Bayesian Epistemology

Beliefs should have a defined degree of certainity rather than a binary true/false(like in classical Epistemology).

Laws of Probability

Probability of a certain statement S is P(S)

Probability Range Principle

For any event S, 0 ≤ P(S) ≤ 1 - Probability is between 0 and 1 inclusive.

Probability Tautology Principle

For all logical truths L, the probability of L = 1.

P(L) = 1

Principle of Finite Additivity

Probability of S OR R happening is probability of S + probability of R. This assumes that S and R - both can't happen together.

S|R = P(S) + P(R)

Sum of the probability of S and NOT S = 1

P(~S) + P(S) = 1

Bayesian Theorem

Pe(H) = (P(H) / P(E)) / Ph(E)

Pe(H) = P(H & E) / P(E)

Pe(H): Probability of H given that E

PS: Pe(H) should be read as PE(H)

Pe(H) is the inverse of Ph(E)

Inductive Argument / Dutch Book Argument

Degree of belief: The probability that someone assigns to a event / how much money(upto 1$) that someone is willing to bet on the event that incase it happens, they get double the money.

Dutch Book - when you have a set of conditions that others will bet on - but the probability will be higher than 1.

Example: In a horse race, chance of Horse 1 winning is .5, Horse 2 is .25 and Horse 3 is .3. Here, the total is 1.05. So no matter the outcome, the house will get some money.

Synchronic Dutch Book: If someone has a set of degree of belief at the same time that will result in a loss no matter the outcome. Eg. Someone bets .51 that horse 1 will win and .51 that horse 1 will lose. They will get back 1$ no matter the outcome - but will lose .02$.

Diachronic Dutch Book: If someone has a set of degree of belief at different times that will result in a loss no matter the outcome. This happens when you don't update your probabilities correctly when a event with related probability is shown to be true.

Confirmation Theory

Paradox of Dogmatism: If you are certain of an E event(P = 1), you can safely ignore all evidence against it. This will lead to un-updated priors.

Simple Principle of Conditionalization: You should update your belief when you find evidence with set confidence about a related condition.

  • Confirmation: Evidence E confirm theory H if and only if our degree of belief in H is increased if E is known to be true.
  • Disconfirmation: Evidence E confirm theory H if and only if our degree of belief in H is DECREASED if E is known to be true.

Entailment

If H entails E, then E confirms H(as long as E != 1 and E != 0 - probability of E can't be certain)

Bayesian Dogmatism

If you are certain anything(Probability = 1), then if you change your belief even with overwhelming evidence that the thing you believe is NOT true, you will be irrational.

Do not have absolute certainty - no Probability = 1 or 0.

Objections to Bayesian Epistemology

Immutability of Logic

To believe in Bayesian Epistemology, you have believe that the Laws of Probability to be certain - which is against the Bayesian Dogmatism principle - you should not be certain of anything.

Problem of Logical Omniscience

Bayesian Epistemology requires us to map our degrees of belief to the probability values. But there will be a lot of things that we don't know - enough to make any probability assignments to be unreliable. Unless we are Omniscient, we can't use Bayesian Epistemology.

Prediction vs Accommodation

If our theory predicts something - and that something actually happens, that should increase our degree of belief in the theory. Compare this to creating a theory that accommodates/explains something that has already happened. The theory that predicted and proven true should have more weight. But acc. to Bayesian Epistemology, both have the same value.

Problem of the Priors

We can use Bayesian Epistemology to determine the Probability of an outcome - but this will only work if the probability assigned to all linked events and theories are correct. Bayesian Epistemology does to validate those probabilities.

  • Uncertain Evidence
  • Problem of Old Evidence
  • Problem of new theories