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A Bayesian analysis of stops and arrests

Let В and В denote, respectively, the events that a person who is stopped is, or is not, Black. Let A and A denote, respectively, the events that a person is, or is not, arrested after a stop. Then, the probabilities that a person is/is not arrested (following a stop), given that he or she is Black are, respectively,

Then the risk ratio (RR) associated with being Black is

where фв _ P(B_]_A)_ js so-called Bayes factor (BF) applied to persons P(B I A)

who are Black.

The RR is the odds of a null hypothesis being true (A: the person is arrested) to another, competing, hypothesis being true (A: person is not arrested) under a particular set of data (the person is Black). On the other hand, the BF is the odds of the data being observed (the person is Black) when the null hypothesis isjrue [A: the person is arrested) to when the alternative hypothesis is true (Л: person is not arrested).

The BF provides a measure of whether the data (the “appearance” of the person) have increased or decreased the odds of the null hypothesis, against the alternative hypothesis, being true: Фв > 1, signifies that the odds of being arrested, relative to not being arrested, have increased if the person is Black; Фв < 1, signifies that these odds have decreased.

However, in cases of stopping potential offenders, the natural question to ask is the following: given that two persons - one Black, the other non- Black - present themselves before a police officer, what is the ratio of their probabilities of being arrested? In other words, the relevant “RR” for the pi A | В)

security officials is ^ ^ What are the odds that the hypothesis (namely,

a person is arrested) is true under two different “pieces” of information - the person is Black, the person is not Black? This ratio may be evaluated as

P(B I A)

where Ur = - . may be termed the inverse Bayes factor (IBF) applied

I (d I Л.)

to persons who are suitable for arrest. Intuitively, the IBF is the odds-ratio of the null hypothesis being true (a person is suitable for arrest) under one set of data (the person is Black) against it being true under an obverse set of data (the person is not Black). If QA > 1, one is more likely to observe the null hypothesis (A: the person is arrested) under one set of data (B: the person is Black) than under the obverse (B: the person is not Black).

Table 5.7 shows the total number of stops, and subsequent arrests, made by the police in the Metropolitan Area over the 13-year period from 2006- 07 to 2018-19: in this period, the Metropolitan police made 4,424,709 stops and, following these stops, made 448,773 arrests, yielding an arrest rate of 10.1%. Of the total number of stops, 46.6% were of White persons and 30.8% were of Blacks while of the total number of post-stop arrests, 45.1% were of White persons and 34.4% were of Blacks.

Using the data in Table 5.7, the BF (Фл) and the IBF (UA) values, and their associated RRs, фв and грА, may be calculated by comparing Blacks with non-Blacks:

Table 5.7 Stops and arrests: London Metropolitan Area, 2006-07 to 2018-19






Proportion of Stops (%)

Proportion of Arrests (%)

Proportion in Area Population




















Mixed Race


















Source: Own calculations from MoJ (2019).

The value of the BF shows that if a person stopped was arrested, the chances of him being Black were 13% higher (Фв = 1.13) compared to the chances of him being Black if he was not arrestedв = P(B I A)/P(B I Л». On the other hand, the value of IBF shows that if a person stopped was arrested, their chances of being Black would be slightly more than a half of the likelihood of an arrested person not being Black (0A = P(B I A)/P{B I A) =0.53). So, in answer to the question facing a police officer - namely, what are the odds that the hypothesis (namely, a person is arrested) is true if the person is Black/not Black - the chances of a Black person being arrested after a stop are 19% higher than for a non-Black person.


The question is whether the fact that the chances of a Black person being arrested after a stop are 19% higher than for a non-Black person is enough to justify the very high stop rates of Black persons noted in Table 5.1. In the absence of such justification, the damage caused to race relations, through the large-scale stopping of innocent Blacks - by an essentially “White” police force - has to be the central point of concern about the implementation of stops in E&W.19 As Borooah (2011) has written, the notion that Black boys and men are more likely to be stopped by the police than their White counterparts has been a feature of British life (Ryder, 2009). In the US, Black motorists reported being stopped more often than White motorists and, when stopped, disproportionately believed that race was the reason for the stop (Engel and Calnon, 2004; Reitzel and Piquero, 2006).20

A reason for persisting with stops, notwithstanding their ineffectiveness and intrusiveness, is provided by McConville et al. (1991, 1997) who argue that “the aims of stops and arrests are often not to enforce the law per se but to secure broader objectives: the imposition of order, the assertion of authority, the acquisition of information” (McConville et ai, 1991, p. 16). On the assertion of authority argument, stops are a valuable tool of policing precisely because they are intrusive and humiliating.

If racial disparity is due to the presence of racist officers, rather than to organisational policies, then the question arises as to the source of prejudice. It could be conscious, explicit prejudice on which officers act through their own volition (Becker, 1957; Fiske, 1998), or it could be unconscious prejudice which is triggered in certain circumstances and on which the protagonists have little control. To paraphrase Graham and Lowery (2004), under unconscious thought processes, actions leading to the disparate treatment of racial minorities are unintentional, involuntary, and effortless. If racial disparity in the treatment of minorities is based on prejudice, conscious or unconscious, then the issue is about the training police officers should receive in order to overcome negative stereotypes: recent findings (Lowery et al., 2001) focus on the important role of good interracial/ethnic social relations in altering negative perceptions.

With the publication of the MacPherson report in 1999 into the death of the Black teenager Stephen Lawrence (Home Office, 1999), the term institutional racism has been applied to the attitudes of the police - and, indeed, several other public bodies - towards minority groups. This form of racism was defined by the report’s author, Sir William MacPherson, as the “collective failure of an organisation to provide an appropriate and professional service to people because of their colour, culture, or ethnic origin ... [which] through unwitting prejudice, ignorance, thoughtlessness, and racist stereotyping disadvantages minority ethnic people”. Ten years after the death of Stephen Lawrence, however, Commander Cressida Dick, then head of the Metropolitan Police’s Diversity Directorate, warned that it might be very difficult to conceive of a situation in which there was no institutional racism (Dick, 2003).

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