Monday, 11 September 2017

An objective prior probability for guilt?

One of the greatest impediments to the use of probabilistic reasoning in legal arguments is the difficulty in agreeing on an appropriate prior probability that the defendant is guilty. The 'innocent until proven guilty' assumption technically means a prior probability of 0 - a figure that (by Bayesian reasoning) can never be overturned no matter how much evidence follows. Some have suggested the logical equivalent of 1/N where N is the number of people in the world. But this probability is clearly too low as N includes too many who could not physically have committed the crime. On the other hand the often suggested prior 0.5 is too high as it stacks the odds too much against the defendant.

Therefore, even strong supporters of a Bayesian approach seem to think they can and must ignore the need to consider a  prior probability of guilt (indeed it is this thinking that explains the prominence of the 'likelihood ratio' approach discussed so often on this blog).

New work - presented at the 2017 International Conference on Artificial Intelligence and the Law (ICAIL 2017) - shows that, in a large class of cases, it is possible to arrive at a realistic prior that is also as consistent as possible with the legal notion of ‘innocent until proven guilty’. The approach is based first on identifying the 'smallest' time and location from the actual crime scene within which the defendant was definitely present and then estimating the number of people - other than the suspect - who were also within this time/area. If there were n people in total, then before any other evidence is considered each person, including the suspect, has an equal prior probability 1/n of having carried out the crime.

The method applies to cases where we assume a crime has definitely taken place and that it was committed by one person against one other person (e.g. murder, assault, robbery). The work considers both the practical and legal implications of the approach and demonstrates how the prior probability is naturally incorporated into a generic Bayesian network model that allows us to integrate other evidence about the case.

Full details:
Fenton, N. E., Lagnado, D. A., Dahlman, C., & Neil, M. (2017). "The Opportunity Prior: A Simple and Practical Solution to the Prior Probability Problem for Legal Cases". In International Conference on Artificial Intelligence and the Law (ICAIL 2017). Published by ACM. Pre-publication draft.
See also

Thursday, 7 September 2017

Recommendations for Dealing with Quantitative Evidence in Criminal Law

From July to December 2016 the Isaac Newton Institute Programme on Probability and Statistics in Forensic Science in Cambridge hosted many of the world's leading figures from the law, statistics and forensics with a mixture of academics (including mathematicians and legal scholar), forensic practitioners, and practicing lawyers (including judges and eminent QCs). Videos of many of the seminars and presentation from the Programme can be seen here.

A key output of the Programme has now been published. It is a very simple set of twelve guiding principles and recommendations for dealing with quantitative evidence in criminal law for the use of statisticians, forensic scientists and legal professionals. The layout consists of one principle per page as shown below.


Monday, 14 August 2017

The likelihood ratio and its use in the 'grooming gangs' news story

This blog has reported many times previously (see links below) about problems with using the likelihood ratio. Recall that the likelihood ratio is commonly used as a measure of the probative value of some evidence E for a hypothesis H; it is defined as the probability of E given H divided by the probability of E given not H.

There is especially great confusion in its use where we have data for the probability of H given E  rather than for the probability of E given H. Look at the somewhat confusing argument here in relation to the offence of 'child grooming' which is taken directly from the book McLoughlin, P. “Easy Meat: Inside Britain’s Grooming Gang Scandal.” (2016):

Given the sensitive nature of the grooming gangs story in the UK and the increasing number of convictions, it is important to get the maths right. The McLoughlin book is the most thoroughly researched work on the subject.  What the author of the book is attempting to determine is the likelihood ratio of the evidence E with respect to the hypothesis H where:

H: “Offence is committed by a Muslim” (so not H means “Offence is committed by a non-Muslim”)

E: “Offence is child grooming”

In this case, the population data cited by McLoughlin provides our priors P(H)=0.05 and, hence, P(not H)=0.95. But we also have the data on child grooming convictions that gives us P(H | E)=0.9 and, hence, P(not H | E)=0.1.

What we do NOT have here is direct data on either P(E|H) or P(E|not H). However, we can still use Bayes theorem to calculate the likelihood ratio since:

So, in the example we get:

Hence, while the method described in the book is flawed, the conclusion arrived at is (almost) correct.

See also

Friday, 11 August 2017

Automatically generating Bayesian networks in analysis of linked crimes

Constructing an effective and complete Bayesian network (BN) for individual cases that involve multiple related pieces of evidence and hypotheses requires a major investment of effort. Hence, generic BNs have been developed for common situations that only require adapting the underlying probabilities. These so called `idioms’ make it practically possible to build and use BNs in casework without spending unacceptable amounts of time constructing the network. However, in some situations both the probability tables and the structure of the network depend on case specific details.

Examples of such situations are where there are multiple linked crimes. In (deZoete2015) a BN structure was produced for evaluating evidence in cases where a person is suspected of being the offender in multiple possibly linked crimes. In (deZoete2017) this work has been expanded to cover situations with multiple offenders for possibly linked crimes. Although the papers present a methodology of constructing such BNs, the workload associated with constructing them together with the possibility of making mistakes in conditional probability tables, still present unnecessary difficulties for potential users.

As part of the BAYES KNOWLEDGE project, we have developed online accessible GUIs that allow the user to select the parameters that reflect their crime linkage situation (both for one and double offender crime linkage cases). The associated BN is then automatically generated according to the structures described in (deZoete2015) and (deZoete2017). It is presented visually in the GUI and is available as download for the user as a .net file which can be opened in AgenaRisk or another BN software package. These applications both serve as a tool for those interested or working with crime linkage problems and as a proof of principle of the added value of such GUIs to make BNs accessible by removing the effort of constructing every network from scratch.

The GUIs are available from the `DEMO’ tab on the BAYES KNOWLEDGE website and is based on R code, a statistical programming language. This automated workflow can reduce the workload for, in this case, forensic statisticians and increase the mutual understanding between researchers and legal professionals.

Jacob deZoete will be presenting this work at the 10th International Conference on Forensic Inference and Statistics (ICFIS 2017) in Minneapolis, September 2017.


Thursday, 29 June 2017

Queen Mary researchers evaluate impact of new regulations on the Buy-To-Let property market using novel AI methods

In 2015 the British government announced major tax reforms for individual landlords that will be in full effect in tax year 2020/21, being introduced gradually after April 2017. The new reforms and regulations have received much media attention as there has been widespread belief that they were sufficiently skewed against landlords that they could signal the end of the Buy-To-Let (BTL) investment era in the UK.

Research by Anthony Constantinou and Norman Fenton of Queen Mary University of London, has now been published that provides the first comprehensive evaluation of the impact of the reforms on the London BTL property market. The results use a novel model (based on revolutionary new work in an AI method called Bayesian networks) that captures multiple uncertainties and allows investors to assess the impact of various factors of interest on their BTL investment, such as changes in interest rates, capital and rental growth. Additionally, the model allows for portfolio risk management through intervention between time steps, such as the effects of different scenarios of re-mortgaging.

The results show that, over a 10-year period, the overall return-on-investment (ROI) will be reduced under the new tax measures, but that the ROI remains good assuming a common BTL London profile. However, there are major differences depending on the investor strategy. For example, for risk-averse investors who choose not to expand their portfolio, the reforms are expected to have only a marginal negative impact, with the overall ROI reducing from 301% under the old regulations to 290% under the new (-3.7%), and this loss comes exclusively from a decrease in net profits from rental income (-32.2%). However, the impact on risk-seeking investors who aim to expand their property portfolio through leveraging is much more significant, since the new tax reforms are projected to decrease ROI from 941% to 590% (-37.3%), over the same 10-year period.

The impact on net profits also poses substantial risks for loss-making returns excluding capital gains, especially in the case of rising interest rates. While this makes it less desirable or even non-viable for some to continue being a landlord, based on the current status of all factors taken into consideration for simulation, investment prospects are still likely to remain good within a reasonable range of interest rate and capital growth rate variations. Further, the results also indicate that the recent trend of property prices in London increasing faster than rents will not continue for much longer; either capital growth rates will have to decrease, rental growth rates will have to increase, or we shall observe a combination of the two events.

The full paper (with open access link):

Constantinou, A. C., & Fenton, N. (2017). The future of the London Buy-To-Let property market: Simulation with temporal Bayesian Networks. PLoS ONE, 12(6): e0179297, 

The research was supported in part by the European Research Council (ERC) through the research project, ERC-2013-AdG339182-BAYES_KNOWLEDGE, while Agena Ltd provided software support. 

Monday, 6 March 2017

Explaining and predicting football team performance over an entire season

When I was presenting the BBC documentary Climate Changes by Numbers and had to explain the idea of a statistical 'attribution study', I used the analogy of determining which factors most affected the performance of Premiership football teams year on year. Because I had to do it in a hurry I and my colleague Dr Anthony Constantinou did a very crude analysis which focused on a very small number of factors and showed, unsurprisingly, that turnover (i.e. mainly spend on transfer and wages) had the most impact of these. 

We weren't happy with the quality of the study and decided to undertake a much more comprehensive analysis as part of the BAYES-KNOWLEDGE project. This project is all about improved decision-making and risk assessment using a probabilistic technique called Bayesian Networks. In particular, the main objective of the project is to produce useful/accurate predictions and assessments in situations where there is not a lot of data available. In such situations the current fad of 'big data' methods using machine learning techniques do not work; instead we use 'smart-data' -  a method that combines the limited data available with expert causal knowledge and real-world ‘facts’. The idea of predicting Premiership teams' long term performance and identifying the key factors explaining changes was a perfect opportunity to both develop and validate the BAYES-KNOWLEDGE method, especially as we had previously done extensive work in predicting individual premiership match results (see links at bottom).

The results of the study have now been published in one of the premier international AI journals Knowledge Based Systems.

The Bayesian Network model in the paper enables us to predict, before a season starts, the total league points a team is expected to accumulate throughout the season (each team plays 38 games in a season with three points per win and one per draw). The model results compare very favourably against a number of other relevant and different types of models, including some which use far more data. As hoped for the results also provide a novel and comprehensive attribution study of the factors most affecting performance (measured in terms of impact on actual points gained/lost per season). For example, although unsurprisingly, the largest improvements in performance result from massive increases in spending on new players (an 8.49 points gain), an even greater decrease (up to 16.52 points) results from involvement in the European competitions (especially the Europa League) for teams that have previous little experience in such competitions. Also, something  that was very surprising and that possibly confounds bookies - and gives punters good potential for exploiting -  is that promoted teams generate (on average) a staggering increase in performance of 8.34 points, relative to the relegated team they are replacing. The results in the study also partly address/explain the widely accepted 'favourite-longshot bias' observed in bookies odds.

The full reference citation is:
Constantinou, A. C. and Fenton, N. (2017). Towards Smart-Data: Improving predictive accuracy in long-term football team performance. Knowledge-Based Systems, In Press, 2017,
The pre-print version of the paper (pdf) can be found at

We acknowledge the financial support by the European Research Council (ERC) for funding research project, ERC-2013-AdG339182-BAYES_KNOWLEDGE, and Agena Ltd for software support.

See also:

Wednesday, 8 February 2017

Helping US Intelligence Analysts using Bayesian networks

Causal Bayesian networks are at the heart of a major new collaborative research project led by Australian University Monash  - funded by the United States' Intelligence Advanced Research Projects Activity (IARPA). The objective is to help intelligence analysts assess the value of their information. IARPA was set up following the failure of the US intelligence agencies to properly assess the correct levels of threat posed by Al Qaeda in 2001 and Iraq in 2003.

The chief investigator at Monash, Kevin Korb, said in an interview in the Australian:
"..quantitative rather than qualitative methods were crucial in judging the value of intelligence.... more quantitative approaches could have helped contain the ebola epidemic by making authorities appreciate the scale of the problem months earlier. They could also build a better assessment of the likelihood of events like gunfire between vessels in the South China Sea, a substantial devaluation of the Venezuelan currency or a new presidential aspirant in Egypt."
Norman Fenton and Martin Neil (both of Agena and Queen Mary University of London) will be working on the project along with colleagues such as David Lagnado and Ulrike Hahn at UCL.  AgenaRisk will be used throughout the project as the Bayesian network platform.

Further information: