We have written many times before (see the links below) about use of the Likelihood Ratio (LR) in legal and forensic analysis.

To recap: the LR is a very good and simple method for determining the extent to which some evidence (such as DNA found at the crime scene matching the defendant) supports one hypothesis (such as "defendant is the source of the DNA") over an alternative hypothesis (such as "defendant is not the source of the DNA"). The previous articles discussed the various problems and misinterpretations surrounding the use of the LR. Many of these arise when the hypotheses are not mutually exclusive and exhaustive. This problem is especially pertinent in the case of 'DNA mixture' evidence, i.e. when some DNA sample relevant to a case comes from more than one person. With modern DNA testing techniques it is common to find DNA samples with multiple (but unknown number of) contributors. In such cases there is no obvious 'pair' of hypotheses that are mutually exclusive and exhaustive, since we have individual hypotheses such as:

- H1: suspect + one unknown
- H2: suspect + one known other
- H3: two unknowns
- H4: suspect + two unknowns
- H5: suspect + one known other + one unknown
- H6: suspect + two known others
- H7: three unknowns
- H8: one known other + two unknowns
- H9: two known others + one unknown
- H10: three known others
- H11: suspect + three unknowns
- etc.

*n*) and then compare the hypothesis "suspect + (

*n*-1) unknowns" with the hypothesis "n unknowns". For example, if there are likely to be 3 contributors then typically the following hypotheses are compared:

- H1: suspect + two unknowns
- H2: three unknowns

- P(E | H1) = 0.00000000000000000001 (10 to the minus 20)
- P(E | H2) = 0.00000000000000000000000001 (10 to the minus 26)

Apart from the

*in court of getting across what this all means (defence lawyers can and do exploit the very low probability of E given H1) and how it is computed, there is an*

**communication problem***with small likelihoods for non-exhaustive hypotheses and I will highlight the problem with two scenarios involving a simple urn example. Superficially, the scenarios seem identical. The first scenario causes no problem but the second one does. The concern is that it is not at all obvious that the DNA mixture problem always corresponds more closely to the first scenario than the second.*

**underlying statistical problem**In both scenarios we assume the following:

There is an urn with 1000 balls – some of which are white. Suppose W is the (unknown) number of white balls. We have 2 hypotheses:

- H1: W=100
- H2: W=90

Scenario 1: We draw 1001 white balls. In this case using standard statistical assumptions we calculate P(E | H1) = 0.013, P(E|H2) = 0.0000036. Both values are small but the LR is large, 3611, strongly favouring H1 over H2.

Scenario 2: We draw 1100 white balls. In this case P(E | H1) = 0.000057, P(E|H2) < 0.00000001. Again both values are very small but the LR is very large, strongly favouring of H1 over H2.

(note: in both cases we could have chosen a much larger sample and got truly tiny likelihoods but these values are sufficient to make the point).

So in what sense are these two scenarios fundamentally different and why is there a problem?

In scenario 1 not only does the conclusion favouring H1 make sense, but the actual number of balls drawn is very close to the

*number we would get if H1 were true (in fact, W=100 is the 'maximum likelihood estimate' for number of balls). So not only does the evidence point to H1 over H2, but also to H1 over any other hypothesis (and there are 1000 different hypotheses W=0, W=1, W=2 etc.).*

**expected**In scenario 2 the evidence is actually even much more supportive of H1 over H2 than in scenario 1.

*.*

**But it is essentially meaningless because it is virtually certain that BOTH hypotheses are false**So, returning to the DNA mixture example, it is certainly not sufficient to compare just two hypotheses. The LR of one million in favour of H1 over H2 may be hiding the fact that neither of these hypotheses is true. It is far better to identify as exhaustive a set of hypotheses as is realistically possible and then determine the individual likelihood value of each hypothesis. We can then identify the hypothesis with the highest likelihood value and consider its LR compared to each of the other hypotheses.

- Confusion over the Likelihood ratio
- Problems with the Likelihood Ratio method for determining probative value of evidence: the need for exhaustive hypotheses
- Misleading DNA evidence
- Barry George case: new insights on the evidence
- Sally Clark revisited: another key statistical oversight?
- Prosecutor fallacy in Stephen Lawrence case?
- Prosecutor fallacy in media reporting of Burgess DNA case
- Flaky DNA: Prosecutors fallacy yet again
- Prosecutors fallacy just will not go away
- Misleading DNA evidence

BRING BACK THE GOOD TIMES, THINK ABOUT THE HAPPIEST TIME IN YOUR RELATIONSHIP, DR OGUDU CAN MAKE IT HAPPEN AGAIN, YOU DESERVE TO BE HAPPY, TODAY BRING BACK LOST LOVE INSTANTLY USING HIS EFFECTIVE SPELLS

ReplyDeleteAre you looking for ultimate spells that works effectively same time on your problems, ranges from Love spells, marriage spells, and carrier related problems, Gay and lesbian extreme love / attraction spells, women of the night Norton accruing powders and oil for extreme ecstasy and protection with in working ordeals. How to get your ex back permanently? Is this what you are asking yourself? Have you been dumped? Can't you just move on? Did he/she chose someone else over you? Your lover shows no interest in you anymore? You might have been asking yourself how to get your ex lover back permanently and you have been looking for unanswered questions about it, Well today is the day for your worries to be eliminated for good. Dr Ogudu have assisted me in getting my ex lover back,Are you going through emotional torture contact him now so he can use his birth gift of casting strong love spells and his super natural magical powers that mean no harm but motivated by peace, wellness and love. The fact that you ask how to get your lover back permanently is the reason why you should no longer face sorrow on your own, and that is because you are in a perfect condition to win over your ex lover and make him or her fall in love with you all over again. It is a great pain to be in love with someone who does not show the same affection back to you as you hope for. However, it is never too late to have all that you wish for and deserve in a lover. Love spells by (DR OGUDU) are pure nature of magic, they are inspired by true mystery of the beauty of the planet earth and they will do well for you as well. CONTACT HIM NOW AND BE HAPPY AGAIN""""

EMAIL--[OGUDUSPELLTEMPLE@YAHOO.COM]

WHATSAPP NUMBER +2348106058254

Website...http://oguduspelltemple.wixsite.com/drogudu