Inferensi Bayes

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Bayesian inference nyaeta inferensi statistik numana sagala kamungkinan di-interpretasi lain salaku frekuensi atawa proporsi atawa sabangsana, tapi leuwih condong kana tingkat kapercayaan. Ngaran ieu asalna kusabab remenna ngagunakeun Bayes' theorem keur matotoskeun hiji perkara. Teorema Bayes, ngaran teori nu dipake sanggeus Thomas Bayes anu mimiti ngawanohkeun ieu metoda.

[édit] Kajadian jeung metoda ilmiah

Ahli statistik Bayes ngaku yen metoda kaputusan Bayes ngarupakeun bentuk formal tina metoda ilmiah kaasup dina ngumpulkeun evidence numana kahareupna atawa jalan keur nangtukeun hiji hypothesis. Dina hal ieu bisa jadi teu pasti salawasna, sanajan kitu lobana kumpulan kajadian bakal ngajadikeun naekna tingkat kapercayaan hipotesa pangluhurna (salawasna 1) atawa panghandapna (salawasna 0). Teorema Bayes nyadiakeun metoda keur naksir tingkat kapercayaan dina waktu informasi anyar ngan saeutik. Bayes' theorem nyaeta

$P(A|e) = P(A)\frac{P(e | A) }{P(e)}$

Keur kaperluan urang, (A) dijadikeun hipotesa induced tina sababaraha susunan observasi. (e) dijadikeun hipotesa konfirmasi tina observasi.

Watesan P(A|e) disebut posterior probability ti A, given e.
Watesan P(A) disebut prior probability ti A.
Watesan P(e) disebut prior probability ti e.

Faktor Likelihood: Pecahan

$\frac{P(e | A) }{P(e)}$

ngarupakeun faktor skala, probabilitas observasi hasil tina hipotesa dibagi ku probabilitas hipotesa observasi nu ngarupakeun kajadian independen dina hipotesa. Hasil ukuran ieu ngakibatkeun yen hipotesa ayana dina probabilitas nu dijieun tina observasi. Kulantaran kitu hasil observasi bakal jadi teu sahade lamun hipotesa bener, sarta faktor skala bakal jadi gede.

Perkalian faktor skala ieu ku probabilitas observasi nu bener bakal ngahasilkeun probabilitas hipotesa nu bener oge, saperti nu diberekeun ku observasi.

Pagawean konci dina nyieun kaputusan tangtuna ngararancang prior probabiliti dina observasi jeung hipotesa. Lamun prior probabiliti nembongkeun nilai objektif, maka bisa digunakeun keur nangtukeun ukuran objektif hipotesa probabiliti. Tapi, taya jalan nu jelas keur nangtukeun objektif probabiliti. Hal anu teu mungkin keur migawe pendekatan dina nangtukeun hiji probabilitas bis nangtukeun sakabeh hipotesa nu mungkin.

Alternatifna, jeung sering dipake, probabiliti dicokot salaku tingkat kapercayaan subjektif ti bagian partisipan. Teori saterusna nangtukeun ukuran rasio kepercayaan tina observasi nu dijadikeun subjek kapercayaan dina hipotesa. Tapi hasil dina kasus ieu masih keneh nyesakeun subjektif dina posterior probabiliti. Sabab kitu teorema bisa digunakeun keur ngarasionalkeun kapercayaan dina sababaraha hipotesa, tapi nolak objektifitas. Sababaraha skema teu bisa dipake, contona, sifat objektif dina nangtukeun konflik paradigma sain.

Dina loba kasus, akibat kajadian bisa disimpulkeun dina rasio likelihood, nu digambarkeun dina the law of likelihood. Hal ieu bisa dikombinasikeun jeung prior probability keur ngagambarkeun tingkat kapercayaan asli sarta kajadian pangtukangna nu dicokot dina perhitungan. Samemeh kaputusan dijieun, loss function oge diperlukeun keur nimbangkeun gambaran akibat tina kasalahan nangtukeun kaputusan.

[édit] Conto sederhana Kaputusan Bayes

[édit] Kueh tina mangkok nu mana?

Keur conto, aya dua mangkok pinuh ku kueh. Dina mangkok ka #1 aya sapuluh coklat 10 hias jeung 30 coklat polos, sedengkeun dina mangkok kadua #2 aya 20 coklat hias jeung 20 coklat polos. Kandar milih dua mangkok eta sacara acak sarta nyokot kue coklat sacara acak oge. Asumsina taya alesan keur percaya yen Kandar milih-milih eta mangkok atawa kue coklat tea. Kue coklat nu kacokot teh coklat polos. Sabaraha kamungkinna yen Kandar nyokot tina mangkok ka #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 is given by Bayes' theorem. Let H₁ corresponds to bowl #1, and H₂ to bowl #2. It is given that the bowls are identical from Fred's point of view, thus P(H₁) = P(H₂), and the two must add up to 1, so both are equal to 0.5. The "data" D consists in the observation of a plain cookie. From the contents of the bowls, we know that P(D | H₁) = 30/40 = 0.75 and P(D | H₂) = 20/40 = 0.5. Bayes' formula then yields

$\begin{matrix} P(H_1 | D) &=& \frac{P(H_1) \cdot P(D | H_1)}{P(H_1) \cdot P(D | H_1) + P(H_2) \cdot P(D | H_2)} \\ \\ \ & =& \frac{0.5 \times 0.75}{0.5 \times 0.75 + 0.5 \times 0.5} \\ \\ \ & =& 0.6 \end{matrix}$

Before observing the cookie, the probability that Fred chose bowl #1 is the prior probability, P(H₁), which is 0.5. After observing the cookie, we revise the probability to P(H₁|D), which is 0.6.

[édit] False positives in a medical test

False positives are a problem in any kind of test: no test is perfect, and sometimes the test will incorrectly report a positive result. For example, if a test for a particular disease is performed on a patient, then there is a chance (usually small) that the test will return a positive result even if the patient does not have the disease. The problem lies, however, not just in the chance of a false positive prior to testing, but determining the chance that a positive result is in fact a false positive. As we will demonstrate, using Bayes' theorem, if a condition is rare, then the majority of positive results may be false positives, even if the test for that condition is (otherwise) reasonably accurate.

Suppose that a test for a particular disease has a very high success rate:

if a tested patient has the disease, the test accurately reports this, a 'positive', 99% of the time (or, with probability 0.99), and
if a tested patient does not have the disease, the test accurately reports that, a 'negative', 95% of the time (i.e. with probability 0.95).

Suppose also, however, that only 0.1% of the population have that disease (i.e. with probability 0.001). We now have all the information required to use Bayes' theorem to calculate the probability that, given the test was positive, that it is a false positive.

Let A be the event that the patient has the disease, and B be the event that the test returns a positive result. Then, using the second alternative form of Bayes' theorem (above), the probability of a true positive is

$\begin{matrix}P(A|B) &= &\frac{0.99 \times 0.001}{0.99\times 0.001 + 0.05\times 0.999}\, ,\\ ~\\ &\approx &0.019\, .\end{matrix}$

and hence the probability of a false positive is about (1 − 0.019) = 0.981.

Despite the apparent high accuracy of the test, the incidence of the disease is so low (one in a thousand) that the vast majority of patients who test positive (98 in a hundred) do not have the disease. (Nonetheless, this is 20 times the proportion before we knew the outcome of the test! The test is not useless, and re-testing may improve the reliability of the result.) In particular, a test must be very reliable in reporting a negative result when the patient does not have the disease, if it is to avoid the problem of false positives. In mathematical terms, this would ensure that the second term in the denominator of the above calculation is small, relative to the first term. For example, if the test reported a negative result in patients without the disease with probability 0.999, then using this value in the calculation yields a probability of a false positive of roughly 0.5.

In this example, Bayes' theorem helps show that the accuracy of tests for rare conditions must be very high in order to produce reliable results from a single test, due to the possibility of false positives. (The probability of a 'false negative' could also be calculated using Bayes' theorem, to completely characterise the possible errors in the test results.)

[édit] In the courtroom

Bayesian inference can be used to coherently assess additional evidence of guilt in a court setting.

Let G be the event that the defendent is guilty.

Let E be the event that the defendent's DNA matches DNA found at the crime scene.

Let p(E | G) be the probability of seeing event E assuming that the defendent is guilty. (Usually this would be taken to be unity.)

Let p(G | E) be the probability that the defendent is guilty assuming the DNA match event E

Let p(G) be the probability that the defendent is guilty, based on the evidence other than the DNA match.

Bayesian inference tells us that if we can assign a probability p(G) to the defendent's guilt before we take the DNA evidence into account, then we can revise this probability to the conditional probability p(G | E), since

p(G | E) = p(G) p(E | G) / p(E)

Suppose, on the basis of other evidence, a juror decides that there is a 30% chance that the defendent is guilty. Suppose also that the forensic evidence is that the probability that a person chosen at random would have DNA that matched that at the crime scene was 1 in a million, or 10^-6.

The event E can occur in two ways. Either the defendent is guilty (with prior probability 0.3) and thus his DNA is present with probability 1, or he is innocent (with prior probability 0.7) and he is unlucky enough to be one of the 1 in a million matching people.

Thus the juror could coherently revise his opinion to take into account the DNA evidence as follows:

p(G | E) = 0.3 × 1.0 /(0.3 × 1.0 + 0.7 × 10^-6) = 0.99999766667.

In the United Kingdom, Bayes' theorem was explained by an expert witness to the jury in the case of Regina versus Denis Adams. The case went to Appeal and the Court of Appeal gave their opinion that the use of Bayes' theorem was inappropriate for jurors.

[édit] Search theory

In May 1968 the US nuclear submarine Scorpion (SSN 589) failed to arrive as expected at her home port of Norfolk, Virginia. The US Navy was convinced that the vessel had been lost off the Eastern seabord but an extensive search failed to discover the wreck. The US Navy's deep water expert, John Craven, believed that it was elsewhere and he organised a search south west of the Azores based on a controversial approximate triangulation by hydrophones. He was allocated only a single ship, the USNS Mizar, and he took advice from a firm of consultant mathematicians in order to maximise his resources. A Bayesian search methodology was adopted. Experienced submarine commanders were interviewed to construct hypotheses about what could have caused the loss of the Scorpion. The sea area was divided up into grid squares and a probability assigned to each square, under each of the hypotheses, to give a number of probability grids, one for each hypothesis. These were then added together to produce an overall probability grid. The probability attached to each square was then the probability that the wreck was in that square. A second grid was constructed with probabilities that represented the probability of successfully finding the wreck if that square were to be searched and the wreck were to be actually there. This was a known function of water depth. The result of combining this grid with the previous grid is a grid which gives the probability of finding the wreck in each grid square of the sea if it were to be searched. This sea grid was systematically searched in a manner which started with the high probability regions first and worked down to the low probability regions last. Each time a grid square was searched and found to be empty its probability was reassessed using Bayes' theorem. This then forced the probabilities of all the other grid squares to be reassessed (upwards), also by Bayes' theorem. The use of this approach was a major computational challenge for the time but it was eventually successful and the Scorpion was found in October of that year. Suppose a grid square has a probability p of containing the wreck and that the probability of successfully detecting the wreck if it is there is q. If the square is searched and no wreck is found then, by Bayes, the revised probability of the wreck being in the square is given by

$p' = \frac{p(1-q)}{(1-p)+p(1-q)}$

[édit] More mathematical examples

[édit] Naive Bayes classifier

See: naive Bayesian classification.

[édit] Posterior distribution of the binomial parameter

In this example we consider the computation of the posterior distribution for the binomial parameter. This is the same problem considered by Bayes in Proposition 9 of his essay.

We are given m observed successes and n observed failures in a binomial experiment. The experiment may be tossing a coin, drawing a ball from an urn, or asking someone their opinion, among many other possibilities. What we know about the parameter (let's call it a) is stated as the prior distribution, p(a).

For a given value of a, the probability of m successes in m+n trials is

$p(m,n|a) = \begin{pmatrix} n+m \\ m \end{pmatrix} a^m (1-a)^n$

Since m and n are fixed, and a is unknown, this is a likelihood function for a. From the continuous form of the law of total probability we have

$p(a|m,n) = \frac{p(m,n|a)\,p(a)}{\int_0^1 p(m,n|a)\,p(a)\,da} = \frac{\begin{pmatrix} n+m \\ m \end{pmatrix} a^m (1-a)^n\,p(a)} {\int_0^1 \begin{pmatrix} n+m \\ m \end{pmatrix} a^m (1-a)^n\,p(a)\,da}$

For some special choices of the prior distribution p(a), the integral can be solved and the posterior takes a convenient form. Dina sabagean, lamun p(a) ngarupakeun sebaran beta nu mibanda parameter m₀ sarta n₀, mangka posterior oge sebaran beta nu mibanda parameter m+m₀ jeung n+n₀.

A conjugate prior is a prior distribution, such as the beta distribution in the above example, which has the property that the posterior is the same type of distribution.

What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the parameter p. That is, not only can one compute probabilities for experimental outcomes, but also for the parameter which governs them, and the same algebra is used to make inferences of either kind. Interestingly, Bayes actually states his question in a way that might make the idea of assigning a probability distribution to a parameter palatable to a frequentist. He supposes that a billiard ball is thrown at random onto a billiard table, and that the probabilities p and q are the probabilities that subsequent billiard balls will fall above or below the first ball. By making the binomial parameter p depend on a random event, he cleverly escapes a philosophical quagmire that he most likely was not even aware was an issue.

[édit] Aplikasi komputer

Kaputusan Bayesian geus dipake dina widang artificial intelligence jeung expert system. Teknik kaputusan Bayesian geus dijadikeun dasar dina sabagean tenik komputer pattern recognition mimiti taun 1950 katompernakeun.

Mimiti tumuwuhna dina migunakeun kaputusan Bayesian keur filter spam. Contona: Bogofilter, SpamAssassin jeung Mozilla.

In some applications fuzzy logic is an alternative to Bayesian inference. Fuzzy logic and Bayesian inference, however, are mathematically and semantically not compatible: You cannot, in general, understand the degree of truth in fuzzy logic as probability and vice versa.

[édit] Tempo oge:

Thomas Bayes
Bayesian model comparison
Bayesian probability
Occam's Razor
Minimum description length
Gaussian process regression

[édit] Tumbu kaluar

On-line textbook: Information Theory, Inference, and Learning Algorithms, by David MacKay, has many chapters on Bayesian methods, including introductory examples; compelling arguments in favour of Bayesian methods; state-of-the-art Monte Carlo methods, message-passing methods, and variational methods; and examples illustrating the intimate connections between Bayesian inference and data compression.
Naive Bayesian learning paper
A Tutorial on Learning With Bayesian Networks