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This chapter defines the concepts of natural sampling, natural frequencies, and reports experimental evidence for the impact of various external representations on statistical thinking. The mental strategies or shortcuts people use, not only their numerical estimates of risks, turn out to be a function of the external representation of numbers we choose. This chapter provides a theoretical framework that specifies why frequency formats should improve Bayesian reasoning and presents two studies that test whether they do. Its goal is to lead research on Bayesian inference out of the present conceptual cul-de-sac and to shift the focus from human errors to human engineering: how to help people reason the Bayesian way without even teaching them.

This chapter reviews evidence showing that people are remarkably sensitive to the rarity of events when making inferences about them. Indeed, people are so attuned to event rarity that their implicit assumptions about rarity guide their thinking even in laboratory tasks where experimenters have implicitly assumed that rarity would not matter. Lack of awareness of this problem has led many experimenters to misinterpret people’s adaptive responses as irrational. Indeed, focusing on data that are rare leads people to behave in a qualitatively Bayesian manner. These points are illustrated using tasks that involve assessing the covariation between variables, evaluating hypotheses after passively receiving data, and actively searching for data to test hypotheses. Participants’ sensitivity to, and assumptions about, rarity have important implications for understanding lay inference.

Cognitive illusions have been linked to perceptual illusions, suggesting that they are “inevitable illusions”. This chapter criticizes the narrow norms that make humans look irrational and shows how to make inevitable illusions “evitable”. These so-called cognitive illusions largely disappear when one pays attention to conceptual distinctions. Cognitive illusions largely disappear due to the following: polysemy — not all probabilities are mathematical probabilities; a mathematical probability refers to a reference class, which may differ depending on the task; and natural frequencies facilitate Bayesian reasoning.

This chapter presents a formal model that captures both case‐based and rule‐based reasoning. The model is general enough to describe Bayesian reasoning, which may be viewed as an extreme example of rule‐based reasoning. It suggests conditions under which Bayesian reasoning will give way to other modes of reasoning, and alternative conditions under which the opposite conclusion holds. It discusses how probabilistic reasoning may emerge periodically, with other modes of reasoning used between the regimes of different probabilistic models.

Subjective Bayesianism is often criticized for a lack of objectivity: (i) it opens the door to the influence of values and biases, (ii) evidence judgments can vary substantially between scientists, (iii) it is not suited for informing policy decisions. We rebut these concerns by bridging the debates on scientific objectivity and Bayesian inference in statistics. First, we show that the above concerns arise equally for frequentist statistical inference. Second, we argue that the involved senses of objectivity are epistemically inert. Third, we show that Subjective Bayesianism promotes other, epistemically relevant senses of scientific objectivity—most notably by increasing the transparency of scientific reasoning.

On 14 October 2011, the Apple Computer Corporation launched the latest generation of the iPhone 4S mobile phone. The iPhone 4S included Siri, a speech interface that allows users to ‘talk to their phone’. As we look closer though, we begin to suspect that Siri is possibly more than ‘merely’ a great speech-to-text conversion tool. Apart from being able to use one’s phone via voice commands instead of one’s fingers, we are also able to interact with other web-based services. We can search the web, for instance, and if we are looking for a restaurant, those nearest our current location are retrieved, unless, of course, we indicated otherwise. Last but not least, Siri talks back, and that too in a surprisingly human fashion. ‘Voice-enabled location-based search–Google has it already, so what?’, we might say. But there is more. Every voice interaction is processed by Apple’s web-based servers; thus Siri runs on the ‘cloud’ rather than directly on one’s phone. So, as Siri interacts with us, it is also continuously storing data about each interaction on the cloud; whether we repeated words while conversing with it, which words, from which country we were speaking, and whether it ‘understands’ us or not in that interaction. As a result, we are told, Siri will, over time, learn from all this data, improve its speech-recognition abilities, and adapt itself to each individual’s needs. We have seen the power of machine learning in Chapter 3. So, regardless of what Siri does or does not do today, let us for the moment imagine what is possible. After all, Siri’s cloud-based back-end will very soon have millions of voice conversations to learn from. Thus, if we ask Siri to ‘call my wife Jane’ often enough, it should soon learn to ‘call my wife’, and fill in her name automatically. Further, since storage is cheap, Siri can remember all our actions, for every one of us: ‘call the same restaurant I used last week’, should figure out where I ate last week, and in case I eat out often, it might choose the one I used on the same day last week.

There is a new Bayesian, or probabilistic, paradigm in the psychology of reasoning, with new psychological accounts of the indicative conditional of natural language and of conditional reasoning. Dorothy Edgington has had a major impact on this new paradigm, through her views on inference from uncertain premises, the relation between the probability of the indicative conditional, P(if p then q), and the conditional probability, P(q|p), and the use of the Ramsey test to evaluate conditionals. Accounts are given in this chapter of the psychological experiments in the new paradigm that confirm empirical hypotheses inspired by her work and other philosophical sources.

Judgment is needed as an additional source of knowledge in measurement. This chapter aims to clarify what kind of judgment is referred to and what it is not. Judgment is defined in the Kantian sense; it presupposes intuition and imagination, and is not a strict deduction of particulars from universals. It requires expertise, that is, real experience with the particulars to which it is applied. But judgments in this sense are subjective. To eliminate this subjectivity, also called “bias,” accounts are developed that ensure that judgments are rational, for example, “unbiased.” A rational judgment, however, is not a Kantian judgment; it is the optimal solution of a model that represents a real-life judgment problem. But a real-life problem can be modeled in various different ways, each with a different rational solution. In the chapter’s appendix it is shown that correct Bayesian deduction can be biased as defined in mathematical statistics.

This chapter details the inner workings of spam filters, algorithmic devices that separate desirable messages from undesirable messages. It argues that such filters are a particularly important kind of sieve insofar as they readily exhibit key features of sieving devices in general, and algorithmic sieving in particular. More broadly, it describes the relation between ontology (assumptions that drive interpretations) and inference (interpretations that alter assumptions) as it plays out in the classification and transformation of identities, types, or kinds. Focusing on the unstable processes whereby identifying algorithms, identified types, and evasive transformations are dynamically coupled over time, it also theorizes various kinds of ontological inertia and highlights various kinds of algorithmic ineffability. Finally, it shows how similar issues underlie a much wider range of processes, such as the Turing Test, Bayesian reasoning, and machine learning more generally.

‘Predicting the future’–the stuff of dreams one might imagine; the province of astrologers and soothsayers, surely. Perhaps not, the scientific mind might retort: after all, is it not the job of science to discover laws of nature, and thereby make precise, verifiable predictions about the future? But what if we were to claim that prediction is neither fanciful nor difficult, and not even rare. Rather, it is commonplace; something that we all accomplish each and every moment of our lives. Some readers may recall the popular video game, pong, where the goal is to ‘keep the puck in play’ using an electronic paddle. Figure 2 shows images of two different pong games in progress. In addition to the paddle and puck, the players’ eye gaze is also being tracked. The image on the left shows the player’s eyes tracking the puck itself. On the other hand, in the right-hand image, the player is already looking at a point where she expects the puck to travel to. The player on the left is reactive; she simply tracks the puck, and as the game gets faster, she eventually misses. The right player, in contrast, is able to predict where the puck will be, and most of the time she gets it right. Further, we often see her eyes dart faster than the puck to multiple regions of the field as she appears to recalculate her prediction continuously. What kind of player do you think you are? As it happens, almost all of us are predictive players. Even if we have never played pong before, we rapidly begin predicting the puck’s trajectory after even a few minutes of playing. The ‘reactive player’ in this experiment was in fact autistic, which apparently affected the person’s ability to make predictions about the puck’s trajectory. (The neurological causes of autism are still not well known or agreed upon; the recent research from which the images in Figure 2 are taken represent new results that might shed some more lightonthisdebilitatingcondition.) So it appears that prediction, as exhibited by most pong players, is far from being a rare and unusual ability. It is in fact a part and parcel of our everyday lives, and is present, to varying degrees, in all conscious life.

In February 2011, IBM’s Watson computer entered the championship round of the popular TV quiz show Jeopardy!, going on to beat Brad Rutter and Ken Jennings, each long-time champions of the game. Fourteen years earlier, in 1997, IBM’s Deep Blue computer had beaten world chess champion Garry Kasparov. At that time no one ascribed any aspects of human ‘intelligence’ to Deep Blue, even though playing chess well is often considered an indicator of human intelligence. Deep Blue’s feat, while remarkable, relied on using vast amounts of computing power to look ahead and search through many millions of possible move sequences. ‘Brute force, not “intelligence”,’ we all said. Watson’s success certainly appeared similar. Looking at Watson one saw dozens of servers and many terabytes of memory, packed into ‘the equivalent of eight refrigerators’, to quote Dave Ferrucci, the architect of Watson. Why should Watson be a surprise? Consider one of the easier questions that Watson answered during Jeopardy!: ‘Which New Yorker who fought at the Battle of Gettysburg was once considered the inventor of baseball?’ A quick Google search might reveal that Alexander Cartwright wrote the rules of the game; further, he also lived in Manhattan. But what about having fought at Gettysburg? Adding ‘civil war’ or even ‘Gettysburg’ to the query brings us to a Wikipedia page for Abner Doubleday where we find that he ‘is often mistakenly credited with having invented baseball’. ‘Abner Doubleday ’ is indeed the right answer, which Watson guessed correctly. However, if Watson was following these sequence of steps, just as you or I might, how advanced would its abilities to understand natural language have to be? Notice that it would have had to parse the sentence ‘is often mistakenly credited with . . .’ and ‘understand’ it to a sufficient degree and recognize it as providing sufficient evidence to conclude that Abner Doubleday was ‘once considered the inventor of baseball’. Of course, the questions can be tougher: ‘B.I.D. means you take and Rx this many times a day’–what’s your guess? How is Watson supposed to ‘know’ that ‘B.I.D.’ stands for the Latin bis in die, meaning twice a day, and not for ‘B.I.D. Canada Ltd.’, a manufacturer and installer of bulk handling equipment, or even Bid Rx, an internet website? How does it decide that Rx is also a medical abbreviation? If it had to figure all this out from Wikipedia and other public resources it would certainly need farmore sophisticated techniques for processing language than we have seen in Chapter 2.