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Lecture 5

Lecture 5 shifts the focus a bit to statistics used in forensics. Much of this will be an introduction--and in a couple of upcoming lectures we'll revisit some of what we've already discussed, but in a more thorough fashion, and applied to forensics.

Statistics can be used for two distinct purposes. First, as Lucy points out, they can be used to count up (and describe) types of events or entities, for economic, social, and scientific purposes. Second, they can be used to provide a guide to an uncertain world.

This is very simliar to the distinction we've already made between descriptive and inferential statistics, and to populations and samples. If you want to describe public opinion in the United States, you could just ask every citizen whether they approved of George Bush's job performance. But of course that wouldn't be feasible--so we take a sample. We are interested in "counting up" the number of citizens who approve of the President's job performance (and the number who do not, and the number who are undecided, etc.), but at the same time, we are using statistics to make some expression about uncertainty.

I. A Bit of History

Population statistics date to John Gaunt's (1620-74) publication of Observations Made Upon the Bills of Mortality (a manuscript that focuses on assessing the mortality caused by the plague in four timepoints: 1592-93, 1603, 1625, and 1636. He puts the mortality data into tables, and performs basic calculations-- comparisons of proportions (proportion of those who died to those who were christened, proportion of those who died from the plague to those who died, and so on).

Likewise, the first common recognition of "probability" as an important concept becomes evident at the same time (during the Scientific Revolution of the late 17th century). For example, correspondence between Pascal and Fermat in the mid 1600s make reference to probability in the context of gambling.

II. Types / Meanings of "Probability"

Several debates existing probability are directly relevant to the question of forensics.

There are several relevant terms here:

• aleatory v. epistemic
  
  • Aleatory simply means "by chance"; aleatory probabilities are those which are known because there is full information. We know the probability--we don't need to determine it by observing repeated events. For example, if you have a six-sided die that you are sure is fair, then you know that the probability of rolling a six is 1/6. This is a deduced probability. Note, however, that in reality, there may never be something like complete knowledge--the die may not be completely fair, etc. So "aleatory probabilities" is really a theoretical construct.
    
    Note that "deductive reasoning" refers to inferences where the conclusion is just as certain as the premises.
    
    
  • Epistemic probabilities, on the other hand, are those that we can by observation induce knowledge of the system. If one were to examine a representative group of students, and discovered that 10% of them had visited FaceBook, one would have some knowledge of the structure of FaceBook use--but, because we did not survey every single person in the population, we would be uncertain about that estimate. That uncertainty can be quantified (recall our discussion about margins of error.)
    
    With epistemic probabilities, we're employing the uniformitarian assumption: that the processes in the present will be similar to the processes in the past.
    
    Note that inductive reasoning produces a conclusion that has less certainty than the premises (that is, we're certain of the probabilities in the sample-- but we're estimating the probabilities in the population.
    
    Recall that we said that there were no true, absolute aleatory probabilities. All probabilities are to some degree based on observation -- all probabilities are essentially frequencies. (Note: this is very similar to previous discussions of samples versus populations).
    
    
• realist v. idealist
  
  • So, given that all probabilities are to some degree frequencies--that they can only be estimated based on observing repeated events-- does it even make sense to talk about the "probability" that a single event will happen?
    
    That is, to use the example of FaceBook described above, does it even make sense to essentially say that "if I ask exactly one student, she will be a facebook visitor one time out of ten?"
    
    In order to think of a single event quantified by a probability, we think about probabilities in an "idealist" sense--or subjective probability. It's an interpretation of probability which allows that the probability really doesn't exist in an objective fashion.
    
    
  • A realist interpretation of probability, on the other hand, relies on frequencies that are derived from long runs of events. Realist probabilities also acknowledge that there is uncertainty--but think of the uncertainty as something that can be modeled, or explained.
    
    In reality, of course, all statisticians are willing to make probabilistic statements about single events-- so all statisticians are to some degree "idealistic".
  
  
  
• Frequentist v. Bayesian
  
  But this debate between "idealism" and "realism" mirrors a conflict between the frequentist and Bayesian approaches.
  
  
  • Frequentists tend to argue against subjective probabilities--and for long-run frequency based interpretations of probability.
    
    
  • Bayesians are in favor of subjective probability--and are more confident about discussing the probability of a single event.
    
    
  An example: Forensic scientists doing repeated experiments in a lab--or pollsters conducting surveys of thousands of individuals--would be "frequentists". But assessing events in a criminal case is a single event which therefore demands a Bayesian approach.
  
  But keep in mind that even those who adopt a Bayesian approach are to some degree almost always relying on what they know about repeated events. So the distinction can be thought of as a philosophical distinction--and a continuum of the degree to which one is relying on what one knows based on previous events.
  
  
• instrumentalist view
  
  The instrumentalist view sees probability as just a useful device to employ when talking about uncertainty.
  
  

III. Questions

Define and explain the significance of the following terms, and provide an example (hypothetical or real):

• descriptive and inferential statistics
• populations and samples (why can almost everything be considered a "sample"?)
• population statistics
• aleatory versus epistemic
• deduction versus induction
• uniformitarian assumption
• realist v. idealist
• subjective probability
• frequentist v. Bayesian
• instrumentalist