Bayesian Epistemology PHIL 218338 Welcome and thank you

Bayesian Epistemology PHIL 218/338

Welcome and thank you!

Outline Part I: What is Bayesian epistemology? Probabilities as credences The axioms of probability Conditionalisation Part II: Applications and problems: Theism Bear with me! Ideally we would discuss these topics over several lectures.

What is Bayesian Epistemology? Bayesianism is our “leading theory of uncertainty” Alan Hájek and Stephan Hartmann It concerns credences, or degrees of belief, which are often uncertain I’m not going to be attacked by a duck tomorrow Bayesianism ≈ a theory about when our credences are rational or justified (one which may complement other theories of justification) There are many varieties of Bayesianism (Irving Good calculated that there at least 46, 656!) Bayesian epistemology is the “application of Bayesian methods to epistemological problems. ”

First component of Bayesianism: Probabilities as credences

Credences Traditional epistemology deals primarily with qualitative concepts Belief/disbelief Knowledge/ignorance In Bayesian epistemology, these binary concepts are arguably less central and therefore receive less attention Bayesian epistemology deals largely with a quantitative concept of credences Credences ≈ degrees of belief or disbelief

First component of Bayesianism: Probabilities as credences In the 17 th century, mathematicians Blaise Pascal and Pierre de Fermat pioneered a representation of uncertainty as probabilities Subjective interpretation of probability: Subjective interpretation: ‘Probability is degree of belief’ But whose degree of belief? Some actual person or Some ideal person This is the subjective or personal interpretation of probability because these probabilities concern the psychological state of a subject or person

Terminology h = hypothesis/proposition ~h = negation of the hypothesis P(h) = probability of the hypothesis Example: h = It will rain tomorrow P(h) = Probability that it will rain tomorrow These terms are on your handout

Quantitative nature of credences Credences (or subjective probabilities) are taken to be associated with a numerical value or an interval P(h) - decimal P(h) in % P(h)=100% P(h) in normal language P(~h) in normal language

Quantitative nature of credences Credences (or subjective probabilities) are taken to be associated with a numerical value or an interval P(h) - decimal P(h) in % P(h) in normal language P(~h) in normal language P(h)=100% h is certainly true ~h is certainly false

Quantitative nature of credences Credences (or subjective probabilities) are taken to be associated with a numerical value or an interval P(h) - decimal P(h) in % P(h) in normal language P(~h) in normal language P(h)=100% h is certainly true ~h is certainly false P(h)=0% h is certainly false ~h is certainly true

Quantitative nature of credences Credences (or subjective probabilities) are taken to be associated with a numerical value or an interval P(h) - decimal P(h) in % P(h) in normal language P(~h) in normal language P(h)=100% h is certainly true ~h is certainly false P(h)=0% h is certainly false ~h is certainly true P(h)=. 8 P(h)=80% h is probably true ~h is probably not true

Quantitative nature of credences Credences (or subjective probabilities) are taken to be associated with a numerical value or an interval P(h) - decimal P(h) in % P(h) in normal language P(~h) in normal language P(h)=100% h is certainly true ~h is certainly false P(h)=0% h is certainly false ~h is certainly true P(h)=. 8 P(h)=80% h is probably true ~h is probably not true P(h)=. 2 P(h)=20% h is probably not true ~h is probably true

Measuring credences Consider your credence that h, the sun will rise tomorrow Consider your credence that you will (after random selection) draw a red marble from an urn containing 5 red marbles 5 black marbles Are you more confident that the sun will rise tomorrow? If yes, then P(h)>. 5

Measuring credences Consider your credence that h, the sun will rise tomorrow Consider your credence that you will (after random selection) draw a red marble from an urn containing 90 red marbles 10 black marbles Are you more confident that the sun will rise tomorrow? If yes, then P(h)>. 9

Measuring credences Consider your credence that h, the sun will rise tomorrow Consider your credence that you will (via random selection) draw a red marble from an urn containing 9, 999 red marbles 1 black marble Are you more confident that the sun will rise tomorrow? If yes, then P(h)>. 9999

Measuring credences What about your credence that: It will rain tomorrow You will be attacked by a duck tomorrow Maybe an interval might represent your credences better If h = It will rain tomorrow Then P(h) = [. 6, . 7] What do you think? Can all of our credences be represented with numerical values?

Objections to the subjective interpretation The probability of h given some evidence e does not mean someone’s actual credence since there may be no actual credence that is relevant It’s not clear that the probability of h given some evidence e is the credence of some epistemically rational agent When is an agent’s credence epistemically rational? When their credence for h given e equals the (inductive) probability of h given e? This is uniformative! (Patrick Maher) When their belief is not blameworthy from an epistemic point of view? But someone might accidentally mistake the probability of h given e to be low and not be blameworthy, but still the probability of h given e might be high (Patrick Maher) Isn’t this just like saying “A proposition is true if and only if an omniscient God were to believe it? ” – It’s uninformative

Alternatives Inductive probabilities are conceptual primitives – they can be understood, but not expressed in terms of other simpler concepts (Patrick Maher) Probabilities are relative frequencies, which we might loosely understand as the proportion of the time that something is true (the frequentist interpretation of probability) 80% of the time when a student sits this course, it is true that they pass 60% of the time when a patient undergoes chemotherapy, it is true that they will recover

Second component of Bayesianism: Credences should conform to the axioms (or rules) of probability

Second component of Bayesianism: Credences should conform to the axioms (or rules) of probability (A 1) All probabilities are between 1 and 0, (A 2) Logical truths have a probability of 1, i. e. 0 ≤ P(h) ≤ 1 for any h. i. e. P(T)=1 for any tautology T (A 3) Where h 1 and h 2 are two mutually exclusive hypotheses, the probability of h 1 or h 2 (h 1 ∨ h 2) is the sum of their respective probabilities, i. e. P(h 1 ∨ h 2) = P(h 1) + P(h 2). These are on your handout

The axioms in action

Arguments for conformity to the axioms

A Dutch book

A Dutch book

A Dutch book

A Dutch book Bet 1 for assignment 1 +$3 -$7 Bet 2 for assignment 2 -$5 +$5 If r occurs, then they win $3 according to the first bet and lose $5 according to the second, so they lose $2 If r does not occur, then they lose $7 according to the first bet and gain $5 according to the second, so they lose $2 Either way, they lose $2.

Dutch book argument 1. If someone violates the probability axioms, then she is vulnerable to having a Dutch book made against her 2. One should avoid being vulnerable to having a Dutch book made against her (because this is a rational defect) 3. Therefore, one should avoid violating the axioms of probability

An objection to the second component Conformity to the axioms requires logical omniscience, but no one is omniscient “You’re right, but the component only sets an ideal standard, irrespective whether any one can meet it”

Questions? Do you think that one’s credences should conform to the axioms of probability?

Third component of Bayesianism: Credences should be updated via conditionalisation

Terminology

Example of a conditional probability

Likelihoods

Prior probabilities

What is the prior probability that Taylor likes you? Suppose you surveyed 100 people and find the following:

What is the probability that Taylor likes you given the evidence? P(h|e) = ? P(h|e) = 9/(9+36) = 9/45 = 1/5 = 20% =. 2

Posterior probabilities

Conditionalisation via Bayes’s theorem: Application to the case: Bayes’s theorem was expressed in a paper by Rev. Thomas Bayes that was published posthumously.

Arguments for the conditionalization norm Case-by-case evidence Bayes’s theorem is used widely in statistics Dutch-book arguments

Part II: Applications and problems

Does God exist?

Multiple partitions problem

Application to theism

The problem of the priors: Subjective and objective Bayesianism

What evidence is there that God exists? Theistic evidence: Atheistic evidence: Fine-tuning of laws and constants Human suffering A universe Animal suffering Moral truths Non-resistant, non-belief in God Miracle reports Scale of the universe Abiogenesis (Origins of life) Contradictory theistic theories Consciousness Theism is less simple (Occam’s razor)

The fine-tuning argument

The fine-tuning argument – Just kidding!

The multiverse objection

The argument from suffering

The argument from suffering

Sceptical theism “God knows a lot more than us and would have reasons to justify his actions which we do not know of” “So if God existed, there was suffering and we did not see any reason that would justify God’s permission of the suffering, then we would not be surprised” More sophisticated defences of versions of sceptical theism are given by Stephen Wykstra and Daniel Howard-Snyder

The problem of the priors

The problem of the priors

The problem of the priors The posterior probability is sensitive to the value of the prior probability Subjective Bayesians often think that the subjectivity of the prior is not a major problem since the subjectivity will be “washed out” as evidence accumulates So two people starting off with different priors will converge on the probable truth given their conditioning on a growing body of evidence However, as Alan Hájek notes: “Indeed, for any range of evidence, we can find in principle an agent whose prior is so pathological that conditionalizing on that evidence will not get him or her anywhere near the truth, or the rest of us. ” And there are other worries So does the problem of the priors render Bayesianism practically useless? Does it eliminate scepticism about the reliability of inductive inference?

Questions?

Thank you!