Chapter 7: The Philosophy of Probability

Question 1

The Interpretations of Probability

Answer 1

Objectivism
Subjectivism
Epistemtic Approach

Question 2

Objectivism

Answer 2

probability statements refer to facts/claims of External World.

(Ex. Probability that coin will land heads up, is a claim about the coin’s propensity to land on heads every other flip).

Question 3

Subjectivism

Answer 3

probability statements refer to the speaker’s degree of belief about something.

Question 4

Epistemic Approach

Answer 4

probability statements refer to the degree of support one statement gets from another statement.

Question 5

3 Versions of Objectivist Interpretation

Answer 5

1) Classical Interpretation
2) Frequenistic Interpretation
3) Propensity Interpretation

Question 6

Classical Interpretation

Answer 6

Probability = Number of Favorable Cases / Number of possible cases

Question 7

Regarding the classical interpretation, the only reason that a you would propose that a ____________ of ____________ rolls/tosses will definitely occur is if you’ve observed a ____________ that would make you think that. But this is _________ because your aim was to actually find ______________ using the __________ of rolls/tosses.

Answer 7

combination; equipossible; probability; circular; probability; combination

Question 8

Problem with the Classical Interpretation

Answer 8

• What if your probabilities aren’t equally possible cases?

When the number of all possible outcomes is infinite, the favorable outcome will always be zero (ration between finite number : infinity = 0)

Question 9

The Frequency Interpretation

Answer 9

Probability = Total number of positive events / Total number of trials.

Question 10

Frequency Interpretation is always defined relative to some _____________ _______________ (the relevant total number of ___________).

Answer 10

reference class; trials

Question 11

Problem with Frequency Interpretation (See Notes)

Answer 11

In deciding what is the relevant reference class

(whether that’s the combination of two [or more] trials, or deciding between separate trials).

Question 12

The Frequency Interpretation comes into even more trouble if the event for which we are trying to decide among the ______________ ______________ is ____________, such that there’s only ___________ event (trial).

Answer 12

reference classes; unique; one

Question 13

Venn’s Idea of the Frequency Interpretation

Answer 13

• Frequency Interpretation would only work if the number of total trials was infinite.
• Should distinguish between the limiting frequency and frequency observed.

Question 14

Limiting Frequency

Answer 14

Total number of positive events (out of total number of trials) that we would get if one and the same experiment was done infinitely many times.

Question 15

Frequency Observed

Answer 15

Total number of positive events (out of total number of trials) observed from a limited total number of trials.

Question 16

Observed v. Limiting Frequency:
We don’t know the ___________ frquency. It’s impossible to get at the _________ frequency because we’re never gonna conduct a trial of events an __________ number if times. But, the idea is that we can know something about the ___________ based the frequency ___________. However, that’s not useful because a __________ number of trials can’t tell us about ____________ (even if it was like 20,000 trials).

Answer 16

limiting; limiting; infinite; limiting; observed; limited; infinity

Question 17

Propensity Interpretation

Answer 17

probability = propensity/tendency of an object, in the external world, to give rise to a certain effect.

Question 18

Problem w/ Propensity Interpretation

Answer 18

it’s not clear what is meant by an object having ‘propensity,’ or ‘tendency.’

Question 19

It’s not possible to _______________ an object’s ______________. Our being able to see the color red is a byproduct of the _______________ of our eyes to visualize an object as red (among a normal group of people). It doesn’t follow that we can see the _____________ itself which causes our ability to visualize red. We can’t observe ________________, so it doesn’t make sense to try to establish the _________________ of _______________.

Answer 19

visualize; propensity; tendency; tendency; propensity; probabilities of propensity

Question 20

You can’t directly observe the ______________ of a thing; you can, however, observe the thing’s _______________. Table Sugar has the ______________ to be ___________ even if you don’t actually dunk it in water. The thing has an objective ____________ structure that causes it to ____________ in certain ways under certain circumstances.

Answer 20

disposition; manifestation; tendency; soluble; physical; behave

Question 21

Humphrey’s Paradox

Answer 21

• propensity has a temporal direction (probability does not).
• propensity interpretation doesn’t work with inverted probabilities (introduced in Bayes’ Theorem Ch. 6)
  -By calculating p(A|B), we can calculate p(B|A), because we had known priors [p(A)].
• While it may make sense to suggest the propensity of A given B, it makes NO sense to suggest the propensity of B given A.

Ex. It makes sense to suggest the propensity of having a disease GIVEN that you tested positive. But it makes no sense to suggest the propensity of testing positive GIVEN that you have a disease.

Question 22

Logical Interpretation

Answer 22

probability = relation between hypothesis (conclusion) and evidence (premises) supporting it.

• It’s essentially representative of deductive logic : premises true, conclusion has to be true.

Question 23

Problem w/ Logical Interpretation

Answer 23

Relies too heavily on evidence, (so that w/o evidence, we wouldn’t be able to find probability).

Sometimes, we want probability to reflect mere guesses.

Question 24

statements of relation between ______________ and _______________ supporting it are called ________________ _______________.

Answer 24

hypotheses; evidence; epistemic probabilities.

Question 25

Epistemic Probabilities

Answer 25

statements of relation between hypothesis and evidence supporting it.

Question 26

Goal of Ecumenical Approach: bieng an _______________ regarding some probabilities, while (coexistingly) being a ______________, and taking an _____________ approach about others.

Answer 26

objectivist; subjectivist; epistemic

Question 27

Subjective Probability

Answer 27

- probability = subjective degree of belief that an event will happen irrespective of what happens in the external world. - created by the mind.

Question 28

Example of Subjective Probability: If you believe the ________________ of rain tomorrow = 0.9, this means your subjective ____________ of ___________ for the _______________ of rain is very high/__________, and can be represented by a number 0.9.

Answer 28

probability; degree; belief; probability; strong

Question 29

While your probability--________________ of _____________--that an event will happen maybe high (i.e. chances of rain tomorrow), another person's ____________ of ___________ could be exceedingly _______ (or vice versa). Just because the ____________ of ____________ are ____________/varying among multiple people (w/ faculties to believe) doesn't necessarily mean one person's ______________ is __________/right whilst another's is ________/ wrong.

Answer 29

degree of belief; degree of belief; low; degrees of belief; different; probability; true; false

Question 30

Savage's Theory (1954) on Subjective Probability

Answer 30

Finds a way to MEASURE subjective probabilities.

Question 31

Steps to Savage's Theory

Answer 31

1) Has decision maker state preferences over a set of events (i.e. A≻B, B≻A, or A~B) a) These events (for example) could include the highest price "X" decision maker is willing to pay to have their car insured "Y" in the scenario (S) that "your car is stolen in one year." Probability of "your car will be stolen in one year" = X/Y.

Question 32

As seen in a hypothetical example of Savage's Theory, we're trying to find a decision maker's _______________ of stated preferences. It's normal in this setting to assume the decision maker's preferences are based on _________________ ______________ ____________ ___________. However, we're in no position to assume the decision maker used this decision rule to create prefernces among events with _______________ probabilities. That, and people's __________ for monetary value has diminished in recent time. We can get around this problem by employing a set of ____________ that each of the events MUST ___________. If the each event follows these _____________, then the decision maker's preferences can be described "AS IF" assigning numerical _____________ and ___________, where the most _________________ events are assigned the __________ utility, and least _____________ events are assigned the ______________ utilities.

Answer 32

probability; maximizing expected monetary value; unknown; utility; axioms; follow; axioms; probabilities; utilities; preferred; highest; preferred; lowest

Question 33

The axioms, which a decision maker's preferences must follow, define what _______________ are allowed between _____________ (i.e. ___≻___ or ___≻___ or ___~___).

Answer 33

combinations; events; A≻B; B≻A; A~B

Question 34

Why don't we assume the decision maker uses maximizing expected monetary value?

Answer 34

We don't assume the decision maker uses maximizing expected monetary value because we can't. Using maximizing expected monetary value assumes one already has established probabilities over events. So now, using the axioms as constraints over stated preferences, we need to extract these probabilities.

Question 35

What does it mean to reason backwards?

Answer 35

Because we have the stated preferences, of which a set of axioms must follow (i.e. A≻B; B≻A; A~B etc...), we can essentially reason backwards. We can use these stated preferences over events, THAT PROBABILITY WOULD'VE TOLD US ABOUT ANYWAY, to find and extract the probabilities of each of thes events (and assign numerical probabilities and utilities).

Question 36

Why is Subjective Probability shaky?

Answer 36

Because there's nothing really subjective about the probabilities. Wer'e using the axioms over stated preferences of things to which we have different degrees of belief of, and by following these axioms we can desribe the stated preferences AS IF assigning SUBJECTIVE numerical probabilities and utilities. There doesnt' seem to be anything actually subjective about these probability values.