A Security Application

My second CEG search was conducted over a much larger class of hypotheses this time about the nature of the process of radicalisation within prisons. My main focus here is to develop methods to identify groups of individuals who are most likely to engage in specific criminal organization in British prisons. As I will show, this example is very challenging because the classes of each variable are remarkably unbalanced and the percentage of radical prisoners - those units of special interest - is tiny. Furthermore, if expressed in terms of a BN (see Figure 5.11) any plausible generating model would need to be highly context-specific: generic BN model selection methods could therefore not be expected to work well. To accommodate all different types of context-specific dependencies involved in prison radicalisation process a more flexible family such as the CEG class really does need to be used. For the purposes of this illustration I have restricted our analyses to consider only six explanatory variables. These have been chosen because they are often hypothesised as playing a key role in the process of radicalisation. These are:

• Gender - a binary variable distinguishing between male (M) and female (F);

• Religion - a nominal variable with three categories: Rel- religious prisoner, NRel- non-religious prisoner and NRec- not recorded;

• Age - an ordinal variable with three categories: A1- age<30, A2- 30≤ age

<40 and A3- age≥40;

• Offence - a nominal variable with five categories: VAP- violence against person, RBT- robbery, burglary or theft, D- drug, SO- sexual offence and O- others;

• Nationality - a binary variable differentiating between British citizens (B) and foreigners (Fo);

• Network - an ordinal variable differentiating groups of prisoners according to their social interactions with well-known members of the target criminal organisation. It has three categories: I- intense; Fr- frequent; and S- sporadic.

Figure 5.11: Generating BN Model for simulation studies about radicalisation within prisons.

Because of the sensitive nature of data in this field, I have based this example on a data set some of whose variables have been simulated. However I have chosen simulations that are calibrated to real figures and real hypotheses currently in the public domain concerning the British prison population (Ministry of Justice (2013)). So the simulations plausibly parallel the likely current scenario. The generating model used was based on an initially elicited BN depicted in Figure 5.11. The real data set enables us to naively estimate the joint distributions for the first five explanatory variables. These are presented in black in this figure. An important point is that several variables have sparse cell counts: for example Gender (F-5%), Religion (NRec-2%) and Nationality (F-10%).

No data was publicly available for the explanatory variable Network and the re- sponse variable Radicalisation. So in this study I instead construct a probabil- ity model over certain developments based on expert judgements (Cuthbertson (2004), Jordan and Horsburgh (2006), Hannah et al. (2008), Neumann (2010), Silke (2011), Rowe (2014)).

To perform the necessary data simulation I needed to specify the 180 conditional probability distributions of variable Network given the first five explanatory vari- ables. Here I assumed that there are only four different social interaction mecha- nisms; see Table 5.3. For example, male, foreign, younger and non-religious (or not recorded) prisoners who are in jail for violence against person, robbery, burglary,

theft or drug offences are hypothesised to have the strongest tendency to become closer to individuals of the target criminal organisation.

Variable Generating Conditional Probability (%) Number of Mechanism (I,Fr,S)/(H,L)∗ Partitions

Network N1 (75,15,10) 6 N2 (45,30,25) 23 N3 (10,40,50) 79 N4 (1,10,89) 72 Radicalisation R1 (30,70) 6 R2 (3,97) 114 R3 (0.1,99.9) 420

*(I,Fr,S) and (H,l) are, respectively, the category vectors of Network and Radicalisation.

Table 5.3 Generating mechanisms assumed for variables Social Network and Radicali- sation.

The response variable - introduced last - distinguishes between individuals at high or low risk of radicalisation. Being the last variable to be sampled for each pris- oner, this has 540 conditioning partitions. In this environment risk assessments are generally coarse. So based on the expert judgements cited above these partitions are clustered into only three different radicalisation classes of risk (Table 5.3). The highest risk prisoners come from only six partitions that corresponds to those pris- oners who are socially more closed to members of the target criminal organisation. Note that from a technical viewpoint these plausible hypotheses introduce several prior context-specific conditional assessments into our model.

The radicalisation risk of the whole prison population is hypothesised to be small in line with the expert judgement and academic literature (Cuthbertson (2004), Jordan and Horsburgh (2006), Hannah et al. (2008), Neumann (2010), Silke (2011)). Here this is set at around 0.7% of the total population. Based on the premises discussed above, I then simulated 100 complete data sets. Each of these

has 85,000 individuals, approximating the recent yearly totals of the British prison population. Assuming my fixed generating model is true I will now investigate the efficacy of various CEG search methods to identify those prisoners most likely to be radicalised in each of these data sets.

CEG Model Searches

Assume that our optimal model is consistent with a variable sequence Gender, Religion, Age, Offence, Nationality, Network and Radicalisation. This simplifies the search space and matches the goals of this work. The CEG model search was performed using a setting of the hyper-parameter α¯ = 5. This corresponds to the maximum number of categories taken by a variable in the problem. This value is also in line with my previous results that suggest that the selection of a hyper- parameter in this region will provide robust results; see above the CHDS simulation example in Section 5.4.1. I note that this was actually confirmed numerically in additional exploratory studies within this example.

The scale of this problem requires us to use a heuristic algorithm like OAHC since the SCEG space contains more than 101105 _{SCEG models even given the chosen}

variable order. Here full model search strategies such as ones using Dynamic Programming will obviously be infeasible.

As expected the results in Table 5.4 indicate that the OAHC algorithm in con- junction with pm-NLPs was prone to select more parsimonious and user-friendly models than those obtained using standard local priors especially for stages near the leaves of the corresponding event tree. The Hellinger pm-NLPs tend to find a slightly simpler models than the Euclidean pm-NLPs in terms of staged complex- ity. NLPs also ensured that the OAHC algorithm selected models with a number of stages associated with the variables Network and Radicalisation closer to the generating model than those achieved using the Dirichlet local priors.

The use of pm-NLPs enabled the OAHC algorithm to find CEG models that clearly better represented the simulated generating process of radicalisation. For example,

Variable Number of Stages Number of Maximum

DLP Euc-NLP Hel-NLP Generating Stages Number of Stages

Gender 1.0 1.0 1.0 1 1 Religion 2.0 2.0 2.0 ≤2 2 Age 4.8 4.1 4.1 ≤6 6 Offence 6.0 5.9 5.9 ≤6 18 Nationality 7.4 5.4 5.1 ≤10 90 Network 10.2 7.2 6.8 4 180 Radicalisation 7.6 5.6 5.3 3 540

Table 5.4: Average of the Numbers of Stages in 100 Radicalisation CEGs selected by the OAHC algorithm using Direchlet Local Priors (DLP), Eucledian pm-NLPs (Euc-NLP) and Hellinger pm-NLPs (Hel-NLP). It was generated 100 different data set.

Eucledian and Hellinger pm-NLPs classified the highest risk population spuriously in only 29 and 28 date sets, respectively, whilst local priors had problems with 39 data sets. So local priors misclassified some of the highest risk individuals in more than 34% and 39% of the data sets than Euclidean pm-NLPs and Hellinger pm-NLPs, respectively. These misclassifications using local priors and pm-NLPs were associated with the highest risk groups whose sample sizes were less than 25 and whose sample proportions of radical prisoners were concentrated around 12%. Furthermore inference using local priors struggled to identify the risk level for a high risk group of 209 individuals where the sample proportion of radical prisoners was 24%.

There were only three levels of risk of radicalisation in the generating model. So for the sake of simplicity the stages that were found by the OAHC algorithm were amalgamated in Table 5.5 according to their corresponding radicalisation risk in five categories. I matched the risks greater than 25%, between 1% and 7% and less than 1% as corresponding to the risk of 30%, 3% and 0.1% in the generating model, respectively.

Although local and non-local priors yield broadly equivalent estimates for the lower two levels of radicalisation risk, Dirichlet local priors lost track of 9 of the highly hazardous individuals on average whilst pm-NLPs only lost about 6. This means an improvement of 33% in favour of pm-NLPs. The Hellinger pm-NLPs are also a little less prone to misclassified high risk prisoners than the Euclidean pm-NLPs. Note also that local priors unlike the pm-NLPs tend to introduce a stage at risk level between 15% and 25%. If we merged the three higher levels of radicalisation risk into one category, we would lose 3 high risk individuals on average regardless of the type of prior used. However in this case local priors would include 50 more medium risk individuals (3%) in the high category. This would correspond to almost 70% more prisoners that as a result of the analysis would be spuriously identified as a danger to the public.

Although the model used here is rather naive and our results are not perfect, this larger example does nevertheless demonstrate the promise of pm-NLPs used in conjunction with a greedy search of CEG models when applied to much larger scale asymmetric populations like the one above.

Dirichlet Local Prior - Errors Eucledian pm-NLP - Errors Hellinger pm-NLP - Errors Number of

SCEG Risk(%) ≥25 (15,25] (7,15] (1,7] ≤1 ≥25 (15,25] (7,15] (1,7] ≤1 ≥25 (15,25] (7,15] (1,7] ≤1 Prisoners

G e ne ra ti ng M o de l R is k 30 -8.9 2.5 3.1 3.0 0.3 -5.5 0 1.6 3.6 0.3 -4.9 0 1.2 3.6 0.3 699 3 16.4 0.2 111 -887 759 19.4 0 57 -844 768 17.8 0 66.3 -853 769 119×102 0.1 0.9 0 3.5 359 -363 1.1 0 1.4 373 -375 1.1 0 1.0 330 -333 724×102

Table 5.5: Average Number of misclassified prisoners in 100 CEGs selected by the AHC algorithm according to their risk of radicalisation in the Generating Model

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