Chapter 7: The Philosophy of Probability Flashcards
The Interpretations of Probability
Objectivism
Subjectivism
Epistemtic Approach
Objectivism
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).
Subjectivism
probability statements refer to the speaker’s degree of belief about something.
Epistemic Approach
probability statements refer to the degree of support one statement gets from another statement.
3 Versions of Objectivist Interpretation
1) Classical Interpretation
2) Frequenistic Interpretation
3) Propensity Interpretation
Classical Interpretation
Probability = Number of Favorable Cases / Number of possible cases
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.
combination; equipossible; probability; circular; probability; combination
Problem with the Classical Interpretation
- 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)
The Frequency Interpretation
Probability = Total number of positive events / Total number of trials.
Frequency Interpretation is always defined relative to some _____________ _______________ (the relevant total number of ___________).
reference class; trials
Problem with Frequency Interpretation (See Notes)
In deciding what is the relevant reference class
(whether that’s the combination of two [or more] trials, or deciding between separate trials).
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).
reference classes; unique; one
Venn’s Idea of the Frequency Interpretation
- Frequency Interpretation would only work if the number of total trials was infinite.
- Should distinguish between the limiting frequency and frequency observed.
Limiting Frequency
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.
Frequency Observed
Total number of positive events (out of total number of trials) observed from a limited total number of trials.
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).
limiting; limiting; infinite; limiting; observed; limited; infinity
Propensity Interpretation
probability = propensity/tendency of an object, in the external world, to give rise to a certain effect.
Problem w/ Propensity Interpretation
it’s not clear what is meant by an object having ‘propensity,’ or ‘tendency.’
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 _______________.
visualize; propensity; tendency; tendency; propensity; probabilities of propensity
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.
disposition; manifestation; tendency; soluble; physical; behave
Humphrey’s Paradox
- 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.
Logical Interpretation
probability = relation between hypothesis (conclusion) and evidence (premises) supporting it.
- It’s essentially representative of deductive logic : premises true, conclusion has to be true.
Problem w/ Logical Interpretation
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.
statements of relation between ______________ and _______________ supporting it are called ________________ _______________.
hypotheses; evidence; epistemic probabilities.
Epistemic Probabilities
statements of relation between hypothesis and evidence supporting it.
Goal of Ecumenical Approach: bieng an _______________ regarding some probabilities, while (coexistingly) being a ______________, and taking an _____________ approach about others.
objectivist; subjectivist; epistemic
Subjective Probability
- probability = subjective degree of belief that an event will happen irrespective of what happens in the external world.
- created by the mind.
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.
probability; degree; belief; probability; strong
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.
degree of belief; degree of belief; low; degrees of belief; different; probability; true; false
Savage’s Theory (1954) on Subjective Probability
Finds a way to MEASURE subjective probabilities.
Steps to Savage’s Theory
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.
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.
probability; maximizing expected monetary value; unknown; utility; axioms; follow; axioms; probabilities; utilities; preferred; highest; preferred; lowest
The axioms, which a decision maker’s preferences must follow, define what _______________ are allowed between _____________ (i.e. ___≻___ or ___≻___ or ___~___).
combinations; events; A≻B; B≻A; A~B
Why don’t we assume the decision maker uses maximizing expected monetary value?
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.
What does it mean to reason backwards?
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).
Why is Subjective Probability shaky?
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.
