Bayes' theorem describes the relationships that exist within an array of simple and conditional probabilities. For example: Suppose there is a certain disease randomly found in one-half of one percent (.005) of the general population. A certain clinical blood test is 99 percent (.99) effective in detecting the presence of this disease; that is, it will yield an accurate positive result in 99 percent of the cases where the disease is actually present. But it also yields false-positive results in 5 percent (.05) of the cases where the disease is not present. The following table shows (in red) the probabilities that are stipulated in the example and (in blue) the probabilities that can be inferred from the stipulated information: P(A) = .005the probability that the disease will be present in any particular person P(~A) = 1—.005 = .995the probability that the disease will not be present in any particular person P(B|A) = .99the probability that the test will yield a positive result [B] if the disease is present [A] P(~B|A) = 1—.99 = .01the probability that the test will yield a negative result [~B] if the disease is present [A] P(B|~A) = .05the probability that the test will yield a positive result [B] if the disease is not present [~A] P(~B|~A) = 1—.05 = .95the probability that the test will yield a negative result [~B] if the disease is not present [~A]

Given this information, Bayes' theorem allows for the derivation of the two simple probabilities P(B) = [P(B|A) x P(A)] + [P(B|~A) x P(~A)]
= [.99 x .005]+[.05 x .995] = .0547the probability of a positive test result [B], irrespective of whether the disease is present [A] or not present [~A] P(~B) = [P(~B|A) x P(A)] + [P(~B|~A) x P(~A)]

= [.01 x .005]+[.95 x .995] = .9453the probability of a negative test result [~B], irrespective of whether the disease is present [A] or not present [~A] which in turn allows for the calculation of the four remaining conditional probabilities P(A|B) =...

...Basic Probability Notes
Probability— the relative frequency or likelihood that a specific event will occur. If the event is A, then the probability that A will occur is denoted P(A). Example: Flip a coin. What is the probability of heads? This is denoted P(heads). Properties of Probability 1. The probability of an event E always lies in the range of 0 to 1; i.e., 0 ≤ P( E ) ≤ 1. Impossible event—an event...

...ConditionalProbability
How to handle Dependent Events
Life is full of random events! You need to get a "feel" for them to be a smart and successful person.
Independent Events
Events can be "Independent", meaning each event is not affected by any other events.
Example: Tossing a coin.
Each toss of a coin is a perfect isolated thing.
What it did in the past will not affect the current toss.
The chance is simply 1-in-2, or 50%, just like ANY toss of the coin....

...Richard C. Carrier, Ph.D.
“Bayes’ Theorem for Beginners: Formal Logic and Its Relevance to Historical Method — Adjunct Materials and Tutorial”
The Jesus Project Inaugural Conference “Sources of the Jesus Tradition: An Inquiry”
5-7 December 2008 (Amherst, NY)
Table of Contents for Enclosed Document
Handout Accompanying Oral Presentation of December 5...................................pp. 2-5 Adjunct Document Expanding on Oral...

...UNIT 2 THEOREMS
Structure
2.1 Introduction
Objectives
PROBABILITY
2.2 Some Elementary Theorems
2.3 General Addition Rule
2.4 ConditionalProbability and Independence
2.4.1 ConditionalProbability 2.4.2 Independent Events and MultiplicationRule 2.4.3 Theorem of Total Probability and BayesTheorem
2.5 Summary
2.1...

...Gaussian observation noise
(see Fig. 1, left panel). Suppose we have N users
and M movies. Let Rij be the rating value of user i
for movie j, Ui and Vj represent D-dimensional userspeciﬁc and movie-speciﬁc latent feature vectors respectively. The conditional distribution over the observed ratings R ∈ RN ×M (the likelihood term) and
the prior distributions over U ∈ RD×N and V ∈
RD×M are given by:
N
M
Iij
N (Rij |UiT Vj , α−1 )
p(R|U, V, α) =
(1)
i=1 j=1
N...

...Conditional Sentences / If-Clauses Type I, II und III
Conditional Sentences are also known as Conditional Clauses or If Clauses. They are used to express that the action in the main clause (without if) can only take place if a certain condition (in the clause with if) is fulfilled. There are three types of Conditional Sentences.
Conditional Sentence Type 1
→ It is possible and also very likely that the condition will be...

...PROBABILITY DISTRIBUTION
In the world of statistics, we are introduced to the concept of probability. On page 146 of our text, it defines probability as "a value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur" (Lind, 2012). When we think about how much this concept pops up within our daily lives, we might be shocked to find the results. Oftentimes, we do not think in these terms, but...

...variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The mean also of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X)...