Probability - Bayes et al Flashcards
Bayesian probability is one of the different interpretations of the concept of probability
What is the other other most common view
What was the classical interpretation
Can you explain the difference
Frequentist view
A Bayesian specifies some prior probability to a hypothesis and then revises it according to new relevant evidence coming in. The probability then is said to be an assignment of a state of knowledge or state of belief.
The frequents don’t hold prior probabilities. They define a probability as the limit of its relative frequency in a large number of trials
In the classical interpretation probability was defined in terms of the principle of indifference, e.g. the probabilities of dice games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry for reasoning.
Can you provide the Bayes Theorem Formula as traditionally presented (a test of memory etc) - green points
Only try and tackle this when you understand Bayes
P(H|E) = P(H)•P(E|H) / ( P(E|H)•P(H) + P(E|¬H)•P(¬H) )
[Terminology: P(_) stands for “probability of _,” H stands for “hypothesis,” E stands for “evidence,” the vertical bar stands for “given,” e.g., P(E|H) is the “probability of E given that H is true”, and finally ¬ means “not.”]
What are odds?
Can you give Bayes theorem in odds form
Odds can be seen as ratios of probabilities. Just as we use P(A) for the “probability of A,” we may talk about O(A), the “odds of A” (where A is some apparently sensible proposition).
In terms of probabilities, O(A) = P(A)/P(~A). So for example, if there is a 66% probability of rain tomorrow, then O(rain) = 0.66/(1-0.66), or more easily 66:33, which finally reduces to 2:1 (usually read “two to one in favour”)
The ‘:’ is equivalent to a ‘/’ sign so you could also write this as a decimal (in this case 2) - the only risk is that you forget it is not a probability if it is less than 1
O(H|E) = O(H) * P(E|H) / P(E|¬H)
O(H) is just the prior odds, and the ratio P(E|H)/P(E|¬H) corresponds to “evidential strength,” although the literature usually calls it a likelihood ratio or a Bayes factor.
In other words.
The Odds Given the new evidence = previous odds * evidential strength of the new evidence (simples)
What is the difference between normative and descriptive statements in philosophy
In philosophy, normative statements make claims about how things should or ought to be, how to value them, which things are good or bad, and which actions are right or wrong
Descriptive (or ‘Positive’) statements are (purportedly-) factual statements that attempt to describe reality. In other words, they are ‘truth-apt’; capable of being factually correct or incorrect.
For example, “children should eat vegetables”, and “those who would sacrifice liberty for security deserve neither” are normative claims.
On the other hand, “vegetables contain a relatively high proportion of vitamins”, “smoking causes cancer”, and “a common consequence of sacrificing liberty for security is a loss of both” are positive claims.
What is the other names for evidential strength
What it is
Bayes Factor or Likelihood Ratio
= P(of evidence if hypothesis is true) / P(of evidence if hypothesis is false)
Application of Bayes in Medicine
What is sensitivity
What is specificity
What happens when you make a test have better sensitivity
Sensitivity is the proportion of people who have a disease who have a positive test result
i.e. the ratio of true positives to (true positives + false negatives)
Specificity is the proportion of patients who don’t have the disease who have a negative result
i.e. the ratio of true negatives to (true negatives + false positives)
More sensitive means its specificity reduces (analagous to say making something more sensitive means you pick up more noise - lots of false positives) - there is usually a trade off, its hard to get tests that are both specific and sensitive
How do you convert Sensitivity into a evidential strength
How do you convert Specificity into a evidential strength
Positive evidential strength is equal to the sensitivity / (1-specificity)
Negative evidential strength is equal to the (1-sensitivity) / specificity
You use which ever is relavent to the question - are you examining whether the person has a disease given the post test result
or whether the person doesn’t have the disease given an negative test result
First thing to remember is that both terms (spec and sens are needed - its not a typo)
Second thing to remember is that sensitivity and specificity are concerned with looking for true positives and true negatives
How do you convert odds to probability
Probability = Odds / (1+Odds)
