Nonparametric Priors on Complete Separable Metric Spaces
... belongs to the same class of distributions as the prior [45]. The limit in Theorem 2.5 preserves conjugacy—part (i) shows that conjugacy of all (pI , qI ) implies conjugacy of (p, q), as in Example 2.6. There is, however, a stronger form of conjugacy which is of greater practical importance, since i ...
... belongs to the same class of distributions as the prior [45]. The limit in Theorem 2.5 preserves conjugacy—part (i) shows that conjugacy of all (pI , qI ) implies conjugacy of (p, q), as in Example 2.6. There is, however, a stronger form of conjugacy which is of greater practical importance, since i ...
A strong nonrandom pattern in Matlab default random number
... Hence, we have to accept that there is always a chance that a new dependence is found in a used generator. However, it is necessary to distinguish different levels of severity of such dependencies. Two important quantitative characteristics of a dependence are the following: 1. The length of the seq ...
... Hence, we have to accept that there is always a chance that a new dependence is found in a used generator. However, it is necessary to distinguish different levels of severity of such dependencies. Two important quantitative characteristics of a dependence are the following: 1. The length of the seq ...
Full text
... (and of course Fn+1 = Fn + Fn−1 ), for if our series began with two 1’s or with a 0 the decompositions of many numbers into non-adjacent summands would not be unique. In 1937 the physicist Frank Benford [2], then working for General Electric, observed that the distributions of the leading digits of ...
... (and of course Fn+1 = Fn + Fn−1 ), for if our series began with two 1’s or with a 0 the decompositions of many numbers into non-adjacent summands would not be unique. In 1937 the physicist Frank Benford [2], then working for General Electric, observed that the distributions of the leading digits of ...
Reality and Probability: Introducing a New Type
... the two experiments in one way or another. This is wrong. It is in fact one of the deep mistakes of classical logic, where one introduces indeed the conjunction of propositions by means of truth tables of both proposition, as if the truth of the conjunction would be defined by verifying the truth of ...
... the two experiments in one way or another. This is wrong. It is in fact one of the deep mistakes of classical logic, where one introduces indeed the conjunction of propositions by means of truth tables of both proposition, as if the truth of the conjunction would be defined by verifying the truth of ...
Random Permutation Statistics and An Improved Slide
... for fk8 . If j8 > 1, our attack can be improved. If we omit Stage 1B, or if j8 = 1 (which is quite infrequent), then the Stage 3 will dominate the attack, which as we will see later can make it up to 211 times slower, depending on the values of tr and j8 . To bridge this gap we wish to exclude a pro ...
... for fk8 . If j8 > 1, our attack can be improved. If we omit Stage 1B, or if j8 = 1 (which is quite infrequent), then the Stage 3 will dominate the attack, which as we will see later can make it up to 211 times slower, depending on the values of tr and j8 . To bridge this gap we wish to exclude a pro ...
Combinatorial Probability
... You should consider this the next time you think about spending $1 for a chance to win $10 million. On the other hand when the probabilities of success are small it is not sensible to think in terms of how much you’ll win on the average. World Series continued. Using (2.3) we can easily compute the ...
... You should consider this the next time you think about spending $1 for a chance to win $10 million. On the other hand when the probabilities of success are small it is not sensible to think in terms of how much you’ll win on the average. World Series continued. Using (2.3) we can easily compute the ...
Probability and Information Theory
... then the discretization makes the robot immediately become uncertain about the precise position of objects: each object could be anywhere within the discrete cell that it was observed to occupy. In many cases, it is more practical to use a simple but uncertain rule rather than a complex but certain ...
... then the discretization makes the robot immediately become uncertain about the precise position of objects: each object could be anywhere within the discrete cell that it was observed to occupy. In many cases, it is more practical to use a simple but uncertain rule rather than a complex but certain ...
Eliciting Subjective Probabilities Through
... equally likely disjoint subevents—the union of which has probability 1/2—should have probability 1/4. It is worth noting that the exchangeability method requires neither reference to the concept of probability nor a direct judgment, but only simple choices between binary prospects. Moreover, this me ...
... equally likely disjoint subevents—the union of which has probability 1/2—should have probability 1/4. It is worth noting that the exchangeability method requires neither reference to the concept of probability nor a direct judgment, but only simple choices between binary prospects. Moreover, this me ...
Probability Models
... red is 50%, so that this is a good bet. However, after mastering conditional probability (Section 1.6), you will know that conditional on one side being red, the probability that the other side is also red is equal to 2/3. So, by the theory of expected values (Chapter 3), you will know that you shou ...
... red is 50%, so that this is a good bet. However, after mastering conditional probability (Section 1.6), you will know that conditional on one side being red, the probability that the other side is also red is equal to 2/3. So, by the theory of expected values (Chapter 3), you will know that you shou ...
pdf
... denote the probability that we extract a type 1 witness, and similarly for ε 2 (k). The above shows that either ε1 (k) or ε2 (k) is not negligible (or possibly both are). We show that either of these lead to an efficient procedure for inverting f with non-negligible probability, and hence a contradi ...
... denote the probability that we extract a type 1 witness, and similarly for ε 2 (k). The above shows that either ε1 (k) or ε2 (k) is not negligible (or possibly both are). We show that either of these lead to an efficient procedure for inverting f with non-negligible probability, and hence a contradi ...
20 More Permutations
... the word, we have 6 letters to arrange and there is just one place for the “E”s, so the answer is ...
... the word, we have 6 letters to arrange and there is just one place for the “E”s, so the answer is ...
Finite and Infinite Sets. Countability. Proof Techniques
... We count first all elements with index 1, then we count all elements with index 2, etc. Important: it is possible to count all the first elements (and then the second, third, etc) because we have only 3 sets: A, B, C, i.e. finite number of sets, so we can put the sets themselves (not the elements of ...
... We count first all elements with index 1, then we count all elements with index 2, etc. Important: it is possible to count all the first elements (and then the second, third, etc) because we have only 3 sets: A, B, C, i.e. finite number of sets, so we can put the sets themselves (not the elements of ...
Public-key Cryptosystems Provably Secure against Chosen
... hnplementations of the notion were suggested by Rivest, Shamir and Adleman [28] and Merkle and Hellman [22]. The exact nature of security of these implementations was not given in a precise form, since an exact definition of security was not known at the time. Rabin [26], nevertheless, has given a s ...
... hnplementations of the notion were suggested by Rivest, Shamir and Adleman [28] and Merkle and Hellman [22]. The exact nature of security of these implementations was not given in a precise form, since an exact definition of security was not known at the time. Rabin [26], nevertheless, has given a s ...
Infinite Sets of Integers Whose Distinct Elements Do Not Sum to a
... There are 2n different subsets I of {1, 2, . . . , n}, so the set {4n − 2n , . . . , 4n − 1, 4n } with 2n + 1 elements contains some two indices j < j ′ for which the corresponding subsets I (and so the values for S(I)) are equal. Subtracting cn + j + S(I) = s2j from cn + j ′ + S(I) = s2j ′ , we ded ...
... There are 2n different subsets I of {1, 2, . . . , n}, so the set {4n − 2n , . . . , 4n − 1, 4n } with 2n + 1 elements contains some two indices j < j ′ for which the corresponding subsets I (and so the values for S(I)) are equal. Subtracting cn + j + S(I) = s2j from cn + j ′ + S(I) = s2j ′ , we ded ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.