Propensity probability
Encyclopedia
The propensity theory of probability is one interpretation
of the concept of probability
. Theorists who adopt this interpretation think of probability
as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome. This kind of objective probability is sometimes called 'chance'.
Propensities, or chances, are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate. A central aspect of this explanation is the Law of large numbers
. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequency of heads will (with high probability) be close to the probability of heads on each single toss. This law suggests that stable long-run frequencies are a manifestation of invariant single-case probabilities. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives. Hence, these single-case probabilities are known as propensities or chances.
In addition to explaining the emergence of stable relative frequencies, the idea of propensity is motivated by the desire to make sense of single-case probability attributions in quantum mechanics, such as the probability of decay
of a particular atom
at a particular time.
The main challenge facing propensity theories is to say exactly what propensity means. (And then, of course, to show that propensity thus defined has the required properties.) At present, unfortunately, none of the well-recognised accounts of propensity comes close to meeting this challenge.
, who had only slight acquaintance with the writings of Charles S. Peirce, however. Popper noted that the outcome of a physical experiment is produced by a certain set of "generating conditions". When we repeat an experiment, as the saying goes, we really perform another experiment with a (more or less) similar set of generating conditions. To say that a set of generating conditions has propensity p of producing the outcome E means that those exact conditions, if repeated indefinitely, would produce an outcome sequence in which E occurred with limiting relative frequency p. For Popper then, a deterministic experiment would have propensity 0 or 1 for each outcome, since those generating conditions would have same outcome on each trial. In other words, non-trivial propensities (those that differ from 0 and 1) only exist for genuinely indeterministic experiments.
Popper's propensities, while they are not relative frequencies, are yet defined in terms of relative frequency. As a result, they face many of the serious problems that plague frequency theories. First, propensities cannot be empirically ascertained, on this account, since the limit of a sequence is a Tail event, and is thus independent of its finite initial segments. Seeing a coin land heads every time for the first million tosses, for example, tells one nothing about the limiting proportion of heads on Popper's view. Moreover, the use of relative frequency to define propensity assumes the existence of stable relative frequencies, so one cannot then use propensity to explain the existence of stable relative frequencies, via the Law of large numbers.
and Donald A. Gillies
, have proposed propensity theories somewhat similar to Popper's, in that propensities are defined in terms of either long-run or infinitely long-run relative frequencies.
Other propensity theorists (e.g. Ronald Giere
)do not explicitly define propensities at all, but rather see propensity as defined by the theoretical role it plays in science. They argue, for example, that physical magnitudes such as electrical charge cannot be explicitly defined either, in terms of more basic things, but only in terms of what they do (such as attracting and repelling other electrical charges). In a similar way, propensity is whatever fills the various roles that physical probability plays in science.
Other theories have been offered by D. H. Mellor, and Ian Hacking
called this the Principal Principle, a term that philosophers have mostly adopted. For example, suppose you are certain that a particular biased coin has propensity 0.32 to land heads every time it is tossed. What is then the correct price for a gamble that pays $1 if the coin lands heads, and nothing otherwise? According to the Principal Principle, the fair price is 32 cents.
Probability interpretations
The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we...
of the concept of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
. Theorists who adopt this interpretation think of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome. This kind of objective probability is sometimes called 'chance'.
Propensities, or chances, are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate. A central aspect of this explanation is the Law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequency of heads will (with high probability) be close to the probability of heads on each single toss. This law suggests that stable long-run frequencies are a manifestation of invariant single-case probabilities. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives. Hence, these single-case probabilities are known as propensities or chances.
In addition to explaining the emergence of stable relative frequencies, the idea of propensity is motivated by the desire to make sense of single-case probability attributions in quantum mechanics, such as the probability of decay
Radioactive decay
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...
of a particular atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
at a particular time.
The main challenge facing propensity theories is to say exactly what propensity means. (And then, of course, to show that propensity thus defined has the required properties.) At present, unfortunately, none of the well-recognised accounts of propensity comes close to meeting this challenge.
Karl Popper
A later propensity theory was proposed by philosopher Karl PopperKarl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
, who had only slight acquaintance with the writings of Charles S. Peirce, however. Popper noted that the outcome of a physical experiment is produced by a certain set of "generating conditions". When we repeat an experiment, as the saying goes, we really perform another experiment with a (more or less) similar set of generating conditions. To say that a set of generating conditions has propensity p of producing the outcome E means that those exact conditions, if repeated indefinitely, would produce an outcome sequence in which E occurred with limiting relative frequency p. For Popper then, a deterministic experiment would have propensity 0 or 1 for each outcome, since those generating conditions would have same outcome on each trial. In other words, non-trivial propensities (those that differ from 0 and 1) only exist for genuinely indeterministic experiments.
Popper's propensities, while they are not relative frequencies, are yet defined in terms of relative frequency. As a result, they face many of the serious problems that plague frequency theories. First, propensities cannot be empirically ascertained, on this account, since the limit of a sequence is a Tail event, and is thus independent of its finite initial segments. Seeing a coin land heads every time for the first million tosses, for example, tells one nothing about the limiting proportion of heads on Popper's view. Moreover, the use of relative frequency to define propensity assumes the existence of stable relative frequencies, so one cannot then use propensity to explain the existence of stable relative frequencies, via the Law of large numbers.
Recent work
A number of other philosophers, including David MillerDavid Miller (philosopher)
David W. Miller is a philosopher and prominent exponent of critical rationalism. He taught in the Department of Philosophy at the University of Warwick in Coventry, UK....
and Donald A. Gillies
Donald A. Gillies
Donald A. Gillies is a British philosopher and historian of science and mathematics. He is an Emeritus Professor in the Department of Science and Technology Studies at University College, London.-Career:...
, have proposed propensity theories somewhat similar to Popper's, in that propensities are defined in terms of either long-run or infinitely long-run relative frequencies.
Other propensity theorists (e.g. Ronald Giere
Ronald Giere
Ronald Giere is an American philosopher of science who is an emeritus professor of philosophy at the University of Minnesota. He is a Fellow of The AAAS, a long-time member of the editorial board of the journal Philosophy of Science, and a past president of the Philosophy of Science Association...
)do not explicitly define propensities at all, but rather see propensity as defined by the theoretical role it plays in science. They argue, for example, that physical magnitudes such as electrical charge cannot be explicitly defined either, in terms of more basic things, but only in terms of what they do (such as attracting and repelling other electrical charges). In a similar way, propensity is whatever fills the various roles that physical probability plays in science.
Other theories have been offered by D. H. Mellor, and Ian Hacking
Ian Hacking
Ian Hacking, CC, FRSC, FBA is a Canadian philosopher, specializing in the philosophy of science.- Life and works :...
Principal Principle of David Lewis
What roles does physical probability play in science? What are its properties? One central property of chance is that, when known, it constrains rational belief to take the same numerical value. David LewisDavid Kellogg Lewis
David Kellogg Lewis was an American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...
called this the Principal Principle, a term that philosophers have mostly adopted. For example, suppose you are certain that a particular biased coin has propensity 0.32 to land heads every time it is tossed. What is then the correct price for a gamble that pays $1 if the coin lands heads, and nothing otherwise? According to the Principal Principle, the fair price is 32 cents.