2 Three Types of Probability
The previous post (Embrace Radical Uncertainty) introduces the topic, and explains how the foundations of probability, laid in early 20th century, were completely derailed by logical positivism. The first section of the paper explains the three main types of probability that MUST be distinguished, in order to understand the flaws in the arguments for personal probabilty. Unfortunately, logical positivism DENIES the existence of the first two types listed below. What is worse, it denies MEANING to concept, making it impossible to talk about, or think about, these concepts. These mental barriers are an important reason why the flaws in the argument for Bayesianity have remained un-detected for so long.
Section 1 — THREE TYPES OF PROBABILITY
De-Finetti’s (1974) famous book on The Theory of Probability starts with the sentence that “Probability does not exist”. The conflict between the title and the first sentence arises because of the failure to distinguish between different notions of probability, as we will clarify. Contrary to standard usage, throughout this paper, probability will refer to the probability of a unique, one-time event — known as a single case probability. Consider an experiment E, the outcome of which can be an event G, or its complement G*. We want to think about meanings which could be attached to the expression p(G) or the probability of the outcome G. We consider three different possible ways of interpreting this expression.
Ontic Probability: This is a probability which is created by the objective circumstances of the experimental setup, which are invariant across observers, and persist whether or not there are observers. Thus, the probability of the event G exists as a feature of external reality, and is not affected by mental states of observers, or by perceptions and observations. We will use p(G) to denote the ontic probability of G.
Epistemic Probability: This refers to an intermediate state of knowledge K of a particular observer A about the experiment E and its outcome G. A has complete knowledge if he can predict exactly what will occur: either G or not G. He has zero knowledge if he knows nothing which can allow him to say anything relevant to predicting whether or not G will occur as an outcome of the experiment E. The intermediate state of knowledge arises when past experience with experiments similar to E allows A to say something useful about the likelihood of occurrence of G. This likelihood will be denoted as PA(G).
Revealed Probability: The epistemic probability PA(G) is an internal mental state of knowledge of A, which is not directly observable by others. Observers can get access to this knowledge by asking A to choose over lotteries. If A believes that the likelihood of occurrence of G is greater than 50%, then he will prefer a lottery which pays $5 on occurrence of G to a lottery which pays $5 with 50% (ontic or epistemic) probability. Several such choices can accurately reveal the internal mental state of knowledge of A to external observers. Knowledge which external observers acquire by observing choices of A over lotteries will be called the revealed probability of G, which is short for the epistemic probability PA(G) as revealed to observer B by actions of A.
Some examples will clarify the distinctions being made. There is a 50% probability that a free neutron will undergo radioactive decay within 10.3 minutes, whether or not there are any observers of this event. This is an ontic probability, which is a feature of external reality, even when there are no observers.
Epistemic probability refers to a state of knowledge intermediate between complete knowledge of what will happen, and complete lack of knowledge. This type of knowledge can arise from several possible sources. One source is knowledge of observed frequency of occurrence of outcome G in similar experiments E. Another source is knowledge of conditions of symmetry under which the experiment is performed, which lead to equal possibility for some types of outcomes. A third sources is expert opinion based on deep knowledge about the experiment, such as in the case of quantum probabilities, or in the case of weather forecasts.
Since epistemic probability is an internal mental state of A, it is not directly observable by external observers. Any method used by external observers to assess the state of knowledge that A has about the probability of occurrence of G as an outcome of the experiment E leads to a revealed probability.
It is clear that the three notions of probability are very different from each other. Ontic probability concerns external reality, epistemic probability is an internal mental state of knowledge, and revealed probability is an assessment by an observer B, of an internal mental state of A, based on observed behavior of A. The argument of De-Finetti for the existence of subjective probabilities can be summarized as follows. Logical Positivism leads De-Finetti to the conclusion that ontic single case probabilities do not exist. Logical positivism also denies existence of unobservable internal mental states of knowledge, so that epistemic probability also does not exist, in the sense defined above. Additionally, there are no probabilistic events in external reality, so there is no knowledge to be had about probabilistic regularities in observed patterns of events. De-Finetti shows that rational human behavior in situations of uncertainty nonetheless satisfies certain rules which are the same as rules which would be followed if the human in question had epistemic probabilities. Thus a revealed probability can be inferred from a set of choices over lotteries. This revealed probability is taken to be an epistemic probability, representing knowledge or belief about the uncertain event. It is this last interpretive step which is wrong.
Our counter-argument to De-Finetti states that if there really are no probabilistic regularities in external reality, then we can only take arbitrary actions. These may appear to conform to an epistemic probability which displays knowledge about external reality, but this match is an illusion. Subjective probabilities arise from arbitrary actions based on ignorance, and these can be clearly distinguished from epistemic probabilities which lead to informed decisions based on knowledge. When A is ignorant, then the actions and choices of A do not actually reflect or reveal the epistemic probabilities PA(G). Substantial elaboration is required to explain this brief sketch of the main argument of this paper
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