ECON30019 Chap.7 Conditional Probability and Bayesian Updating
Conditional Probability and Bayesian Updating
Before any belief can be called wrong, the subject fixes the information environment. Under certainty the result of an action is settled in advance; under risk the probabilities are objective and agreed by everyone that researchers can treat as known; under uncertainty or ambiguity there are none, different people may apply different probabilities, and those beliefs cannot be directly observed.
In neoclassical economics probability theory is used as the normative theory of probabilistic judgement, and that single fact licenses everything in the three chapters that follow, because it makes a systematic gap between belief and axiom a bias rather than a difference of opinion.
The chapter covers the axioms, the difference between mutual exclusivity and independence, conditioning and the law of total probability, and Bayes' rule with its derivation and its use in sequential updating, ending on the convergence result that the next chapter breaks.
What this chapter covers
- 01
Certainty, risk and uncertainty, and which one the subject lives in
- 02
Probability theory as the normative benchmark, and what that licenses
- 03
Random experiment, sample space, event, and the three axioms
- 04
Mutually exclusive against independent, and why they are close to opposites
- 05
The geometric reading of independence as a representative slice
- 06
Conditional probability as an instruction to shrink the sample space
- 07
Partitions and the law of total probability
- 08
Bayes' rule, its derivation in three lines, and its expanded form
- 09
Enumerate and restrict: the slowest method and the safest
- 10
Sequential updating, and the washing out of priors
Two priors, one piece of evidence
- 1Name the events and write both likelihoods: the probability of a report when faulty is 1, and when sound is 0.2. These are shared by both engineers.
- 1Build the first engineer's denominator: one times 0.05 plus 0.2 times 0.95, which is 0.05 plus 0.19, giving 0.24.
- 1Divide: 0.05 over 0.24 is approximately 0.208, up from a prior of 0.05.
- 1Repeat for the second engineer: the denominator is 0.5 plus 0.1, which is 0.6, so the posterior is 0.5 over 0.6, approximately 0.833.
- 1Update again for the first engineer, using 0.208 as the new prior: the denominator is 0.208 plus 0.2 times 0.792, which is 0.366.
- 1Divide: 0.208 over 0.366 is approximately 0.568. Both beliefs rise with every report and the gap between the engineers narrows.
Key terms
- Choice under risk
- A setting where the relevant outcomes carry objective probabilities that everyone agrees on and researchers can treat as observable.
- Choice under uncertainty
- A setting in which no objective probabilities exist, different people may apply different ones, and researchers cannot directly observe those beliefs. Also called ambiguity.
- Mutually exclusive
- A pair of events of which at most one can happen, so their joint probability is zero. With non-zero probabilities such events are maximally dependent rather than independent.
- Independent events
- A pair of events for which the occurrence of one leaves the chance of the other untouched, so the joint probability is the product of the two separate probabilities.
- Partition
- A collection of mutually exclusive events that together cover every possible outcome, with no overlap and nothing left out. It is what the law of total probability sums over.
- Posterior probability
- The probability of a hypothesis after evidence has been observed. In sequential updating it becomes the prior for the next observation.
Conditional Probability and Bayesian Updating FAQ
Why does the subject insist that probability theory is normative here?
Because that status is what turns a gap between belief and calculation into an error. If probability theory is the standard people should meet, a systematic departure from it is a bias that can be named and corrected. Remove the normative label and the same evidence becomes a description of how people happen to think, with nothing to correct and nothing to model.
What is the fastest way to avoid confusing exclusivity with independence?
Condition on one event and see what happens. Under independence the conditional probability of the other equals its unconditional probability, so learning the first event occurred tells you nothing. Under mutual exclusivity the conditional probability is zero, so learning it occurred tells you everything. Those are opposite extremes, not similar conditions.
Why do conditional-probability puzzles surprise people so reliably?
Because the condition usually shrinks the sample space in a way that is not proportional. In the three-card demonstration the condition eliminates faces rather than cards, and one card contributes two faces of the visible colour while another contributes one, so counting faces gives a third rather than a half. The subject notes the game involves no deception at all.
Exam move
Work every question in this chapter with the sample space written down, because the arithmetic is short and the bookkeeping is where marks disappear. For small problems, enumerate the outcomes and cross out the ones the condition rules out; the surviving count is usually the whole answer.
For Bayes questions, write both conditionals out in words before substituting any number, converting any specificity or true-negative rate by subtracting from one, and build the denominator first. Practise sequential updating on two or three rounds, since it is the same operation repeated and it is what the next chapter overturns.
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