Читать книгу Statistics and the Evaluation of Evidence for Forensic Scientists - Franco Taroni - Страница 27

An excellent description of probability and its role in forensic science has been given by Lindley (1991). Lindley's description starts with the idea of a standard for uncertainty. He provides an analogy using the concept of balls in an urn. Initially, the balls are of two different colours, black and white. In all other respects, size, weight, texture, etc., they are identical. In particular, if one were to pick a ball from the urn, without looking at its colour, it would not be possible to tell what colour it was. The two colours of balls are in the urn in proportions and for black and white balls, respectively, such that . For example, if there were 10 balls in the urn of which 6 were black and 4 were white, then , and .

The urn is shaken up and the balls thoroughly mixed. A ball is then drawn from the urn. Because of the shaking and mixing, it is assumed that each ball, regardless of colour, is equally likely to be selected. Such a selection process, in which each ball is equally likely to be selected, is known as a random selection, and the chosen ball is said to have been chosen at random.

The ball, chosen at random, can be either black, an event that will be denoted , or white, an event that will be denoted . There are no other possibilities; one and only one of these two events has to occur. The uncertainty of the event , the drawing of a black ball, is related to the proportion of black balls in the urn. If is small (close to zero), is unlikely. If is large (close to 1), is likely. A proportion close to 1/2 implies that and are about equally likely. The proportion is referred to as the probability of obtaining a black ball on a single random drawing from the urn. In a similar way, the proportion is referred to as the probability of obtaining a white ball on a single random drawing from the urn.

Notice that on this simple model probability is represented by a proportion. As such it can vary between 0 and 1. A value of occurs if there are no black balls in the urn, and it is, therefore, impossible to draw a black ball from the urn. The probability of obtaining a black ball on a single random drawing from the urn is zero. A value of occurs if all the balls in the urn are black. It is certain that a ball drawn at random from the urn will be black. The probability of obtaining a black ball on a single random drawing from the urn is one. All values between these extremes of 0 and 1 are possible (by considering very large urns containing very large numbers of balls).

A ball has been drawn at random from the urn. What is the probability that the selected ball is black? The event is the selection of a black ball. Each ball has an equal chance of being selected. The colours black and white of the balls are in the proportions and , respectively. The proportion, , of black balls corresponds to the probability that a ball, drawn in the manner described (i.e. at random) from the urn is black. It is then said that the probability a black ball is drawn from the urn, when selection is made at random, is . Some notation is needed to denote the probability of an event. The probability of , the drawing of a black ball, is denoted and similarly denotes the probability of the drawing of a white ball. Then it can be written that and . Note that

This concept of balls in an urn can be used as a reference for considering uncertain events. The methodology has been described as follows (Lindley 2006):

Your⁶ probability of the uncertain event of rain tomorrow is the fraction of [black] balls in an urn from which the withdrawal of a [black] ball at random is an event of the same uncertainty for you as that of the event of rain. [] You are invited to compare that event with the standard, adjusting the number of [black] balls in the urn until you have the same beliefs in the event and in the standard. Your probability for the event is then the resulting fraction of [black] balls. (p. 35)

Another example concerns a hypothetical sporting event. Let denote the uncertain event that the England football team will win the next major international football championship. Let denote the uncertain event that a black ball will be drawn from the urn. A choice has to be made between and , and this choice has to be ethically neutral. If is chosen and a black ball is drawn from the urn then a prize is won. If is chosen and England do win the championship the same prize is won. The proportion of black balls in the urn is known in advance. Obviously, if then is the better choice, assuming, of course, that England do have some non‐zero probability of winning the championship. If then is the better choice. Somewhere in the interval , there is a value of , say, where the choice does not matter to You. You are indifferent as to whether or is chosen. If is chosen . Then it said that also. In this way, the uncertainty in relation to any event can be measured by a probability , where is the proportion of black balls, which leads to indifference between the two choices, namely, the choice of drawing a black ball from the urn and the choice of the uncertain event in whose probability one is interested.

Notice, though, that there is a difference between these two probabilities. By counting, the proportion of black balls in the urn can be determined precisely. Probabilities of other events such as the outcome of the toss of a coin or the roll of a die are also relatively straightforward to determine, based on assumed physical characteristics such as fair coins and fair dice. Let denote the event that when a coin is tossed it lands head uppermost. Then, for a fair coin, in which the outcomes of a head or a tail at any one toss are considered as equally likely, the probability the coin comes down head uppermost is 1/2. Let denote the event that when a die is rolled it lands 4 uppermost. Then, for a fair die, in which the outcomes at any one roll are equally likely, the probability the die lands 4 uppermost is 1/6.

Probabilities relating to the outcomes of sporting events, such as football matches or championships or horse races, or to the outcome of a civil or criminal trial, are rather different in nature. It may be difficult to decide on a particular value for . The value may change as evidence accumulates such as the results of particular matches and the fitness or otherwise of particular players, or the fitness of horses, the identity of the jockey, the going of the race track, etc. Also, different people may attach different values to the probability of a particular event.

These kinds of probability – as briefly specified before in Section 1.3 – are sometimes known as subjective or personal probabilities; see de Finetti (1933), Savage (1954), Good (1959), DeGroot (1970), and the more recent publication by Kadane (2011). Another term is measure of belief since the probability may be thought to provide a measure of one's belief in a particular event. A philosophical discussion on the use of those terms is given in Lucena‐Molina (2016, 2017). Despite these difficulties the arguments concerning probability still hold. Given an event whose outcome is uncertain, the probability that occurs, , is defined as the proportion of black balls in the urn such that if one had to choose the outcome (the event that a black ball was chosen) where and the outcome then one would be indifferent to which one was chosen. There are difficulties but the point of importance is that a standard for probability exists. An extended comment on subjective probabilities is given in Sections 1.7.5–1.7.7.

A use of probability as a measure of belief is described in Section 1.7.5 where it is used to represent relevance. The differences and similarities in the two kinds of probability discussed earlier and their ability to be combined have been referred to as a duality (Hacking 1975).

It is helpful also to consider two quotes concerning the relationship amongst probability, logic and consistency, both from Ramsey (1931).

We find, therefore, that a precise account of the nature of partial beliefs reveals that the laws of probability are laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency. They do not depend for their meaning on any degree of belief in a proposition being uniquely determined as the rational one; they merely distinguish those sets of beliefs which obey them as consistent ones. (p. 182)

We do not regard it as belonging to formal logic to say what should be a man's expectation of drawing a white or black ball from an urn; his original expectations may within the limits of consistency be any he likes; all we have to point out is that if he has certain expectations he is bound in consistency to have certain others. This is simply bringing probability into line with ordinary formal logic, which does not criticise premises but merely declares that certain conclusions are the only ones consistent with them. (p. 189)

In brief, a person is entitled to their own measures of belief, but must be consistent with them. Ramsey's remarks relate to the appropriateness of a set of probabilities held by a particular individual. This appropriateness needs to be checked. Probability values need to be expressed in an operational way that will also make clear what coherence means and what coherent conditions are. De Finetti (1976) framed the operational perspective as follows:

However, it must be stated explicitly how these subjective probabilities are defined, i.e. in order to give an operative (and not an empty verbalistic) definition, it is necessary to indicate a procedure, albeit idealised but not distorted, an (effective or conceptual) experiment for its measurement. (p. 212)

Therefore, one should keep in mind the distinction between the definition and the assessment of probability. A description of de Finetti's perspective has been published by Dawid and Galavotti (2009).

One way in which these expressions can be checked is to measure probabilities maintained by an individual in terms of bets the individual is willing to accept. An alternative to consideration of balls in an urn is to consider two lotteries. An individual probability can be determined using a process known as elicitation. In this context, elicitation is the comparison of two lotteries of the same price. Consider a situation in which it is of interest to determine a probability for rain tomorrow. This example can be found in Winkler (1996). There are two lotteries:

 Lottery A: Win £100 with probability 0.5 or win nothing with probability 0.5.

 Lottery B: Win £100 if it rains tomorrow or win nothing if it does not rain tomorrow.

In this situation, it is reasonable to assume a person would choose that lottery which, in their opinion, presents the greater probability of winning the prize. If lottery B is preferred, then this indicates that one considers the probability of rain tomorrow to be greater than 0.5. Similarly, a choice of lottery A implies the probability of rain tomorrow is less than 0.5. Additionally, in a case in which one is indifferent between the two lotteries, one's probability for rain tomorrow equates with the probability of winning the prize in lottery A. Therefore, a procedure can be devised in which the probability of winning lottery A is adjusted so that the individual, whose probability for a proposition of interest is to be elicited, is indifferent with respect to lotteries A and B. In a similar manner, the personal probability of an individual for any event of interest can be elicited.

The possibility that subjective degrees of belief may be represented in terms of betting rates in lotteries or in the relative frequency of balls in an urn is often put forward as support for an argument that requires subjective degrees of belief to satisfy the laws of probability. This requirement is satisfied with the notion of coherence that has the normative role of forcing people to be honest and to make the best assessment of their own measure of belief.

De Finetti (1931a) showed that coherence, a simple economic behavioural criterion, implies that a given individual should avoid a combination of probability assignments that is guaranteed to lead to loss. All that is needed to ensure such avoidance is for uncertainty to be represented and manipulated using the theory of probability. In this context, the possibility of representing subjective degrees of belief in terms of betting odds is often forwarded as part of a line of argument to require that subjective degrees of belief should satisfy the laws of probability. This line of argument takes two parts. The first is that betting odds should be coherent, in the sense that they should not be open to a sure‐loss contract. The second part is that a set of betting odds is coherent if and only if it satisfies the laws of probability. The Dutch Book argument encompasses both parts: the proof that betting odds are not open to a sure loss contract if and only if they are probabilities is called the Dutch book theorem. Thus, if an individual translates their state of knowledge in such a manner that the assigned probabilities, as a whole, do not respect the laws of probability (standard probability axioms), then their assignments are not coherent. In this context, such incoherence is also called logical imprudence. An example can be found in Section 1.7.6.