lecture 8

Question 1

Thee kinds of probability

Answer 1

Subjective probability Interpretation
Frequency probability interpretation
Propensity probability interpretation

Question 2

Epistemic

Answer 2

concerned with rational belief

Question 3

Objective

Answer 3

concerned with features of the worlds

Question 4

Subjective interpretation of probability

Answer 4

The probability of an outcome is a rational degree of belief that the outcome occurs

Question 5

Degree of belief

Answer 5

somebodys level of confidence in the truth of a certain hypothesis

Question 6

How do you know somebodys level of confidence in an hypothesis

Answer 6

Revealed by possible bets about that hypothesis that that individual would accept and reject

Question 7

Degrees of belief and betting behavior

General intuition

Answer 7

look for the odds o na given bet such taht one would be equally willing to take either side of the bet (subjectively fair odd)

If you know a person subjectively fair odds you can “read off” that persons degree of belief in the hypothesis h the bet is about

Question 8

To bet on h at odds of X:1 = to be willing to risk loosing X$ if h is false, in return of 1$ (plus the stake X) if h is true

Answer 8

A large X corresponds to a lot of confidence that h is true

Question 9

if your subjectively fair odds for a bet on h are X:Y, then your degree of belief in h is

Answer 9

X/(X+Y)

This is the ratio between your stake X (=how much you could lose) and the total amount staked X + Y

Question 10

Example degrees of belief calculation

Answer 10

You believe that a 3:1 odd is fair for a bet on the truth of h (e.g. that you will pass this course).

If you bet that h is true, you win 1$ (and get your stake in return) if you are right, and lose 3$ if wrong

Your degree of belief in h is 3/(3+1) = 3/4

Question 11

rational degrees of belief

Answer 11

rational in the context of subjectivist interpretations of probability means coherent with the rules of probability

Question 12

Andrey kolmogorovs axioms of probability

Answer 12

axiom 1: all probabilities are numbers between 0 and 1

axiom 2: If a proposition is certainly true, then it has a probability of 1. If certainly false, then it has prob. 0.

Axiom 3: if h and h* are exclusive alternatives (they cannot both be true at the same time) then P(hor h) = P(h) + P(h)

Question 13

Dutch book arguments

Answer 13

Premise 1: If your degrees of belief in a hypothesis do not conform to the rules of probability, there are possible betting situations where you are guaranteed to lose money (you will fall prey of a dutch book).

Premise 2: YOu want to avoid a guaranteed loss

Therefore your degrees of belief should respect the rules of probability

Question 14

what does being dutch booked actually mean?

Answer 14

whatever you have chosen in the situation you are guaranteed to lose

Question 15

Two problems

Answer 15

If the only constraint of rationality on degrees of belief is they cohere with the axioms of probability, then one may have very weird but still rational degrees of belief

E.g. I may judge the probability that the NL will disappear under water tomorrow is 0. this belief of mine might be wird but still rational

Betting behaviour does not seem to be a generally good guide about ones degrees of belief

e.g. the value I give to money does not increase linearly, or I might just dislike gambling