Statistics and the Evaluation of Evidence for Forensic Scientists - Franco Taroni

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target="_blank" rel="nofollow" href="#fb3_img_img_1833dc40-f090-5ce2-949d-6b4525ab0679.png" alt="images"/>, there is a value of images, images say, where the choice does not matter to You. You are indifferent as to whether images or images is chosen. If images is chosen images. Then it said that images also. In this way, the uncertainty in relation to any event can be measured by a probability images, where images 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 images 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 images or a tail images at any one toss are considered as equally likely, the probability the coin comes down head uppermost is 1/2. Let images denote the event that when a die is rolled it lands 4 uppermost. Then, for a fair die, in which the outcomes images at any one roll are equally likely, the probability the die lands 4 uppermost is 1/6.

      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 images whose outcome is uncertain, the probability that images occurs, images, is defined as the proportion of black balls images in the urn such that if one had to choose the outcome images (the event that a black ball was chosen) where images and the outcome images 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.

      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.

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