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Điều kiện mặc định trong các công thức là hàm ''f'' khả vi và các tích phân công thức tồn tại.
Ứng dụng của định lý Bayes thường dựa trên một giả thiết có tính triết học [[Bayesian probability]] ngầm định rằng độ bất định và kỳ vọng có thể tính toán được giống như là xác xuất. Định lí Bayes được đặt theo tên của Reverend [[Thomas Bayes]] ([[1702]]—[[1761]]), người nghiên cứu cách tính một phân bố với tham số là một [[phân bố nhị phân]]. Người bạn của ông, [[Richard Price]], chỉnh sửa và giới thiệu công trình năm [[1763]], sau khi Bayes mất, với tựa đề ''An Essay towards solving a Problem in the Doctrine of Chances''. [[Pierre-Simon Laplace]] mở rộng kết quả trong bài luận năm [[1774]]. ▼
===Mở rộng của định lý Bayes===
Theorems analogous to Bayes' theorem hold in problems with more than two variables.
These theorems are not given distinct names,
as they may be [[mass production|mass-produced]] by applying the laws of probability.
The general strategy is to work with a decomposition of the joint probability, and to marginalize (integrate) over the variables that are not of interest.
Depending on the form of the decomposition,
it may be possible to prove that some integrals must be 1,
and thus they fall out of the decomposition;
exploiting this property can reduce the computations very substantially.
A [[Bayesian network]] is essentially a mechanism for automatically generating the extensions of Bayes' theorem that are appropriate for a given decomposition of the joint probability.
==Ví dụ==
{{đang dịch|ngôn ngữ=tiếng Anh}}
Ứng dụng của định lý Bayes thường dựa trên một giả thiết có tính triết học [[Bayesian probability]] ngầm định rằng độ bất định và kỳ vọng có thể tính toán được giống như là xác xuất. Ví dụ cụ thể xem ở dưới đây. Để xem thêm các ví dụ cụ thể, bao gồm cả những ví dụ đơn giản hơn, bạn có thể xem bài viết về ví dụ của [[Bayesian inference#Simple examples of Bayesian inference|Bayesian inference]].
We describe the marginal probability distribution of a variable ''A'' as the ''[[prior probability distribution]]'' or simply the ''prior''. The conditional distribution of ''A'' given the "data" ''B'' is the ''[[posterior probability distribution]]'' or just the ''posterior''.
Suppose we wish to know about the proportion '''r''' of voters in a large population who will vote "yes" in a referendum. Let '''n''' be the number of voters in a random sample (chosen with replacement, so that we have [[statistical independence]]) and let '''m''' be the number of voters in that random sample who will vote "yes". Suppose that we observe ''n'' = 10 voters and ''m'' = 7 say they will vote yes. From Bayes' theorem we can calculate the probability distribution function for ''r'' using
:<math> f(r | n=10, m=7) =
\frac {f(m=7 | r, n=10) \, f(r)} {\int_0^1 f(m=7|r, n=10) \, f(r) \, dr}. </math>
From this we see that from the prior probability density function ''f''(''r'') and the likelihood function ''L''(''r'') = ''f''(''m'' = 7|''r'', ''n'' = 10), we can compute the posterior probability density function ''f''(''r''|''n'' = 10, ''m'' = 7).
The prior probability density function ''f''(''r'') summarizes what we know about the distribution of ''r'' in the absence of any observation. We provisionally assume in this case that the prior distribution of ''r'' is uniform over the interval [0, 1]. That is, ''f''(''r'') = 1. If some additional background information is found, we should modify the prior accordingly. However before we have any observations, all outcomes are equally likely.
Under the assumption of random sampling, choosing voters is just like choosing balls from an urn. The likelihood function ''L''(''r'') = ''P''(''m'' = 7|''r'', ''n'' = 10,) for such a problem is just the probability of 7 successes in 10 trials for a [[binomial distribution]].
:<math> P( m=7 | r, n=10) = {10 \choose 7} \, r^7 \, (1-r)^3. </math>
As with the prior, the likelihood is open to revision -- more complex assumptions will yield more complex likelihood functions. Maintaining the current assumptions, we compute the normalizing factor,
:<math> \int_0^1 P( m=7|r, n=10) \, f(r) \, dr = \int_0^1 {10 \choose 7} \, r^7 \, (1-r)^3 \, 1 \, dr = {10 \choose 7} \, \frac{1}{1320} </math>
and the posterior distribution for ''r'' is then
:<math> f(r | n=10, m=7) =
\frac{{10 \choose 7} \, r^7 \, (1-r)^3 \, 1} {{10 \choose 7} \, \frac{1}{1320}} = 1320 \, r^7 \, (1-r)^3 </math>
for ''r'' between 0 and 1, inclusive.
One may be interested in the probability that more than half the voters will vote "yes". The ''prior probability'' that more than half the voters will vote "yes" is 1/2, by the symmetry of the [[uniform distribution]]. In comparison, the posterior probability that more than half the voters will vote "yes", i.e., the conditional probability given the outcome of the opinion poll -- that seven of the 10 voters questioned will vote "yes" -- is
:<math>1320\int_{1/2}^1 r^7(1-r)^3\,dr \approx 0.887</math>
which is about an "89% chance".
==Ghi chú lịch sử==
▲Định lí Bayes được đặt theo tên của Reverend [[Thomas Bayes]] ([[1702]]—[[1761]]), người nghiên cứu cách tính một phân bố với tham số là một [[phân bố nhị phân]]. Người bạn của ông, [[Richard Price]], chỉnh sửa và giới thiệu công trình năm [[1763]], sau khi Bayes mất, với tựa đề ''An Essay towards solving a Problem in the Doctrine of Chances''. [[Pierre-Simon Laplace]] mở rộng kết quả trong bài luận năm [[1774]].
One of Bayes' results (Proposition 5) gives a simple description of [[conditional probability]], and shows that it does not depend on the order in which things occur:
:''If there be two subsequent events, the probability of the second b/N and the probability of both together P/N, and it being first discovered that the second event has also happened, the probability I am right ''[i.e., the conditional probability of the first event being true given that the second has happened]'' is P/b.''
Bayes' main result (Proposition 9 in the essay) is the following: assuming a [[uniform distribution]] for the [[prior distribution]] of the [[binomial]] parameter ''p'', the probability that ''p'' is between two values ''a'' and ''b'' is
:<math>
\frac {\int_a^b {n+m \choose m} p^m (1-p)^n\,dp}
{\int_0^1 {n+m \choose m} p^m (1-p)^n\,dp}
</math>
where ''m'' is the number of observed successes and ''n'' the number of observed failures. His preliminary results, in particular Propositions 3, 4, and 5, imply the result now called Bayes' Theorem (as described below), but it does not appear that Bayes himself emphasized or focused on that result.
What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the parameter ''p''. So, one can compute probability for an experimental outcome, but also for the parameter which governs it, and the same algebra is used to make inferences of either kind.
Bayes 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 escapes a philosophical quagmire of which he most likely was not even aware.
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