By John Maynard Keynes
With this insightful exploration of the probabilistic connection among philosophy and the historical past of technological know-how, the recognized economist breathed new existence into experiences of either disciplines. initially released in 1921, this significant mathematical paintings represented an important contribution to the idea concerning the logical likelihood of propositions, and introduced the “logical-relationist” thought.
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Additional info for A Treatise on Probability (Dover Books on Mathematics)
The plausibility represents the degree to which the hypothesis is not in contradiction with the given knowledge. ✛ ✌❲❂❃❂ We can compute the probability of a logical formula such as ❹✬✽ ☎ ✸✾✽⑤➤ by putting the corresponding logical sentence in disjoint form. Suppose we would like to compute ☞ ☞ ❂ ❂ ❂ the probability of ✯ ❹ ✽ ➴ ➒ ➘ from ✯ ❹ ✽ and ✬ ❹ ♣ ✽ ➘ . The equivalent disjoint form is: ☞ ☞ ❂ ☞ ☞ ❂➮❅ ☞ ☞ ❂➮❅ ☞ ✏ ✏ ❂✑✏ ☞ ❂ ). We ➒➷✽➬➘ ➂ ☞ (because ➞ ➒ ♣ ✽ ➘ ➂ ✽ ➸ ➒ ➘ ✽ ✓ ➒ ➂ ✽ ✓ ➒ ➘ ☞ ☞ ❂❞❂♥❅ ☞ ❂ ☞ ❂ ☞ ❂➓❅ ✏ ✏ have then: , because and ❹✯✽ ➒✓➘ ❹✯✽ ➒➱✽♣➘ ➂ ❹✯✽ ⑦●❹✬✽➬➘ ➂ ✏ ☞ Finally, with☞ the independence assumption ➘ ☞ ❂❞❂ ☞ ❂ ☞ ❂❞❂ before, ✏ ➂ ☞ are ❂❪❅ disjoint❂ ❱ events.
For any hypothesis ➤ , the ✝→ú ✝➛ü set can always be divided into the disjoint sets of those conjunctions which ✝➔üý support ➤ , and of those conjunctions which do not support ➤ . ✝➛üý is of no use to hypothesis ➤ , and the information that must Obviously, ✌ the set ✝→ü ✝→þ➣ÿ❧✝➛ü be added to can only be taken in meaning that any supports ❅ ✂ ✝➛þ ↔ ✝ subset ✝ þý ✝→✁ þ , ✗ ✝ þý ✄ þý ❅ ✝→of ü the hypothesis. Take such that: , .
8). 6). 3). from ✤ ✣ ✌❳Ó➸➠ is ❂ ✣ In retrieval based on inference, a document is retrieved if the query ✣ implied ✽ by . One possible way to implement this idea is to add the clause to the ✤ ✌ knowledge base (D is ”observed”)2 and then to evaluate the hypothesis . Evaluating Õ ➭ 2 For Ô readers familiar with the IR logical approach, the reason for having will be explained in the next section. 3. An example related to information retrieval 27 an hypothesis means finding all supporting and all refuting arguments for this hypothesis, and then computing the numerical reliability of these arguments.
A Treatise on Probability (Dover Books on Mathematics) by John Maynard Keynes