Phrase this only in terms of events/propositions, not in terms of single words like “context”, “situation” or “problem”. Probability only applies to events which occur or don’t occur, or to propositions that are true or false. Otherwise it is unclear what e.g. P(x | Situation) means.
Bayes theorem involves exactly two events (or propositions), so we must make sure we express merely similar sounding events (Situation looks like this / Situation like this arise) with one and the same event, in order to not exceed the total amount of two events overall.
Before you read on, you may want to first try the above approach yourself. The following is my attempt at a solution.
Formalization attempt
Here is a possible formalization which uses only two propositions (albeit with two “free variables” and the indexical “current”, which arguably is another variable):
Idea A applies to the current context.
The current context is of type B.
P(Idea A applies to the current context | The current context is of type B) = P(The current context is of type B | Idea A applies to the current context) * P(The idea A applies to the current context) / P(The current context is of type B)
Or perhaps more abstractly:
∀A∀B∀x(P(A(x)∣B(x))=P(B(x)∣A(x))×P(A(x))P(B(x)))
Not sure whether this accurately captures what you had in mind. It probably needs to be refined.
Thanks for the formalization attempt. After thinking and reading some more, I feel I’ve only restated in a vague manner the Hypothesis and Evidence version of Bayes’ Theorem—https://en.wikipedia.org/wiki/Bayesian_inference. Quoting from that page: ”
, the posterior probability, is the probability of HgivenE, i.e., afterE is observed. This is what we want to know: the probability of a hypothesis given the observed evidence.”
″Idea A applies” would be the Hypothesis in my case, and “current context is of type B” is the Evidence. To restate: P(Idea A applies | current context is of type B) = P (current context is of type B | Idea A applies) * P (Idea A applies) / P (current context is of type B).
If the above version is correct but straightforward, I’m still impressed by two things. Even after you see some evidence the following two things matter: how likely/probable your Hypothesis is true *in general*, and also how likely/probable the Evidence can show up in cases where your Hypothesis is false (as captured in the denominator)!
To make this valid we need two things.
Phrase this only in terms of events/propositions, not in terms of single words like “context”, “situation” or “problem”. Probability only applies to events which occur or don’t occur, or to propositions that are true or false. Otherwise it is unclear what e.g. P(x | Situation) means.
Bayes theorem involves exactly two events (or propositions), so we must make sure we express merely similar sounding events (Situation looks like this / Situation like this arise) with one and the same event, in order to not exceed the total amount of two events overall.
Before you read on, you may want to first try the above approach yourself. The following is my attempt at a solution.
Formalization attempt
Here is a possible formalization which uses only two propositions (albeit with two “free variables” and the indexical “current”, which arguably is another variable):
Idea A applies to the current context.
The current context is of type B.
P(Idea A applies to the current context | The current context is of type B) = P(The current context is of type B | Idea A applies to the current context) * P(The idea A applies to the current context) / P(The current context is of type B)
Or perhaps more abstractly:
∀A∀B∀x(P(A(x)∣B(x))=P(B(x)∣A(x))×P(A(x))P(B(x)))
Not sure whether this accurately captures what you had in mind. It probably needs to be refined.
Thanks for the formalization attempt. After thinking and reading some more, I feel I’ve only restated in a vague manner the Hypothesis and Evidence version of Bayes’ Theorem—https://en.wikipedia.org/wiki/Bayesian_inference. Quoting from that page: ”
″Idea A applies” would be the Hypothesis in my case, and “current context is of type B” is the Evidence. To restate:
P(Idea A applies | current context is of type B) = P (current context is of type B | Idea A applies) * P (Idea A applies) / P (current context is of type B).
If the above version is correct but straightforward, I’m still impressed by two things. Even after you see some evidence the following two things matter: how likely/probable your Hypothesis is true *in general*, and also how likely/probable the Evidence can show up in cases where your Hypothesis is false (as captured in the denominator)!