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 that absolutely cannot occur; probability is zero. Example: Suppose you roll a normal die. What is the probability that you will get a seven? P(7) = 0. Sure event—an event that is certain to occur; probability is one. Example: Suppose you roll a normal die. What is the probability that you will get a number less than 7? P(a number less than 7) = 1. 2. The sum of the probabilities of all simple mutually exclusive events (or final outcomes) that can occur in a population or sample events in an expirement is always 1. Example: Suppose you flip two coins. What are the outcomes? HH, HT, TH, TT. This rule says that the probabilities of each of these outcomes should sum to one. That is, P(HH) + P(HT) + P(TH) + P(TT) = 1 Marginal and ConditionalProbabilities Suppose the faculty at a local school were polled as to their agreement/disagreement with the following statement: Coaches...

...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.
So each toss is an Independent Event.
Dependent Events
But events can also be "dependent" ... which means they can be affected by previous events ...
Example: Marbles in a Bag
2 blue and 3 red marbles are in a bag.
What are the chances of getting a blue marble?
The chance is 2 in 5
But after taking one out you change the chances!
So the next time:
* if you got a red marble before, then the chance of a blue marble next is 2 in 4
* if you got a blue marble before, then the chance of a blue marble next is 1 in 4
See how the chances change each time? Each event depends on what happened in the previous event, and is called dependent.
That is the kind of thing we will be looking at here.
"Replacement"
Note: if you had replaced the marbles in the bag each time, then the chances would not have changed and the events would be independent:
* With Replacement: the events are Independent (the chances don't change)
* Without Replacement: the events...

...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 Presentation.............................................pp. 6-26 Simple Tutorial in Bayes’ Theorem.......................................................................pp. 27-39
NOTE: A chapter of the same title was published in 2010 by Prometheus Press (in Sources of the Jesus Tradition: Separating History from Myth, ed. R. Joseph Hoffmann, 2010) discussing or referring to the contents of this online document. That primary document (to which this document is adjunct) has also been published in advance as “Bayes’ Theorem for Beginners: Formal Logic and Its Relevance to Historical Method” in Caesar: A Journal for the Critical Study of Religion and Human Values 3.1 (2009): 26-35.
1
Richard C. Carrier, Ph.D.
“BayesTheorem for Beginners: Formal Logic and Its Relevance to Historical Method”
December 2008 (Amherst, NY) Notes and Bibliography 1. Essential Reading on “Historicity Criteria” Stanley Porter, The Criteria for...

...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 INTRODUCTION
You have already learnt about probability axioms and ways to evaluate probability of events in some simple cases. In this unit, we discuss ways to evaluate the probability of combination of events. For this, we derive the addition rule which deals with the probability of union of two events and the multiplication rule which deals with thc probability of intersection of two events. Two important concepts namely : ConditionalProbability and independence of events, are introduced and Bayestheorem, which deals with conditionalprobability is presented.
Objectives
After reading this unit, you should be able to
* * * *
evaluate the probability of certain combination of events involving union, intersection and complementation, evaluate conditionalprobability, check independence of...

...assumption of independence between the viewer and movie factors seems unreasonable, and, as our experiments demonstrate, the
distributions over factors in such models turn out to
be non-Gaussian. This conclusion is supported by the
fact that the Bayesian PMF models outperform their
MAP trained counterparts by a much larger margin
than the variationally trained models do.
2. Probabilistic Matrix Factorization
Probabilistic Matrix Factorization (PMF) is a probabilistic linear model with 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
N (Ui |0, α−1 I)
U
(2)
N (Vj |0, α−1 I),
V
p(U |αU ) =
(3)
i=1
M
p(V |αV ) =
j=1
where N (x|µ, α−1 ) denotes the Gaussian distribution
with mean µ and precision α, and Iij is the indicator
variable that is equal to 1 if user i rated movie j and
equal to 0 otherwise.
Learning in this model is performed by maximizing
the log-posterior over the movie and user features with
ﬁxed hyperparameters (i.e. the observation noise variance and prior variances):
ln p(U, V...

...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 fulfilled.
Form: if + Simple Present, will-Future
Example: If I find her address, I’ll send her an invitation.
Conditional Sentence Type 2
→ It is possible but very unlikely, that the condition will be fulfilled.
Form: if + Simple Past, Conditional I (= would + Infinitive)
Example: If I found her address, I would send her an invitation.
Conditional Sentence Type 3
→ It is impossible that the condition will be fulfilled because it refers to the past.
Form: if + Past Perfect, Conditional II (= would + have + Past Participle)
Example: If I had found her address, I would have sent her an invitation.
Exceptions
Sometimes Conditional Sentences Type I, II and III can also be used with other tenses.
Conditional Sentences
Because conditional sentences are quite complex in both form and meaning, they are a problem for most learners of English. If you have a...

...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 imagine what the probability of us getting behind the wheel of a car twice a day, Monday through Friday, and arriving at work and home safely. Thankfully, the probability for me has been 'one'! This means that up to this point I have made it to work and returned home every day without getting into an accident. While probability might have one outcome with one set of circumstances, this does not mean it will always turn out that way. Using the same example, just because I have arrived at work every day without getting into an accident, this does not mean it will always be true. As I confess with my words, and pray it does stay the same, probability tells me there is room for a different outcome.
In business, we often look at the probability of success or financial gain when making a decision. There are several things to take into consideration such as the experiment, potential outcomes, and possible events. An...

...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) is , formally defined by
Variance - The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation - The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
2. Define the following;
a) Binomial Distribution - is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Therewith the probability of an event is defined by its binomial...