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Pearson correlation coefficient. Several sets of ( x , y) points, with the correlation coefficient of x and y for each set. The correlation reflects the strength and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom).
Example graph of a logistic regression curve fitted to data. The curve shows the estimated probability of passing an exam (binary dependent variable) versus hours studying (scalar independent variable). See Example for worked details. In statistics, the logistic model (or logit model) is a statistical model that models the log-odds of an event as a linear combination of one or more independent ...
The Mount Edziza volcanic complex (MEVC) is a group of volcanoes and associated lava flows in northwest British Columbia, Canada. Located on the Tahltan Highland, the MEVC has a broad, steep-sided lava plateau; its highest summit is 2,786 metres (9,140 feet). Its volcanoes formed over the last 7.5 million years during five cycles of magmatic ...
- Statement of Theorem
- Examples
- Interpretations
- Forms
- Correspondence to Other Mathematical Frameworks
- Generalizations
- History
- Use in Genetics
- See Also
- Further Reading
Bayes' theorem is stated mathematically as the following equation: where A {\displaystyle A} and B {\displaystyle B} are events and P ( B ) ≠ 0 {\displaystyle P(B)\neq 0} . 1. P ( A ∣ B ) {\displaystyle P(A\mid B)} is a conditional probability: the probability of event A {\displaystyle A} occurring given that B {\displaystyle B} is true. It is also...
Drug testing
Suppose, a particular test for whether someone has been using cannabis is 90% sensitive, meaning the true positive rate(TPR)=0.90. Therefore it leads to 90% true positive results (correct identification of drug use) for cannabis users. The test is also 80% specific, meaning true negative rate (TNR)=0.80. Therefore the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or false positive rate(FPR)=0.20, for non-users. Assuming 0.05 prevalence, meanin...
Cancer rate
Even if 100% of patients with pancreatic cancer have a certain symptom, when someone has the same symptom, it does not mean that this person has a 100% chance of getting pancreatic cancer. Assuming the incidence rate of pancreatic cancer is 1/100000, while 10/99999 healthy individuals have the same symptoms worldwide, the probability of having pancreatic cancer given the symptoms is only 9.1%, and the other 90.9% could be "false positives" (that is, falsely said to have cancer; "positive" is...
Defective item rate
A factory produces an item using three machines—A, B, and C—which account for 20%, 30%, and 50% of its output, respectively. Of the items produced by machine A, 5% are defective; similarly, 3% of machine B's items and 1% of machine C's are defective. If a randomly selected item is defective, what is the probability it was produced by machine C? Once again, the answer can be reached without using the formula by applying the conditions to a hypothetical number of cases. For example, if the fact...
The interpretation of Bayes' rule depends on the interpretation of probability ascribed to the terms. The two main interpretations are described below. Figure 2 shows a geometric visualization similar to Figure 1. Gerd Gigerenzer and co-authors have pushed hard for teaching Bayes Rule this way, with special emphasis on teaching it to physicians. An...
Random variables
Consider a sample space Ω generated by two random variables X and Y. In principle, Bayes' theorem applies to the events A = {X = x} and B = {Y = y}. 1. P ( X = x | Y = y ) = P ( Y = y | X = x ) P ( X = x ) P ( Y = y ) {\displaystyle P(X{=}x|Y{=}y)={\frac {P(Y{=}y|X{=}x)P(X{=}x)}{P(Y{=}y)}}} However, terms become 0 at points where either variable has finite probability density. To remain useful, Bayes' theorem must be formulated in terms of the relevant densities (see Derivation).
Bayes' rule
Bayes' theorem in odds formis: 1. O ( A 1 : A 2 ∣ B ) = O ( A 1 : A 2 ) ⋅ Λ ( A 1 : A 2 ∣ B ) {\displaystyle O(A_{1}:A_{2}\mid B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\mid B)} where 1. Λ ( A 1 : A 2 ∣ B ) = P ( B ∣ A 1 ) P ( B ∣ A 2 ) {\displaystyle \Lambda (A_{1}:A_{2}\mid B)={\frac {P(B\mid A_{1})}{P(B\mid A_{2})}}} is called the Bayes factor or likelihood ratio. The odds between two events is simply the ratio of the probabilities of the two events. Thus 1. O ( A 1 : A 2 ) = P ( A 1 ) P...
Propositional logic
Bayes' theorem represents a generalisation of contraposition which in propositional logiccan be expressed as: 1. ( ¬ A → ¬ B ) → ( B → A ) . {\displaystyle (\lnot A\to \lnot B)\to (B\to A).} The corresponding formula in terms of probability calculus is Bayes' theorem which in its expanded form is expressed as: 1. P ( A ∣ B ) = P ( B ∣ A ) a ( A ) P ( B ∣ A ) a ( A ) + P ( B ∣ ¬ A ) a ( ¬ A ) . {\displaystyle P(A\mid B)={\frac {P(B\mid A)a(A)}{P(B\mid A)a(A)+P(B\mid \lnot A)a(\lnot A)}}.} In t...
Subjective logic
Bayes' theorem represents a special case of deriving inverted conditional opinions in subjective logicexpressed as: 1. ( ω A | ~ B S , ω A | ~ ¬ B S ) = ( ω B ∣ A S , ω B ∣ ¬ A S ) ϕ ~ a A , {\displaystyle (\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})=(\omega _{B\mid A}^{S},\omega _{B\mid \lnot A}^{S}){\widetilde {\phi }}a_{A},} where ϕ ~ {\displaystyle {\widetilde {\phi }}} denotes the operator for inverting conditional opinions. The argument ( ω B ∣ A S , ω B ∣ ¬ A S ) {...
Conditioned version
A conditioned version of the Bayes' theorem results from the addition of a third event C {\displaystyle C} on which all probabilities are conditioned: 1. P ( A ∣ B ∩ C ) = P ( B ∣ A ∩ C ) P ( A ∣ C ) P ( B ∣ C ) {\displaystyle P(A\mid B\cap C)={\frac {P(B\mid A\cap C)\,P(A\mid C)}{P(B\mid C)}}}
Bayes' rule with 3 events
In the case of 3 events - A, B, and C - it can be shown that:
Bayes' theorem is named after the Reverend Thomas Bayes (/beɪz/; c. 1701 – 1761), who first used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter, published as An Essay towards solving a Problem in the Doctrine of Chances (1763). He studied how to compute a distributi...
In genetics, Bayes' theorem can be used to calculate the probability of an individual having a specific genotype. Many people seek to approximate their chances of being affected by a genetic disease or their likelihood of being a carrier for a recessive gene of interest. A Bayesian analysis can be done based on family history or genetic testing, in...
Why Most Published Research Findings Are False, a 2005 essay in metascienceby John IoannidisGrunau, Hans-Christoph (24 January 2014). "Preface Issue 3/4-2013". Jahresbericht der Deutschen Mathematiker-Vereinigung. 115 (3–4): 127–128. doi:10.1365/s13291-013-0077-z.Gelman, A, Carlin, JB, Stern, HS, and Rubin, DB (2003), "Bayesian Data Analysis," Second Edition, CRC Press.Grinstead, CM and Snell, JL (1997), "Introduction to Probability (2nd edition)," American Mathematical Society (free pdf available) ."Bayes formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994]t. e. In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables ...
Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified.
Sigmund Freud (/ f r ɔɪ d / FROYD, German: [ˈziːkmʊnt ˈfrɔʏt]; born Sigismund Schlomo Freud; 6 May 1856 – 23 September 1939) was an Austrian neurologist and the founder of psychoanalysis, a clinical method for evaluating and treating pathologies seen as originating from conflicts in the psyche, through dialogue between patient and psychoanalyst, and the distinctive theory of mind and ...
