Title of slide
... Prior predictive distribution for statistical test The more important use of a Bayesian ingredient is in computing the distribution of the statistic. One can take this to be the Bayesian averaged model (prior predictive distribution), i.e., Generate x ~ Lm(x|s) to determine f(t(x)|s), Generate x ~ ...
... Prior predictive distribution for statistical test The more important use of a Bayesian ingredient is in computing the distribution of the statistic. One can take this to be the Bayesian averaged model (prior predictive distribution), i.e., Generate x ~ Lm(x|s) to determine f(t(x)|s), Generate x ~ ...
Exercise
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
here
... edges joined by an edge and exactly half of them do not. In fact, for any selection of m distinct pairs of vertices, the set of graphs on n vertices can be split into 2m equallysized classes based on which of the m pairs they connect by an edge and which they ...
... edges joined by an edge and exactly half of them do not. In fact, for any selection of m distinct pairs of vertices, the set of graphs on n vertices can be split into 2m equallysized classes based on which of the m pairs they connect by an edge and which they ...
Peano and Heyting Arithmetic
... • HA ` ∀x, y, z, z 0 (φ_ (x, y, z) ∧ φ_ (x, y, z 0 ) → z = z 0 ), and • HA ` ∀x, y(φπ (x) → ∃zφ_ (x, y, z)). The first two clauses state the HA proves that φ_ correctly identifies π(σ _ hni) for actual sequences σ and natural numbers n. But this isn’t enough to give the last two clauses, because HA ...
... • HA ` ∀x, y, z, z 0 (φ_ (x, y, z) ∧ φ_ (x, y, z 0 ) → z = z 0 ), and • HA ` ∀x, y(φπ (x) → ∃zφ_ (x, y, z)). The first two clauses state the HA proves that φ_ correctly identifies π(σ _ hni) for actual sequences σ and natural numbers n. But this isn’t enough to give the last two clauses, because HA ...
The Compactness Theorem 1 The Compactness Theorem
... For the induction step, suppose that we have constructed assignments A0 , . . . , An such that An satisfies (∗). Consider the two assignments B, B 0 that extend An with dom(B) = dom(B 0 ) = {p1 , p2 , . . . , pn+1 } (say B[[pn+1 ]] = 0 and B 0 [[pn+1 ]] = 1.) Since any proper extension of An is an ...
... For the induction step, suppose that we have constructed assignments A0 , . . . , An such that An satisfies (∗). Consider the two assignments B, B 0 that extend An with dom(B) = dom(B 0 ) = {p1 , p2 , . . . , pn+1 } (say B[[pn+1 ]] = 0 and B 0 [[pn+1 ]] = 1.) Since any proper extension of An is an ...
(A B) |– A
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
com.1 The Compactness Theorem
... c < k in ∆0 have k < K. If we expand Q to Q0 with cQ = 1/K we have that Q0 |= Γ ∪ ∆0 , and so Γ ∪ ∆ is finitely satisfiable (Exercise: prove this in detail). By compactness, Γ ∪ ∆ is satisfiable. Any model S of Γ ∪ ∆ contains an infinitesimal, namely cS . Problem com.2. In the standard model of arit ...
... c < k in ∆0 have k < K. If we expand Q to Q0 with cQ = 1/K we have that Q0 |= Γ ∪ ∆0 , and so Γ ∪ ∆ is finitely satisfiable (Exercise: prove this in detail). By compactness, Γ ∪ ∆ is satisfiable. Any model S of Γ ∪ ∆ contains an infinitesimal, namely cS . Problem com.2. In the standard model of arit ...
Predicate logic. Formal and informal proofs
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • ...
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • ...
Fiducial inference for discrete and continuous distributions 1
... Birnbaum pointed out (1960) that the fiducial method has some similarities with the likelihood method. Let us consider the case of two parameters θ1 and θ2 and distributions that admit symmetry of the likelihood ratio, in fact the case with an unique decomposition into simple experiments. Within tha ...
... Birnbaum pointed out (1960) that the fiducial method has some similarities with the likelihood method. Let us consider the case of two parameters θ1 and θ2 and distributions that admit symmetry of the likelihood ratio, in fact the case with an unique decomposition into simple experiments. Within tha ...
on partially conservative sentences and interpretability
... however, due to Kreisel [7] who observed that if ConP is a "natural" normalization of "P(eano arithmetic) is consistent", then -,ConP £ Consíil?, P). Related results have also been obtained by Hájek [4], Jensen and Ehrenfeucht [6], and Kreisel and Levy [8]. Subsequently the results of Guaspari and S ...
... however, due to Kreisel [7] who observed that if ConP is a "natural" normalization of "P(eano arithmetic) is consistent", then -,ConP £ Consíil?, P). Related results have also been obtained by Hájek [4], Jensen and Ehrenfeucht [6], and Kreisel and Levy [8]. Subsequently the results of Guaspari and S ...
On Gabbay`s temporal fixed point operator
... those subformulas of A that are basic. For example, the basic subformulas of A = ¬Y q → Y Y q are the first Y q and the Y Y q; neither the two occurrences of q nor the subformula Y q of the Y Y q are basic subformulas of A. The normal subformulas of A are just the first Y q, ¬Y q, Y Y q and A. Lemma ...
... those subformulas of A that are basic. For example, the basic subformulas of A = ¬Y q → Y Y q are the first Y q and the Y Y q; neither the two occurrences of q nor the subformula Y q of the Y Y q are basic subformulas of A. The normal subformulas of A are just the first Y q, ¬Y q, Y Y q and A. Lemma ...
Redundancies in the Hilbert-Bernays derivability conditions for
... The term "free variable part" of a logic with quantifiers is made precise by specifying a class of functions and encodings of those functions. Then the cited part of the logic consists of those formulae which can be construed as representing equality between functions and propositional combinations ...
... The term "free variable part" of a logic with quantifiers is made precise by specifying a class of functions and encodings of those functions. Then the cited part of the logic consists of those formulae which can be construed as representing equality between functions and propositional combinations ...
A Paedagogic Example of Cut-Elimination
... Definition 1 The set of symbols of LI consist of the following: 1. variables: x0 , x1 , x2 , . . . , 2. relation symbols: ∩, ∪, ≤, and 3. auxiliary symbols: ( and ). The set of terms of LI is the least set closed under the following formation rules: 1. Every variable is a term. 2. If X and Y are ter ...
... Definition 1 The set of symbols of LI consist of the following: 1. variables: x0 , x1 , x2 , . . . , 2. relation symbols: ∩, ∪, ≤, and 3. auxiliary symbols: ( and ). The set of terms of LI is the least set closed under the following formation rules: 1. Every variable is a term. 2. If X and Y are ter ...
Fuzzy logic and probability Institute of Computer Science (ICS
... In our opinion any serious discussion on the relation between fuzzy logic and probability must start by mak ing clear the basic differences. Admitting some simpli fication, we cotL'>ider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less ...
... In our opinion any serious discussion on the relation between fuzzy logic and probability must start by mak ing clear the basic differences. Admitting some simpli fication, we cotL'>ider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less ...
ON PRESERVING 1. Introduction The
... need is a kind of generalized negation principle: Definition 1. A logic X is said to have denial provided that for every formula α, there is some formula β such that conX ({α, β}). In such a case we shall say that α and β deny each other (in X, which qualification we normally omit when it is clear f ...
... need is a kind of generalized negation principle: Definition 1. A logic X is said to have denial provided that for every formula α, there is some formula β such that conX ({α, β}). In such a case we shall say that α and β deny each other (in X, which qualification we normally omit when it is clear f ...
Existential Definability of Modal Frame Classes
... All of the frame constructions used in the theorem – bounded morphisms, disjoint unions, generated subframes and ultrafilter extensions – are presented in detail in [1] (the same notation is used in this paper). Just to be clear, we say that a class K reflects a construction if its complement Kc , th ...
... All of the frame constructions used in the theorem – bounded morphisms, disjoint unions, generated subframes and ultrafilter extensions – are presented in detail in [1] (the same notation is used in this paper). Just to be clear, we say that a class K reflects a construction if its complement Kc , th ...
On Equivalent Transformations of Infinitary Formulas under the
... originally as a tool for proving a theorem about the logic FO(ID), has been used also to prove a new generalization of Fages’ theorem [4]. One of the reasons why stable models of infinitary formulas are important is that they are closely related to aggregates in answer set programming (ASP). The sem ...
... originally as a tool for proving a theorem about the logic FO(ID), has been used also to prove a new generalization of Fages’ theorem [4]. One of the reasons why stable models of infinitary formulas are important is that they are closely related to aggregates in answer set programming (ASP). The sem ...
Divergence Time Estimation using BEAST v2.1.3 Background
... are appropriate for use as zero-offset calibration priors. When applying these priors on node age, the fossil age is the origin of the prior distribution. Thus, it is useful to consider the fact that the prior is modeling the amount of time that has elapsed since the divergence event (ancestral node ...
... are appropriate for use as zero-offset calibration priors. When applying these priors on node age, the fossil age is the origin of the prior distribution. Thus, it is useful to consider the fact that the prior is modeling the amount of time that has elapsed since the divergence event (ancestral node ...
And this is just one theorem prover!
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
Decidable fragments of first-order logic Decidable fragments of first
... Here fragments of first-order logic are distinguish according to That the set of valid L-sentences is not decidable means that there is no effective procedure that on any input eventually terminates and correctly decides whether the input is valid or not. A sentence is valid if and only if its negat ...
... Here fragments of first-order logic are distinguish according to That the set of valid L-sentences is not decidable means that there is no effective procedure that on any input eventually terminates and correctly decides whether the input is valid or not. A sentence is valid if and only if its negat ...
And this is just one theorem prover!
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
Modus ponens
... While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to el ...
... While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to el ...
Bayesian inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as evidence is acquired. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called ""Bayesian probability"".