Misplaced Pages

#823176

101-517: One of the many applications of Bayes' theorem is Bayesian inference , a particular approach to statistical inference , where it is used to invert the probability of observations given a model configuration (i.e., the likelihood function ) to obtain the probability of the model configuration given the observations (i.e., the posterior probability ). Bayes' theorem is named after the Reverend Thomas Bayes ( / b eɪ z / ), also

202-414: A normal distribution with unknown mean and variance are constructed using a Student's t-distribution . This correctly estimates the variance, due to the facts that (1) the average of normally distributed random variables is also normally distributed, and (2) the predictive distribution of a normally distributed data point with unknown mean and variance, using conjugate or uninformative priors, has

303-436: 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 a confusing term when, as here, the test gives bad news). Based on incidence rate,

404-413: A Student's t-distribution. In Bayesian statistics, however, the posterior predictive distribution can always be determined exactly—or at least to an arbitrary level of precision when numerical methods are used. Both types of predictive distributions have the form of a compound probability distribution (as does the marginal likelihood ). In fact, if the prior distribution is a conjugate prior , such that

505-404: A bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1? Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl #1. The precise answer

606-425: A computable sequence x is the sum of the probabilities of all programs (for a universal computer) that compute something starting with p . Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes' theorem can be used to predict the yet unseen parts of x in optimal fashion. Event (probability theory) In probability theory , an event

707-654: A friend, the minister, philosopher, and mathematician Richard Price . Over two years, Richard Price significantly edited the unpublished manuscript, before sending it to a friend who read it aloud at the Royal Society on 23 December 1763. Price edited Bayes's major work "An Essay Towards Solving a Problem in the Doctrine of Chances" (1763), which appeared in Philosophical Transactions , and contains Bayes' theorem. Price wrote an introduction to

808-552: A fundamental part of computerized pattern recognition techniques since the late 1950s. There is also an ever-growing connection between Bayesian methods and simulation-based Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a graphical model structure may allow for efficient simulation algorithms like the Gibbs sampling and other Metropolis–Hastings algorithm schemes. Recently Bayesian inference has gained popularity among

909-549: 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 , meaning 5% of people use cannabis, what

1010-470: 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 factory produces 1,000 items, 200 will be produced by Machine A, 300 by Machine B, and 500 by Machine C. Machine A will produce 5% × 200 = 10 defective items, Machine B 3% × 300 = 9, and Machine C 1% × 500 = 5, for

1111-967: A sequence of independent and identically distributed observations E = ( e 1 , … , e n ) {\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})} , it can be shown by induction that repeated application of the above is equivalent to P ( M ∣ E ) = P ( E ∣ M ) ∑ m P ( E ∣ M m ) P ( M m ) ⋅ P ( M ) , {\displaystyle P(M\mid \mathbf {E} )={\frac {P(\mathbf {E} \mid M)}{\sum _{m}{P(\mathbf {E} \mid M_{m})P(M_{m})}}}\cdot P(M),} where P ( E ∣ M ) = ∏ k P ( e k ∣ M ) . {\displaystyle P(\mathbf {E} \mid M)=\prod _{k}{P(e_{k}\mid M)}.} By parameterizing

SECTION 10

#1732783051824

1212-399: A set of exclusive and exhaustive propositions, Bayesian inference may be thought of as acting on this belief distribution as a whole. Suppose a process is generating independent and identically distributed events E n ,   n = 1 , 2 , 3 , … {\displaystyle E_{n},\ n=1,2,3,\ldots } , but the probability distribution

1313-446: A site thought to be from the medieval period, between the 11th century to the 16th century. However, it is uncertain exactly when in this period the site was inhabited. Fragments of pottery are found, some of which are glazed and some of which are decorated. It is expected that if the site were inhabited during the early medieval period, then 1% of the pottery would be glazed and 50% of its area decorated, whereas if it had been inhabited in

1414-457: A statistician and philosopher. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. His work was published in 1763 as An Essay Towards Solving a Problem in the Doctrine of Chances . Bayes studied how to compute a distribution for the probability parameter of a binomial distribution (in modern terminology). On Bayes's death his family transferred his papers to

1515-405: A total of 24. Thus, the likelihood that a randomly selected defective item was produced by machine C is 5/24 (~20.83%). This problem can also be solved using Bayes' theorem: Let X i denote the event that a randomly chosen item was made by the i machine (for i  = A,B,C). Let Y denote the event that a randomly chosen item is defective. Then, we are given the following information: If

1616-1034: A uniform prior of f C ( c ) = 0.2 {\textstyle f_{C}(c)=0.2} , and that trials are independent and identically distributed . When a new fragment of type e {\displaystyle e} is discovered, Bayes' theorem is applied to update the degree of belief for each c {\displaystyle c} : f C ( c ∣ E = e ) = P ( E = e ∣ C = c ) P ( E = e ) f C ( c ) = P ( E = e ∣ C = c ) ∫ 11 16 P ( E = e ∣ C = c ) f C ( c ) d c f C ( c ) {\displaystyle f_{C}(c\mid E=e)={\frac {P(E=e\mid C=c)}{P(E=e)}}f_{C}(c)={\frac {P(E=e\mid C=c)}{\int _{11}^{16}{P(E=e\mid C=c)f_{C}(c)dc}}}f_{C}(c)} A computer simulation of

1717-430: A value with the greatest probability defines maximum a posteriori (MAP) estimates: { θ MAP } ⊂ arg ⁡ max θ p ( θ ∣ X , α ) . {\displaystyle \{\theta _{\text{MAP}}\}\subset \arg \max _{\theta }p(\theta \mid \mathbf {X} ,\alpha ).} There are examples where no maximum

1818-423: Is a set of outcomes of an experiment (a subset of the sample space ) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an elementary event or an atomic event ; that is, it

1919-583: Is a singleton set . An event that has more than one possible outcome is called a compound event. An event S {\displaystyle S} is said to occur if S {\displaystyle S} contains the outcome x {\displaystyle x} of the experiment (or trial) (that is, if x ∈ S {\displaystyle x\in S} ). The probability (with respect to some probability measure ) that an event S {\displaystyle S} occurs

2020-485: Is a cannabis user given that they test positive," which is what is meant by PPV. We can write: The denominator P ( Positive ) = P ( Positive | User ) P ( User ) + P ( Positive | Non-user ) P ( Non-user ) {\displaystyle P({\text{Positive}})=P({\text{Positive}}\vert {\text{User}})P({\text{User}})+P({\text{Positive}}\vert {\text{Non-user}})P({\text{Non-user}})}

2121-454: Is a direct application of the Law of Total Probability . In this case, it says that the probability that someone tests positive is the probability that a user tests positive, times the probability of being a user, plus the probability that a non-user tests positive, times the probability of being a non-user. This is true because the classifications user and non-user form a partition of a set , namely

SECTION 20

#1732783051824

2222-421: Is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence , and update it as more information becomes available. Fundamentally, Bayesian inference uses a prior distribution to estimate posterior probabilities. Bayesian inference is an important technique in statistics , and especially in mathematical statistics . Bayesian updating

2323-495: Is a real-valued random variable defined on the sample space Ω , {\displaystyle \Omega ,} the event { ω ∈ Ω ∣ u < X ( ω ) ≤ v } {\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,} can be written more conveniently as, simply, u < X ≤ v . {\displaystyle u<X\leq v\,.} This

2424-626: Is a set of parameters to the prior itself, or hyperparameters . Let E = ( e 1 , … , e n ) {\displaystyle \mathbf {E} =(e_{1},\dots ,e_{n})} be a sequence of independent and identically distributed event observations, where all e i {\displaystyle e_{i}} are distributed as p ( e ∣ θ ) {\displaystyle p(e\mid {\boldsymbol {\theta }})} for some θ {\displaystyle {\boldsymbol {\theta }}} . Bayes' theorem

2525-402: Is about 1 2 {\displaystyle {\tfrac {1}{2}}} , about 50% likely - equally likely or not likely. If that term is very small, close to zero, then the probability of the hypothesis, given the evidence, P ( H ∣ E ) {\displaystyle P(H\mid E)} is close to 1 or the conditional hypothesis is quite likely. If that term

2626-2003: Is applied to find the posterior distribution over θ {\displaystyle {\boldsymbol {\theta }}} : p ( θ ∣ E , α ) = p ( E ∣ θ , α ) p ( E ∣ α ) ⋅ p ( θ ∣ α ) = p ( E ∣ θ , α ) ∫ p ( E ∣ θ , α ) p ( θ ∣ α ) d θ ⋅ p ( θ ∣ α ) , {\displaystyle {\begin{aligned}p({\boldsymbol {\theta }}\mid \mathbf {E} ,{\boldsymbol {\alpha }})&={\frac {p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})}{p(\mathbf {E} \mid {\boldsymbol {\alpha }})}}\cdot p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})\\&={\frac {p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})}{\int p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})\,d{\boldsymbol {\theta }}}}\cdot p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }}),\end{aligned}}} where p ( E ∣ θ , α ) = ∏ k p ( e k ∣ θ ) . {\displaystyle p(\mathbf {E} \mid {\boldsymbol {\theta }},{\boldsymbol {\alpha }})=\prod _{k}p(e_{k}\mid {\boldsymbol {\theta }}).} P X y ( A ) = E ( 1 A ( X ) | Y = y ) {\displaystyle P_{X}^{y}(A)=E(1_{A}(X)|Y=y)} Existence and uniqueness of

2727-472: Is attained, in which case the set of MAP estimates is empty . There are other methods of estimation that minimize the posterior risk (expected-posterior loss) with respect to a loss function , and these are of interest to statistical decision theory using the sampling distribution ("frequentist statistics"). The posterior predictive distribution of a new observation x ~ {\displaystyle {\tilde {x}}} (that

2828-467: Is finite (see above section on asymptotic behaviour of the posterior). A decision-theoretic justification of the use of Bayesian inference was given by Abraham Wald , who proved that every unique Bayesian procedure is admissible . Conversely, every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures. Wald characterized admissible procedures as Bayesian procedures (and limits of Bayesian procedures), making

2929-402: Is given by Bayes' theorem. Let H 1 {\displaystyle H_{1}} correspond to bowl #1, and H 2 {\displaystyle H_{2}} to bowl #2. It is given that the bowls are identical from Fred's point of view, thus P ( H 1 ) = P ( H 2 ) {\displaystyle P(H_{1})=P(H_{2})} , and

3030-831: Is independent of previous observations) is determined by p ( x ~ | X , α ) = ∫ p ( x ~ , θ ∣ X , α ) d θ = ∫ p ( x ~ ∣ θ ) p ( θ ∣ X , α ) d θ . {\displaystyle p({\tilde {x}}|\mathbf {X} ,\alpha )=\int p({\tilde {x}},\theta \mid \mathbf {X} ,\alpha )\,d\theta =\int p({\tilde {x}}\mid \theta )p(\theta \mid \mathbf {X} ,\alpha )\,d\theta .} Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks

3131-467: Is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the 𝜎-algebra. Even though events are subsets of some sample space Ω , {\displaystyle \Omega ,} they are often written as predicates or indicators involving random variables . For example, if X {\displaystyle X}

Bayes' theorem - Misplaced Pages Continue

3232-419: Is often desired to use a posterior distribution to estimate a parameter or variable. Several methods of Bayesian estimation select measurements of central tendency from the posterior distribution. For one-dimensional problems, a unique median exists for practical continuous problems. The posterior median is attractive as a robust estimator . If there exists a finite mean for the posterior distribution, then

3333-401: Is particularly important in the dynamic analysis of a sequence of data . Bayesian inference has found application in a wide range of activities, including science , engineering , philosophy , medicine , sport , and law . In the philosophy of decision theory , Bayesian inference is closely related to subjective probability, often called " Bayesian probability ". Bayesian inference derives

3434-464: Is performed many times. P ( A ) is the proportion of outcomes with property A (the prior) and P ( B ) is the proportion with property B . P ( B  |  A ) is the proportion of outcomes with property B out of outcomes with property A , and P ( A  |  B ) is the proportion of those with A out of those with  B (the posterior). The role of Bayes' theorem is best visualized with tree diagrams. The two diagrams partition

3535-414: Is predicted by the current state of belief. The reverse applies for a decrease in belief. If the belief does not change, P ( E ∣ M ) P ( E ) = 1 ⇒ P ( E ∣ M ) = P ( E ) {\textstyle {\frac {P(E\mid M)}{P(E)}}=1\Rightarrow P(E\mid M)=P(E)} . That is, the evidence is independent of

3636-406: Is raised to 100% and specificity remains at 80%, the probability of someone testing positive really being a cannabis user only rises from 19% to 21%, but if the sensitivity is held at 90% and the specificity is increased to 95%, the probability rises to 49%. 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

3737-450: Is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice. In the general measure-theoretic description of probability spaces , an event may be defined as an element of a selected 𝜎-algebra of subsets of the sample space. Under this definition, any subset of the sample space that

3838-534: Is the probability that a random person who tests positive is really a cannabis user? The Positive predictive value (PPV) of a test is the proportion of persons who are actually positive out of all those testing positive, and can be calculated from a sample as: If sensitivity, specificity, and prevalence are known, PPV can be calculated using Bayes theorem. Let P ( User | Positive ) {\displaystyle P({\text{User}}\vert {\text{Positive}})} mean "the probability that someone

3939-415: Is the probability that S {\displaystyle S} contains the outcome x {\displaystyle x} of an experiment (that is, it is the probability that x ∈ S {\displaystyle x\in S} ). An event defines a complementary event , namely the complementary set (the event not occurring), and together these define a Bernoulli trial : did

4040-449: Is then P X , Y ( d x , d y ) = P Y x ( d y ) P X ( d x ) {\displaystyle P_{X,Y}(dx,dy)=P_{Y}^{x}(dy)P_{X}(dx)} . The conditional distribution P X y {\displaystyle P_{X}^{y}} of X {\displaystyle X} given Y = y {\displaystyle Y=y}

4141-469: Is then determined by P X y ( A ) = E ( 1 A ( X ) | Y = y ) {\displaystyle P_{X}^{y}(A)=E(1_{A}(X)|Y=y)} Existence and uniqueness of the needed conditional expectation is a consequence of the Radon–Nikodym theorem . This was formulated by Kolmogorov in his famous book from 1933. Kolmogorov underlines

Bayes' theorem - Misplaced Pages Continue

4242-553: Is treated in more detail in the article on the naïve Bayes classifier . Solomonoff's Inductive inference is the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. The only assumption is that the environment follows some unknown but computable probability distribution . It is a formal inductive framework that combines two well-studied principles of inductive inference: Bayesian statistics and Occam's Razor . Solomonoff's universal prior probability of any prefix p of

4343-422: Is unknown. Let the event space Ω {\displaystyle \Omega } represent the current state of belief for this process. Each model is represented by event M m {\displaystyle M_{m}} . The conditional probabilities P ( E n ∣ M m ) {\displaystyle P(E_{n}\mid M_{m})} are specified to define

4444-779: Is very large, much larger than 1, then the hypothesis, given the evidence, is quite unlikely. If the hypothesis (without consideration of evidence) is unlikely, then P ( H ) {\displaystyle P(H)} is small (but not necessarily astronomically small) and 1 P ( H ) {\displaystyle {\tfrac {1}{P(H)}}} is much larger than 1 and this term can be approximated as P ( E ∣ ¬ H ) P ( E ∣ H ) ⋅ P ( H ) {\displaystyle {\tfrac {P(E\mid \neg H)}{P(E\mid H)\cdot P(H)}}} and relevant probabilities can be compared directly to each other. One quick and easy way to remember

4545-606: The Bayes factor . Since Bayesian model comparison is aimed on selecting the model with the highest posterior probability, this methodology is also referred to as the maximum a posteriori (MAP) selection rule or the MAP probability rule. While conceptually simple, Bayesian methods can be mathematically and numerically challenging. Probabilistic programming languages (PPLs) implement functions to easily build Bayesian models together with efficient automatic inference methods. This helps separate

4646-486: The interpretation of probability ascribed to the terms. The two predominant interpretations are described below. In the Bayesian (or epistemological) interpretation , probability measures a "degree of belief". Bayes' theorem links the degree of belief in a proposition before and after accounting for evidence. For example, suppose it is believed with 50% certainty that a coin is twice as likely to land heads than tails. If

4747-477: The phylogenetics community for these reasons; a number of applications allow many demographic and evolutionary parameters to be estimated simultaneously. As applied to statistical classification , Bayesian inference has been used to develop algorithms for identifying e-mail spam . Applications which make use of Bayesian inference for spam filtering include CRM114 , DSPAM , Bogofilter , SpamAssassin , SpamBayes , Mozilla , XEAMS, and others. Spam classification

4848-589: The posterior probability as a consequence of two antecedents : a prior probability and a " likelihood function " derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem : P ( H ∣ E ) = P ( E ∣ H ) ⋅ P ( H ) P ( E ) , {\displaystyle P(H\mid E)={\frac {P(E\mid H)\cdot P(H)}{P(E)}},} where For different values of H {\displaystyle H} , only

4949-488: The Bayesian formalism a central technique in such areas of frequentist inference as parameter estimation , hypothesis testing , and computing confidence intervals . For example: Bayesian methodology also plays a role in model selection where the aim is to select one model from a set of competing models that represents most closely the underlying process that generated the observed data. In Bayesian model comparison,

5050-841: The Bayes–Price rule. Price discovered Bayes's work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes's name alone is unfair but so entrenched that anything else makes little sense. Bayes' theorem is stated mathematically as the following equation: P ( A | B ) = P ( B | A ) P ( A ) P ( B ) {\displaystyle P(A\vert B)={\frac {P(B\vert A)P(A)}{P(B)}}} where A {\displaystyle A} and B {\displaystyle B} are events and P ( B ) ≠ 0 {\displaystyle P(B)\neq 0} . Bayes' theorem may be derived from

5151-511: The behaviour of a belief distribution as it is updated a large number of times with independent and identically distributed trials. For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials, the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph L. Doob in 1948, namely if

SECTION 50

#1732783051824

5252-415: The changing belief as 50 fragments are unearthed is shown on the graph. In the simulation, the site was inhabited around 1420, or c = 15.2 {\displaystyle c=15.2} . By calculating the area under the relevant portion of the graph for 50 trials, the archaeologist can say that there is practically no chance the site was inhabited in the 11th and 12th centuries, about 1% chance that it

5353-458: The coin is flipped a number of times and the outcomes observed, that degree of belief will probably rise or fall, but might even remain the same, depending on the results. For proposition A and evidence B , For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see Bayesian inference . In the frequentist interpretation , probability measures a "proportion of outcomes". For example, suppose an experiment

5454-409: The cookie, the probability we assigned for Fred having chosen bowl #1 was the prior probability, P ( H 1 ) {\displaystyle P(H_{1})} , which was 0.5. After observing the cookie, we must revise the probability to P ( H 1 ∣ E ) {\displaystyle P(H_{1}\mid E)} , which is 0.6. An archaeologist is working at

5555-444: The definition of conditional density : Therefore, Let P Y x {\displaystyle P_{Y}^{x}} be the conditional distribution of Y {\displaystyle Y} given X = x {\displaystyle X=x} and let P X {\displaystyle P_{X}} be the distribution of X {\displaystyle X} . The joint distribution

5656-551: The definition of conditional probability : where P ( A ∩ B ) {\displaystyle P(A\cap B)} is the probability of both A and B being true. Similarly, Solving for P ( A ∩ B ) {\displaystyle P(A\cap B)} and substituting into the above expression for P ( A | B ) {\displaystyle P(A\vert B)} yields Bayes' theorem: For two continuous random variables X and Y , Bayes' theorem may be analogously derived from

5757-540: The discussion, and we wish to consider the impact of its having been observed on our belief in various possible events A . In such a situation the denominator of the last expression, the probability of the given evidence B , is fixed; what we want to vary is A . Bayes' theorem then shows that the posterior probabilities are proportional to the numerator, so the last equation becomes: Bayesian inference Bayesian inference ( / ˈ b eɪ z i ə n / BAY -zee-ən or / ˈ b eɪ ʒ ən / BAY -zhən )

5858-461: The distribution of a new, unobserved data point. That is, instead of a fixed point as a prediction, a distribution over possible points is returned. Only this way is the entire posterior distribution of the parameter(s) used. By comparison, prediction in frequentist statistics often involves finding an optimum point estimate of the parameter(s)—e.g., by maximum likelihood or maximum a posteriori estimation (MAP)—and then plugging this estimate into

5959-420: The effects of the initial choice, and especially for large (but finite) systems the convergence might be very slow. In parameterized form, the prior distribution is often assumed to come from a family of distributions called conjugate priors . The usefulness of a conjugate prior is that the corresponding posterior distribution will be in the same family, and the calculation may be expressed in closed form . It

6060-561: The equation would be to use rule of multiplication : P ( E ∩ H ) = P ( E ∣ H ) P ( H ) = P ( H ∣ E ) P ( E ) . {\displaystyle P(E\cap H)=P(E\mid H)P(H)=P(H\mid E)P(E).} Bayesian updating is widely used and computationally convenient. However, it is not the only updating rule that might be considered rational. Ian Hacking noted that traditional " Dutch book " arguments did not specify Bayesian updating: they left open

6161-409: The event occur or not? Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite . So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of

SECTION 60

#1732783051824

6262-411: The example events above. Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions , such as the normal distribution , the sample space is the set of real numbers or some subset of the real numbers . Attempts to define probabilities for all subsets of

6363-430: The factors P ( H ) {\displaystyle P(H)} and P ( E ∣ H ) {\displaystyle P(E\mid H)} , both in the numerator, affect the value of P ( H ∣ E ) {\displaystyle P(H\mid E)}  – the posterior probability of a hypothesis is proportional to its prior probability (its inherent likeliness) and

6464-431: The first rule to the event "not M {\displaystyle M} " in place of " M {\displaystyle M} ", yielding "if 1 − P ( M ) = 0 {\displaystyle 1-P(M)=0} , then 1 − P ( M ∣ E ) = 0 {\displaystyle 1-P(M\mid E)=0} ", from which the result immediately follows. Consider

6565-427: The following table presents the corresponding numbers per 100,000 people. Which can then be used to calculate the probability of having cancer when you have the symptoms: A factory produces items 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

6666-412: The formula for the distribution of a data point. This has the disadvantage that it does not account for any uncertainty in the value of the parameter, and hence will underestimate the variance of the predictive distribution. In some instances, frequentist statistics can work around this problem. For example, confidence intervals and prediction intervals in frequentist statistics when constructed from

6767-457: The importance of conditional probability by writing "I wish to call attention to ... and especially the theory of conditional probabilities and conditional expectations ..." in the Preface. The Bayes theorem determines the posterior distribution from the prior distribution. Uniqueness requires continuity assumptions. Bayes' theorem can be generalized to include improper prior distributions such as

6868-444: The item is defective, the probability that it was made by machine C is 5/24. Although machine C produces half of the total output, it produces a much smaller fraction of the defective items. Hence the knowledge that the item selected was defective enables us to replace the prior probability P ( X C ) = 1/2 by the smaller posterior probability P (X C  |  Y ) = 5/24. The interpretation of Bayes' rule depends on

6969-436: The item was made by the first machine, then the probability that it is defective is 0.05; that is, P ( Y  |  X A ) = 0.05. Overall, we have To answer the original question, we first find P (Y). That can be done in the following way: Hence, 2.4% of the total output is defective. We are given that Y has occurred, and we want to calculate the conditional probability of X C . By Bayes' theorem, Given that

7070-2753: The late medieval period then 81% would be glazed and 5% of its area decorated. How confident can the archaeologist be in the date of inhabitation as fragments are unearthed? The degree of belief in the continuous variable C {\displaystyle C} (century) is to be calculated, with the discrete set of events { G D , G D ¯ , G ¯ D , G ¯ D ¯ } {\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}} as evidence. Assuming linear variation of glaze and decoration with time, and that these variables are independent, P ( E = G D ∣ C = c ) = ( 0.01 + 0.81 − 0.01 16 − 11 ( c − 11 ) ) ( 0.5 − 0.5 − 0.05 16 − 11 ( c − 11 ) ) {\displaystyle P(E=GD\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))} P ( E = G D ¯ ∣ C = c ) = ( 0.01 + 0.81 − 0.01 16 − 11 ( c − 11 ) ) ( 0.5 + 0.5 − 0.05 16 − 11 ( c − 11 ) ) {\displaystyle P(E=G{\bar {D}}\mid C=c)=(0.01+{\frac {0.81-0.01}{16-11}}(c-11))(0.5+{\frac {0.5-0.05}{16-11}}(c-11))} P ( E = G ¯ D ∣ C = c ) = ( ( 1 − 0.01 ) − 0.81 − 0.01 16 − 11 ( c − 11 ) ) ( 0.5 − 0.5 − 0.05 16 − 11 ( c − 11 ) ) {\displaystyle P(E={\bar {G}}D\mid C=c)=((1-0.01)-{\frac {0.81-0.01}{16-11}}(c-11))(0.5-{\frac {0.5-0.05}{16-11}}(c-11))} P ( E = G ¯ D ¯ ∣ C = c ) = ( ( 1 − 0.01 ) − 0.81 − 0.01 16 − 11 ( c − 11 ) ) ( 0.5 + 0.5 − 0.05 16 − 11 ( c − 11 ) ) {\displaystyle P(E={\bar {G}}{\bar {D}}\mid C=c)=((1-0.01)-{\frac {0.81-0.01}{16-11}}(c-11))(0.5+{\frac {0.5-0.05}{16-11}}(c-11))} Assume

7171-400: The literature on " probability kinematics ") following the publication of Richard C. Jeffrey 's rule, which applies Bayes' rule to the case where the evidence itself is assigned a probability. The additional hypotheses needed to uniquely require Bayesian updating have been deemed to be substantial, complicated, and unsatisfactory. If evidence is simultaneously used to update belief over

7272-416: The model building from the inference, allowing practitioners to focus on their specific problems and leaving PPLs to handle the computational details for them. See the separate Misplaced Pages entry on Bayesian statistics , specifically the statistical modeling section in that page. Bayesian inference has applications in artificial intelligence and expert systems . Bayesian inference techniques have been

7373-406: The model with the highest posterior probability given the data is selected. The posterior probability of a model depends on the evidence, or marginal likelihood , which reflects the probability that the data is generated by the model, and on the prior belief of the model. When two competing models are a priori considered to be equiprobable, the ratio of their posterior probabilities corresponds to

7474-750: The model. If the model were true, the evidence would be exactly as likely as predicted by the current state of belief. If P ( M ) = 0 {\displaystyle P(M)=0} then P ( M ∣ E ) = 0 {\displaystyle P(M\mid E)=0} . If P ( M ) = 1 {\displaystyle P(M)=1} and P ( E ) > 0 {\displaystyle P(E)>0} , then P ( M | E ) = 1 {\displaystyle P(M|E)=1} . This can be interpreted to mean that hard convictions are insensitive to counter-evidence. The former follows directly from Bayes' theorem. The latter can be derived by applying

7575-421: The models. P ( M m ) {\displaystyle P(M_{m})} is the degree of belief in M m {\displaystyle M_{m}} . Before the first inference step, { P ( M m ) } {\displaystyle \{P(M_{m})\}} is a set of initial prior probabilities . These must sum to 1, but are otherwise arbitrary. Suppose that

7676-427: The needed conditional expectation is a consequence of the Radon–Nikodym theorem . This was formulated by Kolmogorov in his famous book from 1933. Kolmogorov underlines the importance of conditional probability by writing "I wish to call attention to ... and especially the theory of conditional probabilities and conditional expectations ..." in the Preface. The Bayes theorem determines the posterior distribution from

7777-1696: The newly acquired likelihood (its compatibility with the new observed evidence). In cases where ¬ H {\displaystyle \neg H} ("not H {\displaystyle H} "), the logical negation of H {\displaystyle H} , is a valid likelihood, Bayes' rule can be rewritten as follows: P ( H ∣ E ) = P ( E ∣ H ) P ( H ) P ( E ) = P ( E ∣ H ) P ( H ) P ( E ∣ H ) P ( H ) + P ( E ∣ ¬ H ) P ( ¬ H ) = 1 1 + ( 1 P ( H ) − 1 ) P ( E ∣ ¬ H ) P ( E ∣ H ) {\displaystyle {\begin{aligned}P(H\mid E)&={\frac {P(E\mid H)P(H)}{P(E)}}\\\\&={\frac {P(E\mid H)P(H)}{P(E\mid H)P(H)+P(E\mid \neg H)P(\neg H)}}\\\\&={\frac {1}{1+\left({\frac {1}{P(H)}}-1\right){\frac {P(E\mid \neg H)}{P(E\mid H)}}}}\\\end{aligned}}} because P ( E ) = P ( E ∣ H ) P ( H ) + P ( E ∣ ¬ H ) P ( ¬ H ) {\displaystyle P(E)=P(E\mid H)P(H)+P(E\mid \neg H)P(\neg H)} and P ( H ) + P ( ¬ H ) = 1. {\displaystyle P(H)+P(\neg H)=1.} This focuses attention on

7878-717: The paper which provides some of the philosophical basis of Bayesian statistics and chose one of the two solutions offered by Bayes. In 1765, Price was elected a Fellow of the Royal Society in recognition of his work on the legacy of Bayes. On 27 April a letter sent to his friend Benjamin Franklin was read out at the Royal Society, and later published, where Price applies this work to population and computing 'life-annuities'. Independently of Bayes, Pierre-Simon Laplace in 1774, and later in his 1812 Théorie analytique des probabilités , used conditional probability to formulate

7979-400: The pattern. The rare subspecies is 0.1% of the total population. How likely is the beetle having the pattern to be rare: what is P (Rare | Pattern)? From the extended form of Bayes' theorem (since any beetle is either rare or common), For events A and B , provided that P ( B ) ≠ 0, In many applications, for instance in Bayesian inference , the event B is fixed in

8080-612: The possibility that non-Bayesian updating rules could avoid Dutch books. Hacking wrote: "And neither the Dutch book argument nor any other in the personalist arsenal of proofs of the probability axioms entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour." Indeed, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in

8181-405: The posterior mean is a method of estimation. θ ~ = E ⁡ [ θ ] = ∫ θ p ( θ ∣ X , α ) d θ {\displaystyle {\tilde {\theta }}=\operatorname {E} [\theta ]=\int \theta \,p(\theta \mid \mathbf {X} ,\alpha )\,d\theta } Taking

8282-427: The prior and posterior distributions come from the same family, it can be seen that both prior and posterior predictive distributions also come from the same family of compound distributions. The only difference is that the posterior predictive distribution uses the updated values of the hyperparameters (applying the Bayesian update rules given in the conjugate prior article), while the prior predictive distribution uses

8383-506: The prior distribution. Uniqueness requires continuity assumptions. Bayes' theorem can be generalized to include improper prior distributions such as the uniform distribution on the real line. Modern Markov chain Monte Carlo methods have boosted the importance of Bayes' theorem including cases with improper priors. Bayesian theory calls for the use of the posterior predictive distribution to do predictive inference , i.e., to predict

8484-602: The probability P {\displaystyle P} of an event A {\displaystyle A} is the following formula : P ( A ) = | A | | Ω |   ( alternatively:   Pr ( A ) = | A | | Ω | ) {\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)} This rule can readily be applied to each of

8585-907: The process is observed to generate E ∈ { E n } {\displaystyle E\in \{E_{n}\}} . For each M ∈ { M m } {\displaystyle M\in \{M_{m}\}} , the prior P ( M ) {\displaystyle P(M)} is updated to the posterior P ( M ∣ E ) {\displaystyle P(M\mid E)} . From Bayes' theorem : P ( M ∣ E ) = P ( E ∣ M ) ∑ m P ( E ∣ M m ) P ( M m ) ⋅ P ( M ) . {\displaystyle P(M\mid E)={\frac {P(E\mid M)}{\sum _{m}{P(E\mid M_{m})P(M_{m})}}}\cdot P(M).} Upon observation of further evidence, this procedure may be repeated. For

8686-498: The random variable has an infinite but countable probability space (i.e., corresponding to a die with infinite many faces) the 1965 paper demonstrates that for a dense subset of priors the Bernstein-von Mises theorem is not applicable. In this case there is almost surely no asymptotic convergence. Later in the 1980s and 1990s Freedman and Persi Diaconis continued to work on the case of infinite countable probability spaces. To summarise, there may be insufficient trials to suppress

8787-417: The random variable in consideration has a finite probability space . The more general results were obtained later by the statistician David A. Freedman who published in two seminal research papers in 1963 and 1965 when and under what circumstances the asymptotic behaviour of posterior is guaranteed. His 1963 paper treats, like Doob (1949), the finite case and comes to a satisfactory conclusion. However, if

8888-477: The real numbers run into difficulties when one considers 'badly behaved' sets, such as those that are nonmeasurable . Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities , to work, it is necessary to use a σ-algebra , that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra

8989-428: The relation of an updated posterior probability from a prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work. The Bayesian interpretation of probability was developed mainly by Laplace. About 200 years later, Sir Harold Jeffreys put Bayes's algorithm and Laplace's formulation on an axiomatic basis, writing in a 1973 book that Bayes' theorem "is to

9090-403: The same outcomes by A and B in opposite orders, to obtain the inverse probabilities. Bayes' theorem links the different partitionings. An entomologist spots what might, due to the pattern on its back, be a rare subspecies of beetle . A full 98% of the members of the rare subspecies have the pattern, so P (Pattern | Rare) = 98%. Only 5% of members of the common subspecies have

9191-440: The sample space from being events (see § Events in probability spaces , below). If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any singleton set (an elementary event ), the empty set (an impossible event, with probability zero) and

9292-413: The sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include: Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams . In the situation where each outcome in the sample space Ω is equally likely,

9393-590: The set of people who take the drug test. This combined with the definition of conditional probability results in the above statement. In other words, even if someone tests positive, the probability that they are a cannabis user is only 19%—this is because in this group, only 5% of people are users, and most positives are false positives coming from the remaining 95%. If 1,000 people were tested: The 1,000 people thus yields 235 positive tests, of which only 45 are genuine drug users, about 19%. The importance of specificity can be seen by showing that even if sensitivity

9494-415: The space of models, the belief in all models may be updated in a single step. The distribution of belief over the model space may then be thought of as a distribution of belief over the parameter space. The distributions in this section are expressed as continuous, represented by probability densities, as this is the usual situation. The technique is, however, equally applicable to discrete distributions. Let

9595-493: The term ( 1 P ( H ) − 1 ) P ( E ∣ ¬ H ) P ( E ∣ H ) . {\displaystyle \left({\tfrac {1}{P(H)}}-1\right){\tfrac {P(E\mid \neg H)}{P(E\mid H)}}.} If that term is approximately 1, then the probability of the hypothesis given the evidence, P ( H ∣ E ) {\displaystyle P(H\mid E)} ,

9696-495: The theory of probability what the Pythagorean theorem is to geometry". Stephen Stigler used a Bayesian argument to conclude that Bayes' theorem was discovered by Nicholas Saunderson , a blind English mathematician, some time before Bayes; that interpretation, however, has been disputed. Martyn Hooper and Sharon McGrayne have argued that Richard Price 's contribution was substantial: By modern standards, we should refer to

9797-1359: The two must add up to 1, so both are equal to 0.5. The event E {\displaystyle E} is the observation of a plain cookie. From the contents of the bowls, we know that P ( E ∣ H 1 ) = 30 / 40 = 0.75 {\displaystyle P(E\mid H_{1})=30/40=0.75} and P ( E ∣ H 2 ) = 20 / 40 = 0.5. {\displaystyle P(E\mid H_{2})=20/40=0.5.} Bayes' formula then yields P ( H 1 ∣ E ) = P ( E ∣ H 1 ) P ( H 1 ) P ( E ∣ H 1 ) P ( H 1 ) + P ( E ∣ H 2 ) P ( H 2 )   = 0.75 × 0.5 0.75 × 0.5 + 0.5 × 0.5   = 0.6 {\displaystyle {\begin{aligned}P(H_{1}\mid E)&={\frac {P(E\mid H_{1})\,P(H_{1})}{P(E\mid H_{1})\,P(H_{1})\;+\;P(E\mid H_{2})\,P(H_{2})}}\\\\\ &={\frac {0.75\times 0.5}{0.75\times 0.5+0.5\times 0.5}}\\\\\ &=0.6\end{aligned}}} Before we observed

9898-656: The uniform distribution on the real line. Modern Markov chain Monte Carlo methods have boosted the importance of Bayes' theorem including cases with improper priors. Bayes' rule and computing conditional probabilities provide a solution method for a number of popular puzzles, such as the Three Prisoners problem , the Monty Hall problem , the Two Child problem and the Two Envelopes problem . Suppose,

9999-416: The values of the hyperparameters that appear in the prior distribution. P ( E ∣ M ) P ( E ) > 1 ⇒ P ( E ∣ M ) > P ( E ) {\textstyle {\frac {P(E\mid M)}{P(E)}}>1\Rightarrow P(E\mid M)>P(E)} . That is, if the model were true, the evidence would be more likely than

10100-474: The vector θ {\displaystyle {\boldsymbol {\theta }}} span the parameter space. Let the initial prior distribution over θ {\displaystyle {\boldsymbol {\theta }}} be p ( θ ∣ α ) {\displaystyle p({\boldsymbol {\theta }}\mid {\boldsymbol {\alpha }})} , where α {\displaystyle {\boldsymbol {\alpha }}}

10201-531: Was inhabited during the 13th century, 63% chance during the 14th century and 36% during the 15th century. The Bernstein-von Mises theorem asserts here the asymptotic convergence to the "true" distribution because the probability space corresponding to the discrete set of events { G D , G D ¯ , G ¯ D , G ¯ D ¯ } {\displaystyle \{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}}

#823176