讲解 → 贝叶斯定理依据Candidate from reviewed en material v2: Two ways to measure the same overlap lead directly to Bayes’ theorem. This complete visual proof starts with conditional proportions and rearranges the joint-probability identity, then tests the common shortcut of multiplying marginal probabilities. Coin, die and sibling examples explain why independence is an assumption to check. The final testing illustration combines a prior, a likelihood and the probability of a positive result. The accompanying notes state positive-probability conditions and distinguish an illustrative model from real medical data.
内容位置 → 贝叶斯定理依据Candidate from reviewed zh material v2: 贝叶斯定理可由联合概率恒等式直接变形得到：A、B 同时发生的交集概率，可以按两种条件顺序计算。
内容位置 → 条件概率依据Candidate from reviewed zh material v2: P(B|A) 被描述为 A 区域中也属于 B 的分数，P(A|B) 被描述为 B 区域中也属于 A 的分数。正方形图通过将一个比例嵌套在另一个比例中来使这一点精确化。
内容位置 → 条件概率依据Candidate from reviewed en material v2: P(B|A) is described as the fraction of the A-region that also belongs to B, and P(A|B) as the fraction of the B-region that also belongs to A. The square diagrams make this precise by nesting one proportion inside another.
内容位置 → 独立性依据Candidate from reviewed en material v2: Events are independent when their joint probability equals the product of their marginal probabilities. When A has positive probability, this is equivalent to P(B|A)=P(B). Independence is not confined to coins or dice, and fairness alone does not establish independence.
内容位置 → 独立性依据Candidate from reviewed zh material v2: 当事件的联合概率等于其边缘概率的乘积时，这些事件是独立的。当 A 具有正概率时，这等价于 P(B|A)=P(B)。独立性不仅限于硬币或骰子，仅凭公平性并不能确立独立性。
讲解 → 贝叶斯定理的简洁证明｜3Blue1Brown依据Candidate from reviewed en material v2: Two ways to measure the same overlap lead directly to Bayes’ theorem. This complete visual proof starts with conditional proportions and rearranges the joint-probability identity, then tests the common shortcut of multiplying marginal probabilities. Coin, die and sibling examples explain why independence is an assumption to check. The final testing illustration combines a prior, a likelihood and the probability of a positive result. The accompanying notes state positive-probability conditions and distinguish an illustrative model from real medical data.
讲解 → 贝叶斯定理依据Candidate from reviewed zh material v2: 贝叶斯定理可由联合概率恒等式直接变形得到：A、B 同时发生的交集概率，可以按两种条件顺序计算。
讲解 → 条件概率依据Candidate from reviewed zh material v2: P(B|A) 被描述为 A 区域中也属于 B 的分数，P(A|B) 被描述为 B 区域中也属于 A 的分数。正方形图通过将一个比例嵌套在另一个比例中来使这一点精确化。
讲解 → 条件概率依据Candidate from reviewed en material v2: P(B|A) is described as the fraction of the A-region that also belongs to B, and P(A|B) as the fraction of the B-region that also belongs to A. The square diagrams make this precise by nesting one proportion inside another.
内容位置 → 贝叶斯定理的简洁证明｜3Blue1Brown依据Candidate from reviewed zh material v2: P(B|A) 被描述为 A 区域中也属于 B 的分数，P(A|B) 被描述为 B 区域中也属于 A 的分数。正方形图通过将一个比例嵌套在另一个比例中来使这一点精确化。
讲解 → 条件概率依据Candidate from reviewed zh material v2: P(B|A) 被描述为 A 区域中也属于 B 的分数，P(A|B) 被描述为 B 区域中也属于 A 的分数。正方形图通过将一个比例嵌套在另一个比例中来使这一点精确化。
内容位置 → 贝叶斯定理的简洁证明｜3Blue1Brown依据Candidate from reviewed en material v2: P(B|A) is described as the fraction of the A-region that also belongs to B, and P(A|B) as the fraction of the B-region that also belongs to A. The square diagrams make this precise by nesting one proportion inside another.
讲解 → 条件概率依据Candidate from reviewed en material v2: P(B|A) is described as the fraction of the A-region that also belongs to B, and P(A|B) as the fraction of the B-region that also belongs to A. The square diagrams make this precise by nesting one proportion inside another.
讲解 → 独立性依据Candidate from reviewed en material v2: Events are independent when their joint probability equals the product of their marginal probabilities. When A has positive probability, this is equivalent to P(B|A)=P(B). Independence is not confined to coins or dice, and fairness alone does not establish independence.
讲解 → 独立性依据Candidate from reviewed zh material v2: 当事件的联合概率等于其边缘概率的乘积时，这些事件是独立的。当 A 具有正概率时，这等价于 P(B|A)=P(B)。独立性不仅限于硬币或骰子，仅凭公平性并不能确立独立性。
内容位置 → 贝叶斯定理的简洁证明｜3Blue1Brown依据Candidate from reviewed en material v2: Events are independent when their joint probability equals the product of their marginal probabilities. When A has positive probability, this is equivalent to P(B|A)=P(B). Independence is not confined to coins or dice, and fairness alone does not establish independence.
讲解 → 独立性依据Candidate from reviewed en material v2: Events are independent when their joint probability equals the product of their marginal probabilities. When A has positive probability, this is equivalent to P(B|A)=P(B). Independence is not confined to coins or dice, and fairness alone does not establish independence.
内容位置 → 贝叶斯定理的简洁证明｜3Blue1Brown依据Candidate from reviewed zh material v2: 当事件的联合概率等于其边缘概率的乘积时，这些事件是独立的。当 A 具有正概率时，这等价于 P(B|A)=P(B)。独立性不仅限于硬币或骰子，仅凭公平性并不能确立独立性。
讲解 → 独立性依据Candidate from reviewed zh material v2: 当事件的联合概率等于其边缘概率的乘积时，这些事件是独立的。当 A 具有正概率时，这等价于 P(B|A)=P(B)。独立性不仅限于硬币或骰子，仅凭公平性并不能确立独立性。
math