I use Bayes theorem: posterior odds equals prior odds times likelihood ratio. For an introduction, please read this nice blog post https://entropicthoughts.com/bayes-rule-odds-form
I use this rule, Bayes’ rule, repeatedly, each time taking account of another part of the evidence. It is named for Thomas Bayes, a presbyterian minister and mathematician, who was interested in using it to find a mathematical proof of the existence of God. https://en.wikipedia.org/wiki/Thomas_Bayes
The likelihood ratio for the question at hand, based on some part of the evidence, is the ratio of the probabilities of that part of the evidence under the two competing hypotheses. More precisely, one uses the conditional probabilities of that fact given previously incorporated evidence. We have to start somewhere and we start by describing two alternative hypotheses and our probabilities or degrees of belief or personal betting odds for those two hypotheses, before further evidence is taken into account.
Let’s start with the news reports of a police investigation of a possible killer nurse at a neonatology unit in the UK; the investigation being triggered by a disturbing spike in the death rate on that unit.
I think that in the last fifty years there simply hasn’t been been a case in the UK of a killer nurse on a neonatal ward, except possibly the case of Beverley Allitt. One might argue that there do exist doubts as to the safety of her conviction, or one might argue that there can have been serial killer nurses who completely evaded detection. Did Alittit work in an intensive care unit? I also think that in recent years, every year has seen a scandalous calamity in a UK neonatal ward, leading to avoidable deaths of quite a few babies. So a priori: the relative chances of a killer nurse being responsible for the spike, or simply poor care, is in my estimation 50:1 in favour of poor care in a failing hospital unit rather than activity of a killer. If you disagree, give me your arguments for both those rates and hence their ratio. If you would like to take a different starting point, try that. Eg, what is the chance a random nurse is a serial killer? At some point one will have to use the information that this was a neonatal unit and one will have to take account of the “normal” rate of deaths on the unit. I think my choice is reasonably specific. One could argue that the prior odds should be 10 to 1, or 100 to 1, instead of 50 to 1. I expect that most people will at least agree that killer nurses on neonatal units are very rare, disastrously poor care on a neonatal unit in the UK is not rare at all.
So we are back in 2017 and hear the news and rightly we should be sceptical that there really is a case here. But clearly there are grounds to investigate what is the cause of that spike, and maybe there is more information which the police already have.
Then, many years go by. A particular nurse is detained for questioning in two successive years; and finally arrested in a third year. Two more years go by (Corona). At last, a trial begins. It turns out that roughly seven years of police investigation has uncovered no direct evidence at all (neither medical evidence, toxicological evidence, witness testimony or CCTV recordings, finger prints or DNA) of unlawful action by the nurse who has been under intensive investigation all that time. And not just no evidence against that nurse – no direct strong evidence of malevolent activity by anyone.
One might want to argue that the insulin evidence is strong toxicological evidence. We could argue about that for a long time. Even if one or two babies were given unauthorised doses of insulin there is no direct proof that Lucy Letby did that herself. There is the possibility of accidental administration (twins in adjacent cots). The argument that Lucy did administer insulin seems to have been that we know at some point she carried out other murderous attacks and it is unlikely that there were two murderous nurses working in the unit. But why do we believe there are murderous nurses working on the unit? This argument can only be made after hearing all the other evidence in the case.
So we have to estimate the probability of a 7 year police hunt for evidence of murder by a particular nurse finding no direct evidence of any malevolent activity at all by anyone, if Lucy Letby actually was innocent, and if she truly was a serial killer. In my opinion ,what we actually observed is much more likely under the innocence hypothesis than under the guilty hypothesis. If she truly is innocent the chance of finding powerful directly incriminating evidence must be rather small; if she truly is a serial killer then it must be unlikely that that no baby can be definitely proven to have been murdered or attacked. I guess the two probabilities of no hard evidence to be 95% and 5% respectively. These are probabilities of 19/20 and 1/20 respectively, so a likelihood ratio of 19. I’ll be a bit more cautious and call it 10.
We already had odds of 50:1 in favour of innocence. We have a likelihood ratio of 10:1 in favour of innocence, having learnt that police investigation uncovered no strong and direct proof of malevolent harm to any baby. The odds on Lucy being innocent are therefore now 50 times 10, or 500 to 1.
Let’s now bring in the evidence from psychology. Are there reasons to believe Lucy is a psychopath? Which surely she must be, if she is a serial killer of babies in her care. It seems there is no reason at all to suspect she is a psychopath. I think that there very likely would be strong independent signs of psychopathy in her life history if she really is a serial killer, but obviously not so likely if she is completely innocent. [Clearly she could be both a psychopath but did not actually harm or try to harm any baby. I don’t think this is an interesting hypothesis to explore. I will also not pay attention to the Munchhausen by proxy idea, that she was trying to attract the attention of an older male doctor. All the evidence says that he was more romantically interested in her, than vice versa.]
Put the likelihood ratio at 2, ie twice as likely to see no evidence for psychopathy if innocent, than if a serial killer. Actually I think it should be closer to 10. We should ask some psychologists. Lucy Letby did not sadistically kill little animals when she was a child. By all accounts, she was a dedicated nurse and cared deeply for her work.
We were at 500 to 1 for innocence. Factor in a likelihood ratio of 2 for psychological evidence. Now it’s 1000 to 1. But we are not done yet.
Next, I would like to take account of the statistical evidence that the spike in deaths is quite adequately explained by the acuity of the patients being treated in those 18 months. I would say that this is exactly what we would expect if Lucy is innocent but very unlikely if she’s a serial killer. I think this hypothesis is very adequately supported by published MBRRACE-UK statistics, and what we know about the acuity of the babies in the case. We know why acuity went up in around 2014 and we know why it went down midway in 2017. The spike seems to have been caused by hospital policy which was being made and implemented by the consultants on that unit. They should have expected it.
Say a likelihood ratio of 10. That brings us to 10,000 to 1 she’s innocent; a posterior probability of 99.99%. I haven’t yet brought in the facts of an investigation driven by tunnel vision and coached by doctors who, as we now know, were making quite a few deadly mistakes themselves. I haven’t brought in yet the innocent explanation of the post-it note. In my opinion, the post-it note is powerful evidence for innocence; it makes absolutely no sense under the hypothesis of guilt. The irrelevance of the handover notes and the notations in her diary. Facebook searches? Her alleged lies (about what she was wearing when she was arrested). Anything else?
Anyway, I am now well above 99.99% sure that Lucy is innocent and since the press conference and the report of Shoo Lee and his colleagues, we can all be even more sure that that is the case.
Me too !
God bless the experts, statisticians, neonatologists, and any other expert types who are making a contribution leading to the overturning of guilty verdict against Lucy Letby. This non-expert relied on gut instinct and believed from the very outset that Lucy had been fitted up. However my gut instinct wouldn’t be of much value in a court of law, expert evidence would. It will happen (in time), Lucy Letby will be set free. I would like to see police officers CPS officials, doctors, expert prosecution witnesses charged, prosecuted and jailed for their parts in the most serious miscarriage of justice ever. Something else, Sally Clark was never able to overcome her prison sentence for murder of her two young babies even after being exonerated and set free. She died from acute alcohol poisoning four years after her release from prison. What of Lucy and her future life?
I wholeheartedly agree with you with the emphasis being on exonerating and releasing Lucy asap. Ultimately I think that everyone involved with this appalling case will find it difficult to manage any personal impact . We might never hear about what goes on behind the scenes but conscience , alone, can be hard to bear Immediately, I would like to see that Lucy L., is addressed with respect and all attempts to dehumanise her by calling her “LETBY” cease. She has been the subject of dreadful indignity , baseless accusations etc., and loss of freedom. She will not be impervious to the effects ( likewise for others who have been misjudged and incarcerated) and I hope that she receives, at the very least, a full, sincere and relevant apology and suitable / acceptable support both now and following her release..
Maybe, hopefully, it makes a difference whether you are suspected/convicted of killing your own children, or being accused of killing babies at your place of work. Lucy’s parents never stopped believing in her, and the same goes for a number of her friends. Furthermore thanks to “social media” she can see how a large number of people never believed her to be a murderer. And some really went to bat for her.
Her case may very well be the start of some reforms in the UK judicial system. That said, she and her parents better emigrate to a place where her name is not associated with horrible deeds. The Consultants, CoCH management, Chester PD, Johnson KC c.s., Judge Goss, Justice Thirlwall, Dr. Evans and his fanclub, most UK “journalists” and their editors, not only took away several years of her life, they also took away her future in the UK. She and her family should be given restart in different surroundings, under a different name. I hope she becomes a lawyer.
I agree. Canada might be a good choice but she will need support wherever she settles I hope that she is strengthened by the knowledge that many of us never doubted her innocence and this includes her parents and friends plus her legal teams. Best Wishes.
We need a reform (reshaping) of human nature more than reforms of our judicial system. Those with malign intent will always find ways of executing their evil whatever the system.
I hesitated to click on “like” because, unfortunately, what you say is true.
Dr Shoo Lee’s meeting earlier this week pretty well confirmed that the level of all kinds of care led to the lethal consequences for NICU babies at CoCH. Inadequacy of resources can lead to poor outcomes even when there is a will to do everything as well as possible.
I am no expert but in my experience budgets in public service are usually a consultative process with suitable. senior departmental staff invited to provide comments where a reduction on previous funding levels are proposed. This can result in the need to make a larger reduction elsewhere and further comments would then be sought and so on. If there is an ongoing, year on year requirement for budget reductions there is a real possibility of damage to the service at the sharp end .
Had CoCH reached the point where funding had to be spread very thinly – was there any opportunity for review?
PS.,
I have watched the press conference again as I recalled that infection was mentioned frequently. The 17 cases of which Lucy was accused were placed in 6 different groups for the purpose of examination by the panel of experts. The findings based on 2 cases from each group were presented to the conference for the sake of limiting time. I decided to make a note of the incidences of any mention of the word infection ( which includes infection, named infection, bacteria and fever) and I found 50 approx., A figure very significantly higher than I have heard elsewhere including the SI., or prosecution evidence. And , it appears that anti biotics were not always given – were appropriate tests applied ?
I do not know how much detail was given when Lucy was charged with the murder and attempted murder of certain babies at CoCH NICU. Was she arrested and held., under the broad headings ( murder / attempted murder etc.) or were there more specific charges made such as overfeeding, inappropriate use of unsulin and so on?
I do not know enough about the court systems but it does seem to me that if Lucy and her defence team were not aware of the full details of the charges until the EW., evidence was delivered at the trial , it would probably be impractical to find a suitably qualified Expert Witness for her defence mid proceedings.
A thought -the J.B.Priestley play “An Inspector Calls” is about conscience and the individual and collective responsibility of a family group who each played a part in the downfall of a person known to them. It happens.
“A man who has served 38 years in prison for the murder of a woman has had his conviction quashed by the Court of Appeal after new DNA evidence emerged”. Perhaps the CCRC does have equally (to Lucy Letby) important cases to review and Lucy might well have to wait her turn. Lucy has gathered the headlines and many supporters, it would appear there are others like Lucy, these anonymous individuals have not gathered the headlines nor the supporters.
It seems to me that the moment that there appears to be good reason to think that a miscarriage of justice may have occurred steps should be taken to examine the case. It is almost unbelievable that Andrew Malkinson ( sp) could have been exonerated and released from prison many years ago and another case has been in the news today. It is a good thing that these people have now been freed but at what cost ? And how can and should they be compensated ? Lack of freedom, lack of opportunity to follow career choices, impact on physical and mental health , expectations of a family life are all ambitions / personal wishes denied along with destruction of reputations. I am sure there are more aspects.
I was once told that it is better that a guilty person goes free than an innocent one is imprisoned- I haven’t seen much evidence of that lately. Surely that statement must mean that Guilt must be proven beyond all reasonable doubt.
PS.,
I note that currently there is a serious shortage of prison cells and a surfeit of imprisoned miscarriage of justice cases. Action regarding the latter group might ease the occupancy rate a little !