Start by defining the hypothesis (H) and the observation (O) you are considering.
The main statement or event you want to estimate the probability of.
Example: "It will rain tomorrow" or "This patient has disease X".
A new piece of evidence or data you have observed that might affect your belief in the hypothesis.
Example: "The sky is cloudy" or "The test result is positive".
How Surprising is Your Hypothesis?
Consider the statement: ""
Indicate how surprising this statement is to you *before* considering any new observation.
This is your initial belief in the hypothesis H before you consider the observation O.
"Not At All Surprising" means you think H is very likely (Probability close to 1).
"Super Surprising" means you think H is very unlikely (Probability close to 0).
Super Surprising (P ≈ 0)Not At All Surprising (P ≈ 1)
Probability: --
How Surprising is the Observation, IF Hypothesis is True?
Consider the statement: ", assuming is true."
Indicate how surprising this assumption makes the observation.
This is the probability of seeing the observation O, *given that the hypothesis H is true*.
"Not At All Surprising" means O is very likely if H is true (Probability close to 1).
"Super Surprising" means O is very unlikely if H is true (Probability close to 0).
Super Surprising (P ≈ 0)Not At All Surprising (P ≈ 1)
Probability: --
How Surprising is the Observation, IF Hypothesis is False?
Consider the statement: ", assuming is false."
Indicate how surprising this assumption makes the observation.
This is the probability of seeing the observation O, *given that the hypothesis H is false (~H)*.
"Not At All Surprising" means O is very likely if H is false (Probability close to 1).
"Super Surprising" means O is very unlikely if H is false (Probability close to 0).
This value is crucial for determining how much the observation O updates your belief in H.
Super Surprising (P ≈ 0)Not At All Surprising (P ≈ 1)
Probability: --
Bayes' Theorem Update
Based on your belief about "" (Hypothesis H) and the surprisingness of "" (Observation O) under different assumptions, here is the updated probability:
Results for "" given ""
Prior Probability P(H):--
Posterior Probability P(H|O):--
Likelihood P(O|H):--
Likelihood P(O|~H):--
Prior Odds:--?
Prior Odds = P(H) / P(~H). This is your initial belief ratio of H being true vs. H being false.
Likelihood Ratio:--?
Likelihood Ratio = P(O|H) / P(O|~H). This measures how much more (or less) likely the observation O is under hypothesis H compared to under the alternative hypothesis ~H. It's the "Bayes Factor".
Posterior Odds:--?
Posterior Odds = P(H|O) / P(~H|O). This is your updated belief ratio of H being true vs. H being false, *after* observing O. Bayes' Theorem in odds form is: Posterior Odds = Likelihood Ratio * Prior Odds.