Bayesians and Classical Statisticians (Frequentists) have different concepts of probability and, as a result, have fundamentally different views on the nature of statistical inference.

In a Nutshell

• There are two concepts of probability: epistemic and objective.
• Bayesians use epistemic probabilities (in addition to objective probabilities).
• Classical statisticians (frequentists) use only objective probabilities. Their argument:
  • Probabilities in science must be testable.
  • Only objective probabilities are testable.
  • Therefore only objective probabilities may be used in Statistics.

Two Kinds of Probability

Objective Probability

• Objective (physical) probability is the kind of probability that can be supported or refuted by the corresponding relative frequency. The kind of probability used in statements such as:
  1. The probability of rolling a seven with two dice is 1/6.
  2. If both parent pea plants have yellow-green alleles for seed color, the probability that a child plant has yellow peas is 3/4.
  3. The probability that a free neutron decays within 611 seconds is 1/2.
  4. There’s a 95% probability that between 57 and 63 percent of 1,000 randomly selected adult Americans support legalizing marijuana.
• For statement #1, for example, the corresponding relative frequency is that, in about 1/6 of tosses, the outcome of rolling two dice is 7. Which can tested by rolling the dice hundreds of times.
• For statement #4, the corresponding relative frequency is that in about 95 out of 100 polls of 1000 Americans, between 57 and 63 percent of participants in the poll support legalizing marijuana. Which can be tested by conducting a hundreds of such polls.

Epistemic Probability

• Epistemic (or evidential) probability is the kind of probability relative to evidence. The kind used in statements such as:
  1. The Tunguska Event of 1908 was most likely caused by a meteor.
  2. The probability of rain tomorrow is 90 percent.
  3. It’s very likely the defendant is guilty based on the evidence presented at trial.
  4. It’s more probable than not that smoking causes lung cancer.
  5. There’s a 95% probability that between 57 and 63 percent of Americans support legalizing marijuana
• For these statements there’s no corresponding relative frequency which can be tested. For statement #5, for example, the corresponding relative frequency would have the form:
  • In about 95 out of 100 Xs, between 57 and 63 percent of X support legalizing marijuana.
• But what is X? Statement #5 is about Americans in general. But it doesn’t make sense to say that in about 95 out of 100 American populations, between 57 and 63 percent of the population supports legalizing marijuana. Yet there’s evidence for statement #5: polls in which 60 percent of participants support legalizing marijuana. (For which polls, there are corresponding, testable relative frequencies).
• View Epistemic Probability

Two Kinds of Inference, an Example

• In a random poll of 1,000 Americans, 600 support legalizing the recreational use of marijuana.
• Using Bayes Theorem, a Bayesian infers that:
  • There’s a 95 percent probability that 60 percent of Americans support legalizing pot, with a margin of error of ±3 percentage points.
    • View Calculation.
• A Frequentist infers that:
  • 60 percent of Americans support legalizing pot, with a confidence interval of ±3 percentage points at the 95% confidence level.
    • View Calculation.
• Which means, eliminating the technical terms “confidence interval” and “confidence level”:
  • 60 percent of Americans support legalizing pot, based on there being a 95 percent probability that in a random sample of 1000 Americans 60 percent support legalizing pot, with a margin of error of  ±3 percentage points.

• Thus, a Bayesian infers from the poll that it’s very likely that about 60 percent of Americans support legalizing pot. A frequentist, by contrast, infers that 60 percent of Americans support legalizing pot, based on it’s being very likely that you’ll get the same results if the poll is repeated.
• A frequentist does not draw the conclusion that it’s probable that 60 percent of American support legalizing pot. For a frequentist, only objective probabilities are acceptable in science in general and in statistics in particular. The probability that 60 percent of American support legalizing is not objective, since there is no corresponding relative frequency.

The Frequentist’s Argument

• The argument:
  • Probabilities in science must be directly testable by relative frequencies.
  • Only objective probabilities are directly testable by relative frequencies.
  • Therefore only objective probabilities may be used in Statistics.

• The first premise is questionable.
  • Probabilities predicted by Mendel’s Theory of Inheritance and Quantum Mechanics are directly testable by relative frequencies. Quantum Mechanics, for example, predicts that a silver atom passing through a Stern Gerlach magnet will be deflected upwards, rather than downwards, with a probability of ½. The prediction is confirmed by shooting millions of silver atoms through the magnet.
  • But the first premise makes a normative claim: scientific probabilities must (or should be) testable by relative frequencies. A Bayesian can agree that probabilities in science should be testable and argue that epistemic probabilities are testable, albeit not directly by relative frequencies. Thus the proposition that the probability that 60 percent of Americans support legalizing pot, though not directly testable by relative frequency, can be tested by taking any number of polls.

The Question of Justification

• A (philosophic) question arises in connection with the legalizing-pot example: what justifies the inference from a group of a thousand people to a population of millions?
• A Bayesian has a ready answer:
  • That there’s a 95 percent probability that 60 percent of Americans support legalizing pot is the result of a mathematical calculation on the data, that is, plugging the results of the poll into Bayes Theorem. The result, that is, is a valid deductive argument.
• A Frequentist replies that the hypothesis that 60 percent of Americans support legalizing pot is confirmed in the same way that scientific theories in general are confirmed, that is, by testing their logical consequences. (That is, hypothetico-deductive confirmation.)
  • For example, a logical consequence of the postulates of Quantum Mechanics (in conjunction with auxiliary hypotheses and assumptions) is that there’s a one-half probability that a silver atom passing through a Stern-Gerlach Magnet is deflected upwards. The entailed probability prediction is verified by shooting millions of electrons through the magnet.  The verification of the prediction, then, tends to confirm the postulates of Quantum Mechanics.
  • In the same way, a logical consequence of the hypothesis that 60 percent of Americans support legalizing pot (in conjunction with the Central Limit Theorem) is that there’s a certain probability distribution (called the sampling distribution) for the the results of random polls of 1000 Americans.  The entailed probability distribution can be verified by conducting any number of random polls. The verification of the predicted distribution, then, tends to confirm the hypothesis.