measurement was delayed until 12 hours after the discovery of the body, limiting its usefulness.
So are you claiming that Meredith’s weight before losing blood was 57kg, or just pointing out that a weight of 50-55 kg only shifts us by about 10:1?
On the other hand, if we do take systemic uncertainty into account (as we ultimately must), a shift of 15:1 or even 5:1 would be significant, given your estimate of .95 probability of guilt, or 19:1 odds.
The problem is that systemic uncertainty works both ways.
So to be absolutely clear, then: taking into account all the information you are aware of, and adjusting for systematic uncertainty, what are your current probabilities of guilt conditioned on death having occurred during the following intervals?:
(Be sure to check for consistency with your probability distribution for time-of-death and your overall probability of guilt.)
To look at it another way, I expect that if we examine ten pieces of evidence as to whether the Earth is flat, on average one of the pieces can easily point to the Earth being flat at a 10:1 ratio by chance. You would need to either have a much stronger piece of evidence among the first ten pieces, or else have more than one of the pieces point to the Earth being flat, to show that something is true given the first ten pieces of evidence.
That sounds like a point about priors, rather than systemic uncertainty. What I want to know is the following: if I could show that the time of death was before 21:30, or before 22:00 (etc.), how far would that reduce your current guilt-probability of 95%? (Obviously, if the answer is “negligibly”, then there isn’t any point in discussing gastric lag time.)
I’ll add that a search for ‘”empty duodenum” forensics’ suggests to me that, as far as I can tell, almost nobody except for Amanda Knox’s defense has ever cared whether a duodenum was empty or not.
On the contrary, see here for example. (By the way, it’s actually Sollecito’s defense; the matter is not discussed in Knox’s appeal document.)
The literature often emphasizes that gastric contents are of limited reliability in determining time of death. However, there is a specific circumstance in this case that make it atypically informative: the fact that the duodenum was completely empty, which by default implies that the entire meal was still in the stomach (modulo slippage issue discussed below). This puts a tighter bound on the time of death than in a more typical situation with some smaller fraction of the meal in the stomach.
Vacant Duodenum Hypothesis: “An empty duodenum is not, by itself, definitive proof for or against any time-of-death. The main reason to search the duodenum is in hopes of actually finding food there; no matter what the time-of-death scenario, there is always at least a 1⁄10 chance that the duodenum will be empty when examined.”
I’m not sure how to make sense of this. What matters here is not the emptiness of the duodenum by itself, but rather the conjunction of the empty duodenum with the non-empty stomach. In other words, the phase of digestion—which is clearly time-dependent, with some phases carrying more information about time than others. See for instance the above-cited textbook, which observes as follows:
At autopsy, if 50% of the volume of the last meal is found in the stomach, the last food intake was about 3-4 hours prior to death, with 98% confidence limits not shorter than 1 and not greater than 10 hours.
When 90% of the last meal is found in the stomach, the last food intake was probably within the hour prior to death, with 98% confidence limits not more than 3-4 hours.
If only 30% of the last meal is found, the last food intake was around 4-5 hours previous to death, with 98% confidence limits not shorter than 1-2 and not longer than 10-11 hours prior to death
In the situation at hand, we have 100% of the last meal in the stomach, as revealed by the empty duodenum. This places us in the second bullet, except with even stronger bounds and higher confidence. (And note, by the way, that 3-4 hours is an upper bound on the 98% confidence interval, not the confidence interval itself. I claim that the 98% confidence interval in this case should actually be more like 2.5 hours.)
So to be absolutely clear, then: taking into account all the information you are aware of, and adjusting for systematic uncertainty, what are your current probabilities of guilt conditioned on death having occurred during the following intervals?:
.95 for all the scenarios mentioned, maybe a little less for the 21:00-21:30.
On the contrary, see here for example.
Good find, and it slightly bolsters the case against Knox: contents don’t pass into the duodenum after death (which I expected), and other unspecified parts of digestion continue after death (which I would have bet against). This information slightly increases the probability that the duodenum can empty after death through digestion processes, in which case the duodenum would remain empty no matter what state the stomach is in.
The literature often emphasizes that gastric contents are of limited reliability in determining time of death. However, there is a specific circumstance in this case that make it atypically informative: the fact that the duodenum was completely empty, which by default implies that the entire meal was still in the stomach (modulo slippage issue discussed below)
(snip)
In the situation at hand, we have 100% of the last meal in the stomach, as revealed by the empty duodenum.
But that’s exactly one of the points I’m not confident of. Also even if there is 100% of the meal in the stomach, I still don’t agree that analysis can exclude 21:00-21:30 but include 21:30-22:00 to any large degree of confidence. A model should be robust in its conclusions for us to have confidence in the conclusions; if small, reasonable changes to the model change the conclusions, then we have to limit our level of confidence and weight it with or against corraborating information from elsewhere.
There’s a ton of factors here, I’ll guess that if there’s slippage, it’s about 50% that the entire contents would slip; probably our digestion process is evolutionary designed to make the food pass through easily by that stage. Probably another 50% that a suspiciously large amount of food would be found in the small intenstine.
As it happens, in the present case, the only material found in the small intestine was at the very end, near the ileocecal valve. At least, that is the implication of the wording of Ronchi’s speculation (combined with the absence of any mention by Massei and Cristiani of material nearer to the duodenum):
Prof. Umani Ronchi, at the hearings of 04-19-2008 and 9-19-2009, never discussed “an imperfect application of the ligatures” at the duodenal level, but rather the [supposed] failure to ligature the duodenum on the part of Dr. Lalli during the autopsy (p. 23, hearing of 9-19-2009: “given that the ligatures were not applied, given that without the ligatures this sliding toward the bottom can happen, and that an amount of food that had maybe already passed into the duodenum, could even have, due to gravity, could have gotten all the way to the ileocecal valve.”)
The missing ligature, in fact, allowed Prof. Ronchi to conclude that the gastric contents, at least in part, had slipped in the duodenum or that the contexts, having already passed into the duodenum, could have slid due to gravity all the way to the ileocecal valve after traveling 5 meters of small intestine. From this, the Court deduced that the autopsy finding regarding the objective fact that the duodenum was empty was unreliable.
(Sollecito appeal, p. 165)
Now, if your mistrust of the defense is sufficiently high, perhaps you’re not willing to draw the same inference I have from this passage. However, I’m still interested in the impact it would have on your probability estimates if it were true. Suppose for the sake of argument that there wasn’t anything in the small intestine, save a small amount at the ileocecal valve. How would that affect your beliefs? Are you willing to acknowledge a significant dependence of your opinion on the presence of material in “earlier” parts of the small intestine?
Apart from this, another thing this passage implies is that Ronchi’s speculation about slippage was confined to the possibility of it having occurred at the autopsy, with the intestines uncoiled, in a situation where ligatures had not been applied (which we know to be contrary to the actual situation). He wasn’t suggesting, in other words, that there may have been slippage due to the body having been moved by the killer(s). And if in fact the only material in the small intestine was at the ileocecal valve, then it is very unlikely indeed that material could have slipped through 5 meters of coiled intestine inside the victim’s body, as the slippage hypothesis would in that case require.
So far, we don’t have data either way about lag times (not median) for a pizza, nor how a follow-up snack affects it.
But we do have data for other situations, and those data are what my prior is based on. What’s your prior, and why is it better?
Incidentally, I was able to obtain a copy of the Hellmig et al. paper. Here is the study protocol:
For preparation of the solid test meal, 100 mg of 13C-octanoate was dissolved in an egg. After addition of 50 mL of low-fat milk, the egg was scrambled and fried in a pan. The solid test meal was completed by a piece of brown bread (50 g) and butter (20 g). After an overnight fast a breath sample was collected to define the basic value before the test meal was administered within 10 min. Breath samples were collected every 15 min for the first 120 min, then at 150, 180, 210 and 240 min after ingestion. Patients were again instructed not to drink, eat, smoke or exercise during the test.
\
So far there’s no indication of >180 or even >120 either, right?
The range in the Hellmig et al. study was 29-203 minutes.
Is the main point of disagreement that if you see the numbers:
10, 25, 23, 82, 48
and if a genie tells you the next number is above 150, then you’re saying “it’s almost certainly between 150 and 180!” and I’m saying “these numbers are all over the place, it’s more likely to be near 150 than near 300, but there’s a signficant chance it’s a lot bigger than 150.”
Obviously, it depends crucially on what we know about the process that generated the numbers. Here we’re talking about the duration of a physiological process, which is likely to be distributed approximately normally modulo specific pathological conditions. Of those numbers, the most relevant is the 82 (due to the use of a larger test meal with mixed food groups, and its taking place after the phenomenon described below was discovered).
Beyond differences in the test meal, the shorter times may be accounted for by a phenomenon known as “interdigestive duodenogastric reflux”, which is a “sieving” process involving the shuffling of food back and forth between the stomach and duodenum, that takes place during the lag phase. This phenomenon was not known when some of the earlier studies were published, and so there is a significant possibility that those studies detected duodenal activity that the investigators mistook for the end of the lag phase. (HT to LondonJohn at JREF for this observation.)
But furthermore, we also have to reason about the hypothetical sequences of numbers that we didn’t see. If the numbers had been
110, 125, 123, 182, 148
to say nothing of
100, 250, 230, 820, 480
-- or even if the studies consistently had extreme data points in the range of 300, regardless of their averages—then the Massei-Cristiani theory would be in significantly better shape.
So are you claiming that Meredith’s weight before losing blood was 57kg, or just pointing out that a weight of 50-55 kg only shifts us by about 10:1?
I was actually pointing out that an earlier temperature measurement would probably have permitted a narrower confidence interval.
But, since you mention it, 50-55 kg was just Lalli’s eyeballed guess; the body was not actually weighed. Standard formulas predict 57-60 kg from Meredith’s age, sex and height.
As it happens, in the present case, the only material found in the small intestine was at the very end, near the ileocecal valve.
I agree, but I don’t know whether other material would have been found if present. Is searching the entire small intestine feasible, and if so was such a search performed? Would food in the middle of the small intenstine after death have continued to digest?
Prof. Umani Ronchi, at the hearings of 04-19-2008 and 9-19-2009, never discussed “an imperfect application of the ligatures” at the duodenal level, but rather the [supposed] failure to ligature the duodenum on the part of Dr. Lalli during the autopsy (p. 23, hearing of 9-19-2009: “given that the ligatures were not applied, given that without the ligatures this sliding toward the bottom can happen, and that an amount of food that had maybe already passed into the duodenum, could even have, due to gravity, could have gotten all the way to the ileocecal valve.”)
Right, like the court, I understand that Umani Ronchi was incorrect about the ligatures.
The missing ligature, in fact, allowed Prof. Ronchi to conclude that the gastric contents, at least in part, had slipped in the duodenum or that the contexts, having already passed into the duodenum, could have slid due to gravity all the way to the ileocecal valve after traveling 5 meters of small intestine. From this, the Court deduced that the autopsy finding regarding the objective fact that the duodenum was empty was unreliable.
From Massei, it appears that Umani Ronchi didn’t “conclude that the gastric contents had slipped”; he concluded instead that either the gastric contents might have slipped, or the stomach had not emptied: “He [Umani Ronchi] could not, however, say whether it [the stomach] had partially emptied” (147) and still gave an overall TOD of 20:50 and 4:50. Thus, logically Umani Ronchi didn’t find a TOD of 20:50+ as proving that the stomach has partially emptied. Of course, you can speculate that Umani Ronchi might have been simply being illogical, but to the degree we trust him as the court-appointed expert, we should weigh his conclusion appropriately.
That said, he obviously did make a mistake for some unexplored reason in concluding the ligatures were absent; and I agree we should lower the estimation of his overall reliability, remembering of course to similarly lower the reliability of experts who you do like every time they make a mistake.
Now, if your mistrust of the defense is sufficiently high, perhaps you’re not willing to draw the same inference I have from this passage.
I do trust the defense; I trust them to not unethically stab their client in the back by drawing attention to any inconvenient pro-prosecution facts in their defense appeal document. Pointing out pro-prosecution facts is the prosecution’s and court’s job, not the defense’s, even in inquisitorial systems, and anyway the defense’s checks are signed by the defendant. That said, where the defense appeal document contains a direct quote, then I’d agree that’s pretty reliable.
Are you willing to acknowledge a significant dependence of your opinion on the presence of material in “earlier” parts of the small intestine?
I don’t think it’s a question of significance, it’s more that we’re dealing with a conjunction: that stomach emptying had to have started, that the full small intenstines were searched, and that post-emptying digestion processes would not have emptied the earlier parts of the small intestine. If we can establish that material isn’t present, and that digestion wouldn’t have destroyed the evidence, and that stomach emptying had to have started, then that would establish the TOD you want (even if slippage can’t be ruled out), but so far I can’t agree by more than an order of magnitude that any of those three are true. Given that I didn’t even know until today that digestion processes continue after death, the odds that I’m going to become more confident than that without reliable sources are pretty small.
Apart from this, another thing this passage implies is that Ronchi’s speculation about slippage was confined to the possibility of it having occurred at the autopsy, with the intestines uncoiled, in a situation where ligatures had not been applied (which we know to be contrary to the actual situation).
Yeah, I’ll have to again pass on giving weight to a defense appeal document’s spin about what must have been going through the mind of a court expert for them to be able to say such incriminating-sounding things against their client. It’s the defense’s job to interpret all testimony in as positive a light for the client as possible.
But we do have data for other situations, and those data are what my prior is based on. What’s your prior, and why is it better?
My prior is much vaguer, and reflects my believing I don’t have enough knowledge to have a more narrow prior. I don’t have an answer to “why it’s better”, it’s the one my brain came up with, and I don’t have anyone else’s brain handy to think with.
Incidentally, I was able to obtain a copy of the Hellmig et al. paper. Here is the study protocol:
For preparation of the solid test meal, 100 mg of 13C-octanoate was dissolved in an egg. After addition of 50 mL of low-fat milk, the egg was scrambled and fried in a pan. The solid test meal was completed by a piece of brown bread (50 g) and butter (20 g).
That’s a bit unexpected to me, that looks less than 300 Calories! I would have expected large lag times to be associated with a large meal.
After an overnight fast a breath sample was collected to define the basic value before the test meal was administered within 10 min. Breath samples were collected every 15 min for the first 120 min, then at 150, 180, 210 and 240 min after ingestion. Patients were again instructed not to drink, eat, smoke or exercise during the test.
Right, so that tends to confirm that there’s no exercise or drinking, and they fast before the test. We already agreed drinking probably doesn’t have a huge effect, but Meredith didn’t fast and she might have gotten some physical activity in.
The range in the Hellmig et al. study was 29-203 minutes.
I assume you mean lag time?
Beyond differences in the test meal, the shorter times may be accounted for by a phenomenon known as “interdigestive duodenogastric reflux”, which is a “sieving” process involving the shuffling of food back and forth between the stomach and duodenum, that takes place during the lag phase. This phenomenon was not known when some of the earlier studies were published, and so there is a significant possibility that those studies detected duodenal activity that the investigators mistook for the end of the lag phase. (HT to LondonJohn at JREF for this observation.)
Doesn’t that paradoxically decrease your confidence that you know everything that’s going on with digestive processes and can accurately predict TOD?
-- or even if the studies consistently had extreme data points in the range of 300, regardless of their averages—then the Massei-Cristiani theory would be in significantly better shape.
I agree, the TOD theory would be in even better shape if that were the case.
Standard formulas predict 57-60 kg from Meredith’s age, sex and height.
Different “standard” formulas give different results. Also, the standard deviation of weight based on a/s/h has to be considered. I’ll go with the estimate of the expert who actually saw the body and what her build was, and consider it unlikely that Lalli’s first estimate of weight would be off by more than 15 pounds.
If we model the meal start-time as a normal distribution, then it’ll be simple to add it to the model and combine it with the other sources of uncertainty, since two normal distributions sum to a new normal distribution with a variance equal to the sum of the variances. Though now that I mention it, a lot of the other bits of uncertainty might be somewhat log-normal because they might multiply the time rather than add to it.
To give two contrasting examples, something like female heights (http://www.johndcook.com/blog/2008/07/20/why-heights-are-not-normally-distributed/) would work against Raffaele because outliers are few and extreme, while a gently bimodal distribution like human heights (http://www.johndcook.com/mixture_distribution.html) might work in Raffaele’s favor because of a concentration in the center.
Good question. Let me look here at some more papers. One source of uncertainty is that I don’t know if we care in this case about 2% or 10% or something else.
The first completely-ungated study I found in Google shows 10 minutes for a 2% decrease (http://jnm.snmjournals.org/content/32/7/1349.full.pdf).
Second study shows 25 minutes for a 10% decrease (http://jnm.snmjournals.org/content/32/7/1349.full.pdf).
Third study shows 23 minutes using multiple methods (http://jnm.snmjournals.org/content/37/10/1639.full.pdf).
The gated study you cited shows 81.5 minutes using unknown-to-me methods, perhaps the meal was larger or different from the other studies.
So I guess I would reluctantly discard the concept of attempting solely normal distributions, since this already is looking too right-tailed. So this is too complex for me to easily model, I can only say that intuitively even if we use 19:00, then if a genie tells me it’s at least 120 minutes, then I wouldn’t be much more surprised by 150 minutes or 180 minutes. The first three studies above looked like they were behaving at 10, 25, and 23, and then your example jumped to more than 3x the highest figure so far. So jumping again to even 3x of your number wouldn’t be more than a one-in-ten surprise, especially given the numerous factors I’ve itemized.
If we’re not taking systemic uncertainty into account, then it’s still going to be quite a large probability of guilt. However, I would say that, compared with 23:00, (a) would shift me by about 15:1 on the grounds that the computer evidence would have to be mis-analyzed, or (more likely) Raffaele would have had to manufacture the computer alibi (recall Raffaele is a computer engineer), and (b) by 5:1 on the grounds that the timetable gets a bit tighter than in the 23:00 case. Keep in mind that I’m currently not yet bothering to weigh the eyewitness testimony at all in my assessment of guilt.
I believe the independent court expert more than hearsay that an unknown FRCPath claimed that, even without ligatures, complete slippage is “well-nigh impossible”.
That’s a good point, so I hereby drop the alcohol point altogether for the non-slippage case.
This seems to be for small easily-digested test meals, as far as I can tell. No hospital is going to serve a patient a pizza to determine how well their diabetes is under control. ;-)
I see that large, fried, and/or starchy meals have much larger T(50) times than other meals, and I don’t have any lag times for those. Since T(50) times are frequently unexpectedly large, and since lag times correlate in some large but unknown way with T(50) times, I infer a significant probability that lag times are frequently unexpectedly large as well.
Let me float one scenario. I’d presume that starch increases the T(50) time so much because it can take a long time for large amounts of starch to convert to sugar in the stomach. Does almost the entire portion of starch need to get converted to sugar before any starch can go to the duodenum? If so, then the lag time for a large starch meal would be close to the T(50) time.
Sounds like a whole other discussion.
Based on just stomach evidence, and ignoring expert testimony, I’d have to say it most likely happened around 19:00. So that’s not very useful.
If we take a leap of faith and use the 317 minutes T(50) for 700 kcal fried pasta but don’t believe the starch needs to convert first, then I’d revert to a 1⁄4 guess for lag time on the basis the ratio decreases as T(50) grows, resulting in 80 +/- 6 minutes, so that model fails for me as well, dang it.
Factoring in that it wasn’t before 21:00, but still ignoring expert testimony, I’ll have to take an “inside view” and try to generate hypotheses as to why it took so long. I’ll currently guess that for to get us out to 21:00, either the starch needs to convert to sugar first (40%), or else there was slippage after the body was discovered (5%), or that there was slippage when the body was moved by one or more perps before being discovered by the police and “ligatured” (55%). I’m open to other suggestions. Unfortunately the gated 81-minute median study isn’t currently helpful in this regard, because I have to ask myself, why was this study 81 minutes, instead of the others that were 25 or 10 or 40 minutes? But if we can find out whatever X factor increased it to 81 minutes, then might be able to judge how much of that X factor we had in our case, and whether we had more or less X factor than in the study. Anyway, overall I’ll guess 30% for 21:00-21:30, 20% for 21:30-22:00, 25% for 22:00-23:00, 7% for 23:00-23:30.
Now let’s factor in expert testimony. Since none of our models are working very well, and since the literature that I’ve seen doesn’t converge on a single simple model anyway, I think in the end I’ll go with the independent expert testimony. The experts have access to gated medical journals and even some kind of summary chart of different times under different situations in the literature, as well as forensic experience, which I don’t have. They also get to factor in the body temperature, which I’ve been ignoring.
I’m not following this discussion in detail, but I’m glad you guys are having it—I think it’s a worthwhile case study, and the flavour of the way it’s discussed is informative without getting into the detail of the subject matter.
If we don’t take systemic uncertainty into account, then 15:1 isn’t much of a shift in the face of numbers like 2500:1 or 200000:1 that you were giving for the knife. On the other hand, if we do take systemic uncertainty into account (as we ultimately must), a shift of 15:1 or even 5:1 would be significant, given your estimate of .95 probability of guilt, or 19:1 odds. Crudely approximating this as 20:1, it would take you down to 4:3 (p = 4⁄7 = 0.57) or 4:1 (p= 4⁄5 = 0.8) respectively. I imagine that if you were to believe the 21:26 computer interaction, the pre-22:00 odds could potentially go down to something like 4:3 as well. This indicates that it may be worthwhile to keep pursuing the time-of-death issue.
Well, I was already assuming that you didn’t believe Curatolo (for one thing, he gives Amanda and Raffaele an alibi for 21:30-23:00!). If you do, that will have to be dealt with separately. But in general, not only is eyewitness testimony among the weakest forms of evidence, but it cuts both ways in this case, as the broken-down car illustrates.
Keep in mind, however, that Ronchi’s speculation about slippage was based on the mistaken assumption that ligatures had not been applied. So there may not be as much contradiction here as it appears (especially since it isn’t clear to me that the FRCPath’s opinion necessarily pertained to the non-ligature case, depending on how standard ligatures are).
How much slippage do you think may have occurred?
If you can find a reference to support the idea that a lag time in excess of four or even three hours would not be highly unusual for a small-to-moderate pizza meal eaten by a healthy adult human, I will update appropriately. Conversely, if you can’t (and I haven’t been able to so far), I don’t see how you can derive the level of uncertainty you need to make the Massei theory plausible in the face of all the other data. Even acknowledging the wide variation in lag times depending on the type of meal used in the studies, they are all on the short end; there is no indication, anywhere (that I have come across), of the kind of extremes that we would need at the long end. Not so much as a passing remark. Isn’t this a bit suspicious?
I agree, of course; I just don’t agree that “frequently unexpectedly large” translates to anything like “regularly in excess of four hours”.
I’ll ask Yvain (LW’s medical student extraordinaire) if he knows anything about the mechanisms involved and the plausibility of your proposed scenario. At the very least I expect he’ll know someone who knows something. (Update: Yvain says he doesn’t know any more than we do.)
I’m confused about the notation T(50): does this refer to half-time, or total emptying time? Because the 317 minutes for fried pasta was total emptying time.
I’ll try to find this out from people who have read the study.
Not that it’s surprising, given the roughness of all these estimates, but there seems to be an inconsistency with your other probabilities: if you believe p = 95% for guilt overall, p = 80% conditioned on before 22:00, and p = 57% conditioned on before 21:30, then you have 40 percentage points’ worth of guilt to distribute among the 50% before 22:00, and only 17 of those are taken from the 30% before 21:30; leaving you with 23% to be distributed among the 20% between 21:30 and 22:00, which is impossible.
(EDIT: And also, 55% worth of guilt-probability to be distributed among the 50% post-22:00 probability, likewise impossible.)
Which testimony? Ronchi didn’t give any testimony about lag time, as far as I know.
Unfortunately, one of the many scandals of this case is that the body temperature measurement was delayed until 12 hours after the discovery of the body, limiting its usefulness.
Thanks for another well-researched reply, let’s have a couple more posts on this, and then turn to the DNA for a bit.
The problem is that systemic uncertainty works both ways. If I see there being, say, 10 times as much evidence for guilt then there is for innocence, I’ll still cap the probability of guilt at .95 anyway, due to systemic uncertainty. If I change my mind and decided there was 5, or 20, times as much evidence for guilt, the basic conclusion won’t change.
To look at it another way, I expect that if we examine ten pieces of evidence as to whether the Earth is flat, on average one of the pieces can easily point to the Earth being flat at a 10:1 ratio by chance. You would need to either have a much stronger piece of evidence among the first ten pieces, or else have more than one of the pieces point to the Earth being flat, to show that something is true given the first ten pieces of evidence.
There’s a ton of factors here, I’ll guess that if there’s slippage, it’s about 50% that the entire contents would slip; probably our digestion process is evolutionary designed to make the food pass through easily by that stage. Probably another 50% that a suspiciously large amount of food would be found in the small intenstine. I could narrow it down more if I knew how large the volume of the first part of the small intestine to the first bend is, whether the first part of the small intestine was searched, whether the rest of the small intestine was searched, how fast food is evacuated from the duodenum, whether food keeps getting evacuated from the duodenum after the stress of being threatened with a knife occurs, whether peristalsis continues to push food through the small intestine after stress, whether diffusion of food through the small intestine walls continues after stress or even death, and how fast peristalsis and diffusion work.
I’ll add that a search for ‘”empty duodenum” forensics’ suggests to me that, as far as I can tell, almost nobody except for Amanda Knox’s defense has ever cared whether a duodenum was empty or not. That probably puts an upper-bound on how useful this evidence is; if it were reliable, I would expect it come up more often in online appeals-courts decisions, and in trial reporting. I also can’t find any literature on this, which is odd if it’s a useful way to narrow down time of death. So based on the “evidence of absence”, let me propose the following hypothesis:
Vacant Duodenum Hypothesis: “An empty duodenum is not, by itself, definitive proof for or against any time-of-death. The main reason to search the duodenum is in hopes of actually finding food there; no matter what the time-of-death scenario, there is always at least a 1⁄10 chance that the duodenum will be empty when examined.”
So far, we don’t have data either way about lag times (not median) for a pizza, nor how a follow-up snack affects it. BTW do you know something I don’t about the size of the pizza?
So far there’s no indication of >180 or even >120 either, right? Is the main point of disagreement that if you see the numbers:
10, 25, 23, 82, 48
and if a genie tells you the next number is above 150, then you’re saying “it’s almost certainly between 150 and 180!” and I’m saying “these numbers are all over the place, it’s more likely to be near 150 than near 300, but there’s a signficant chance it’s a lot bigger than 150.”
My bad, I misread the abstract. Doesn’t significantly change the scenarios though.
So are you claiming that Meredith’s weight before losing blood was 57kg, or just pointing out that a weight of 50-55 kg only shifts us by about 10:1?
So to be absolutely clear, then: taking into account all the information you are aware of, and adjusting for systematic uncertainty, what are your current probabilities of guilt conditioned on death having occurred during the following intervals?:
(1) 21:00 − 21:30 (2) 21:30 − 22:00 (3) 22:00 − 23:00 (4) 23:00 − 23:30 (5) after 23:30
(Be sure to check for consistency with your probability distribution for time-of-death and your overall probability of guilt.)
That sounds like a point about priors, rather than systemic uncertainty. What I want to know is the following: if I could show that the time of death was before 21:30, or before 22:00 (etc.), how far would that reduce your current guilt-probability of 95%? (Obviously, if the answer is “negligibly”, then there isn’t any point in discussing gastric lag time.)
On the contrary, see here for example. (By the way, it’s actually Sollecito’s defense; the matter is not discussed in Knox’s appeal document.)
The literature often emphasizes that gastric contents are of limited reliability in determining time of death. However, there is a specific circumstance in this case that make it atypically informative: the fact that the duodenum was completely empty, which by default implies that the entire meal was still in the stomach (modulo slippage issue discussed below). This puts a tighter bound on the time of death than in a more typical situation with some smaller fraction of the meal in the stomach.
I’m not sure how to make sense of this. What matters here is not the emptiness of the duodenum by itself, but rather the conjunction of the empty duodenum with the non-empty stomach. In other words, the phase of digestion—which is clearly time-dependent, with some phases carrying more information about time than others. See for instance the above-cited textbook, which observes as follows:
In the situation at hand, we have 100% of the last meal in the stomach, as revealed by the empty duodenum. This places us in the second bullet, except with even stronger bounds and higher confidence. (And note, by the way, that 3-4 hours is an upper bound on the 98% confidence interval, not the confidence interval itself. I claim that the 98% confidence interval in this case should actually be more like 2.5 hours.)
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.95 for all the scenarios mentioned, maybe a little less for the 21:00-21:30.
Good find, and it slightly bolsters the case against Knox: contents don’t pass into the duodenum after death (which I expected), and other unspecified parts of digestion continue after death (which I would have bet against). This information slightly increases the probability that the duodenum can empty after death through digestion processes, in which case the duodenum would remain empty no matter what state the stomach is in.
(snip)
But that’s exactly one of the points I’m not confident of. Also even if there is 100% of the meal in the stomach, I still don’t agree that analysis can exclude 21:00-21:30 but include 21:30-22:00 to any large degree of confidence. A model should be robust in its conclusions for us to have confidence in the conclusions; if small, reasonable changes to the model change the conclusions, then we have to limit our level of confidence and weight it with or against corraborating information from elsewhere.
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Now, to the slippage issue:
As it happens, in the present case, the only material found in the small intestine was at the very end, near the ileocecal valve. At least, that is the implication of the wording of Ronchi’s speculation (combined with the absence of any mention by Massei and Cristiani of material nearer to the duodenum):
(Sollecito appeal, p. 165)
Now, if your mistrust of the defense is sufficiently high, perhaps you’re not willing to draw the same inference I have from this passage. However, I’m still interested in the impact it would have on your probability estimates if it were true. Suppose for the sake of argument that there wasn’t anything in the small intestine, save a small amount at the ileocecal valve. How would that affect your beliefs? Are you willing to acknowledge a significant dependence of your opinion on the presence of material in “earlier” parts of the small intestine?
Apart from this, another thing this passage implies is that Ronchi’s speculation about slippage was confined to the possibility of it having occurred at the autopsy, with the intestines uncoiled, in a situation where ligatures had not been applied (which we know to be contrary to the actual situation). He wasn’t suggesting, in other words, that there may have been slippage due to the body having been moved by the killer(s). And if in fact the only material in the small intestine was at the ileocecal valve, then it is very unlikely indeed that material could have slipped through 5 meters of coiled intestine inside the victim’s body, as the slippage hypothesis would in that case require.
But we do have data for other situations, and those data are what my prior is based on. What’s your prior, and why is it better?
Incidentally, I was able to obtain a copy of the Hellmig et al. paper. Here is the study protocol:
\
The range in the Hellmig et al. study was 29-203 minutes.
Obviously, it depends crucially on what we know about the process that generated the numbers. Here we’re talking about the duration of a physiological process, which is likely to be distributed approximately normally modulo specific pathological conditions. Of those numbers, the most relevant is the 82 (due to the use of a larger test meal with mixed food groups, and its taking place after the phenomenon described below was discovered).
Beyond differences in the test meal, the shorter times may be accounted for by a phenomenon known as “interdigestive duodenogastric reflux”, which is a “sieving” process involving the shuffling of food back and forth between the stomach and duodenum, that takes place during the lag phase. This phenomenon was not known when some of the earlier studies were published, and so there is a significant possibility that those studies detected duodenal activity that the investigators mistook for the end of the lag phase. (HT to LondonJohn at JREF for this observation.)
But furthermore, we also have to reason about the hypothetical sequences of numbers that we didn’t see. If the numbers had been
110, 125, 123, 182, 148
to say nothing of
100, 250, 230, 820, 480
-- or even if the studies consistently had extreme data points in the range of 300, regardless of their averages—then the Massei-Cristiani theory would be in significantly better shape.
I was actually pointing out that an earlier temperature measurement would probably have permitted a narrower confidence interval.
But, since you mention it, 50-55 kg was just Lalli’s eyeballed guess; the body was not actually weighed. Standard formulas predict 57-60 kg from Meredith’s age, sex and height.
I agree, but I don’t know whether other material would have been found if present. Is searching the entire small intestine feasible, and if so was such a search performed? Would food in the middle of the small intenstine after death have continued to digest?
Right, like the court, I understand that Umani Ronchi was incorrect about the ligatures.
From Massei, it appears that Umani Ronchi didn’t “conclude that the gastric contents had slipped”; he concluded instead that either the gastric contents might have slipped, or the stomach had not emptied: “He [Umani Ronchi] could not, however, say whether it [the stomach] had partially emptied” (147) and still gave an overall TOD of 20:50 and 4:50. Thus, logically Umani Ronchi didn’t find a TOD of 20:50+ as proving that the stomach has partially emptied. Of course, you can speculate that Umani Ronchi might have been simply being illogical, but to the degree we trust him as the court-appointed expert, we should weigh his conclusion appropriately.
That said, he obviously did make a mistake for some unexplored reason in concluding the ligatures were absent; and I agree we should lower the estimation of his overall reliability, remembering of course to similarly lower the reliability of experts who you do like every time they make a mistake.
I do trust the defense; I trust them to not unethically stab their client in the back by drawing attention to any inconvenient pro-prosecution facts in their defense appeal document. Pointing out pro-prosecution facts is the prosecution’s and court’s job, not the defense’s, even in inquisitorial systems, and anyway the defense’s checks are signed by the defendant. That said, where the defense appeal document contains a direct quote, then I’d agree that’s pretty reliable.
I don’t think it’s a question of significance, it’s more that we’re dealing with a conjunction: that stomach emptying had to have started, that the full small intenstines were searched, and that post-emptying digestion processes would not have emptied the earlier parts of the small intestine. If we can establish that material isn’t present, and that digestion wouldn’t have destroyed the evidence, and that stomach emptying had to have started, then that would establish the TOD you want (even if slippage can’t be ruled out), but so far I can’t agree by more than an order of magnitude that any of those three are true. Given that I didn’t even know until today that digestion processes continue after death, the odds that I’m going to become more confident than that without reliable sources are pretty small.
Yeah, I’ll have to again pass on giving weight to a defense appeal document’s spin about what must have been going through the mind of a court expert for them to be able to say such incriminating-sounding things against their client. It’s the defense’s job to interpret all testimony in as positive a light for the client as possible.
My prior is much vaguer, and reflects my believing I don’t have enough knowledge to have a more narrow prior. I don’t have an answer to “why it’s better”, it’s the one my brain came up with, and I don’t have anyone else’s brain handy to think with.
Incidentally, I was able to obtain a copy of the Hellmig et al. paper. Here is the study protocol:
That’s a bit unexpected to me, that looks less than 300 Calories! I would have expected large lag times to be associated with a large meal.
Right, so that tends to confirm that there’s no exercise or drinking, and they fast before the test. We already agreed drinking probably doesn’t have a huge effect, but Meredith didn’t fast and she might have gotten some physical activity in.
I assume you mean lag time?
Doesn’t that paradoxically decrease your confidence that you know everything that’s going on with digestive processes and can accurately predict TOD?
I agree, the TOD theory would be in even better shape if that were the case.
Different “standard” formulas give different results. Also, the standard deviation of weight based on a/s/h has to be considered. I’ll go with the estimate of the expert who actually saw the body and what her build was, and consider it unlikely that Lalli’s first estimate of weight would be off by more than 15 pounds.