and “Random” to Meager, Shy, etc.
... an appropriate formal language. The corresponding sets are called definable. A set is definable if it can be uniquely described by a formula in an appropriate language. Since there are more than continuum many sets and only countably many formulas (and thus, only countably many definable sets), it foll ...
... an appropriate formal language. The corresponding sets are called definable. A set is definable if it can be uniquely described by a formula in an appropriate language. Since there are more than continuum many sets and only countably many formulas (and thus, only countably many definable sets), it foll ...
Topic 4
... As a result, some people who are attracted to the empirically-based nature of relative frequencies shift to a conception of probability that equates the probability that a coin will come down heads, not with the relative frequency, but, instead, with the limit of the relative frequency with which th ...
... As a result, some people who are attracted to the empirically-based nature of relative frequencies shift to a conception of probability that equates the probability that a coin will come down heads, not with the relative frequency, but, instead, with the limit of the relative frequency with which th ...
Slides
... Bayes’ rule (cont’d) • Well, we might say that doctors know that a stiff neck implies meningitis in 1 out of 5000 cases; – That is the doctor has quantitative information in the diagnostic direction from symptoms (effects) to causes. – Such a doctor has no need for Bayes’ rule?! • Unfortunately, di ...
... Bayes’ rule (cont’d) • Well, we might say that doctors know that a stiff neck implies meningitis in 1 out of 5000 cases; – That is the doctor has quantitative information in the diagnostic direction from symptoms (effects) to causes. – Such a doctor has no need for Bayes’ rule?! • Unfortunately, di ...
Generalization Error Bounds for Bayesian Mixture Algorithms
... 2003) provided a data-dependent bound for the so-called Gibbs algorithm, which selects a hypothesis at random from H based on the posterior distribution π(h|S). Essentially, this result provides a bound on the average error ∑ j q j L(h j ) rather than a bound on the error of the averaged hypothesis, ...
... 2003) provided a data-dependent bound for the so-called Gibbs algorithm, which selects a hypothesis at random from H based on the posterior distribution π(h|S). Essentially, this result provides a bound on the average error ∑ j q j L(h j ) rather than a bound on the error of the averaged hypothesis, ...
Conditional Probability
... =1/2. People who receive the morning paper are less likely to receive the evening paper than people who do not receive the morning paper. Since the probability the event E occurs depends on whether or not M occurred, we call these events dependent. There are many events of this form. Any events with ...
... =1/2. People who receive the morning paper are less likely to receive the evening paper than people who do not receive the morning paper. Since the probability the event E occurs depends on whether or not M occurred, we call these events dependent. There are many events of this form. Any events with ...
(c) Suppose two chips are randomly selected without replacement
... (c) Suppose two chips are randomly selected without replacement. Let A be the event that a 7 is obtained on the first draw, and B be the event that a red is obtained on the second draw. What is P (B∣A)? (d) Suppose two chips are randomly selected without replacement. Let A be the event that a 7 is ob ...
... (c) Suppose two chips are randomly selected without replacement. Let A be the event that a 7 is obtained on the first draw, and B be the event that a red is obtained on the second draw. What is P (B∣A)? (d) Suppose two chips are randomly selected without replacement. Let A be the event that a 7 is ob ...
What Conditional Probability Must (Almost) Be
... (A) is defined to be the conditional probability of the event B under the condition A.”2 (Throughout this paper, I will use the more standard notation P (B|A) in place of Kolmogorov’s PA (B).) However, since this equation gives no answer for conditional probabilities when the antecedent has probabil ...
... (A) is defined to be the conditional probability of the event B under the condition A.”2 (Throughout this paper, I will use the more standard notation P (B|A) in place of Kolmogorov’s PA (B).) However, since this equation gives no answer for conditional probabilities when the antecedent has probabil ...
Interpretations of Probability.pdf
... meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest. However, there is no single formal system that is ‘probability’, but rather a host of such systems. To be sure, Kolmogorov's axiomatization, which we w ...
... meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest. However, there is no single formal system that is ‘probability’, but rather a host of such systems. To be sure, Kolmogorov's axiomatization, which we w ...
12.9 EVEL OF SIGNIFICANCE AND HYPOTHESIS TESTING
... Once we have used a specified level of significance to determine a critical region, we can ask ourselves how powerful the test is, in the sense of how probable it is to reject the null hypothesis for a particular specified value of the parameter of interest for which the alternative hypothesis holds ...
... Once we have used a specified level of significance to determine a critical region, we can ask ourselves how powerful the test is, in the sense of how probable it is to reject the null hypothesis for a particular specified value of the parameter of interest for which the alternative hypothesis holds ...
Uncertainty
... • Even if we know all the rules, we might be uncertain about a particular patient because not all the necessary tests have been or can be run. ...
... • Even if we know all the rules, we might be uncertain about a particular patient because not all the necessary tests have been or can be run. ...
Chap 2-Basic Concepts in Probability and Statistics
... every frequency series—deciding which frequency series to use for an estimate, choosing which part of the frequency series to use, and so on. For example, should the insurance company use only its records from last year, which will be too few to provide as much data as is preferable, or should it al ...
... every frequency series—deciding which frequency series to use for an estimate, choosing which part of the frequency series to use, and so on. For example, should the insurance company use only its records from last year, which will be too few to provide as much data as is preferable, or should it al ...
2. Criteria of adequacy for the interpretations of
... various probability-like concepts that purportedly do. Be all that as it may, we will follow common usage and drop the cringing scare quotes in our survey of what philosophers have taken to be the chief interpretations of probability. Whatever we call it, the project of finding such interpretations ...
... various probability-like concepts that purportedly do. Be all that as it may, we will follow common usage and drop the cringing scare quotes in our survey of what philosophers have taken to be the chief interpretations of probability. Whatever we call it, the project of finding such interpretations ...
Probability And Statistics Throughout The Centuries
... the concept of the “probable”. It was impossible for them to accept that things could happen on earth in contradiction to the behavior of the heavenly bodies. So, there was no room for probability, which results from causality rather than being result of logical relations amongst ideas. To this, one ...
... the concept of the “probable”. It was impossible for them to accept that things could happen on earth in contradiction to the behavior of the heavenly bodies. So, there was no room for probability, which results from causality rather than being result of logical relations amongst ideas. To this, one ...
Eliciting Subjective Probabilities Through
... method. First, this method only deals with events from the same variable. In the above example, I did not refer to anything other than the temperature in Paris. With the lottery method, one would have had to propose two different kinds of events: the unknown variable under consideration (e.g., the t ...
... method. First, this method only deals with events from the same variable. In the above example, I did not refer to anything other than the temperature in Paris. With the lottery method, one would have had to propose two different kinds of events: the unknown variable under consideration (e.g., the t ...
an application of information theory to the problem - Philsci
... m], where A is the partition and m the normalized measure function. In information theory, this structure is often referred to as a source, and this same term will be used later in the text to indicate the model of an experiment, since [A, m] is a theoretical description of an experiment. Whenever a ...
... m], where A is the partition and m the normalized measure function. In information theory, this structure is often referred to as a source, and this same term will be used later in the text to indicate the model of an experiment, since [A, m] is a theoretical description of an experiment. Whenever a ...
Rare Probability Estimation under Regularly Varying Heavy Tails
... concentration results for the missing mass to all of the rare probabilities, and then using the regular variation property to show multiplicative concentration. Additionally, we construct new families of estimators that address some of the other shortcomings of the Good-Turing estimator. For example ...
... concentration results for the missing mass to all of the rare probabilities, and then using the regular variation property to show multiplicative concentration. Additionally, we construct new families of estimators that address some of the other shortcomings of the Good-Turing estimator. For example ...
here
... satisfying these axioms is necessarily a space with conditional belief types that induce the operators. The axioms are of two kinds. The first five axioms, similar to the ones used in Samet (2000), imply that for each condition C , the operator B p . j C / is derived from a mapping assigning a prob ...
... satisfying these axioms is necessarily a space with conditional belief types that induce the operators. The axioms are of two kinds. The first five axioms, similar to the ones used in Samet (2000), imply that for each condition C , the operator B p . j C / is derived from a mapping assigning a prob ...
Conditionals, Conditional Probabilities, and
... functions. Statistically speaking, those sentence denotations are indicator variables – a special kind of random variables whose range is restricted to the set {0, 1}. In the present context this perspective was first proposed by Jeffrey (1991) and further developed by Stalnaker and Jeffrey (1994). ...
... functions. Statistically speaking, those sentence denotations are indicator variables – a special kind of random variables whose range is restricted to the set {0, 1}. In the present context this perspective was first proposed by Jeffrey (1991) and further developed by Stalnaker and Jeffrey (1994). ...
Rationality and the Bayesian Paradigm
... minor restrictions such as monotonicity or concavity. In a sense, rationality ceased to be a matter of content, and became a matter of form. Towards the end of the 20th century, partly due to attacks based on psychological findings, economics started questioning the axioms that defined rationality. ...
... minor restrictions such as monotonicity or concavity. In a sense, rationality ceased to be a matter of content, and became a matter of form. Towards the end of the 20th century, partly due to attacks based on psychological findings, economics started questioning the axioms that defined rationality. ...
Proving Facts: Belief versus Probability
... the common law’s focus on likelihood or probability is based on a failure of those on each side of the conflict to appreciate that, while it is true that all proof involves probabilities, there are different types of probabilities, only one of which – which is not statistical in nature – is sufficie ...
... the common law’s focus on likelihood or probability is based on a failure of those on each side of the conflict to appreciate that, while it is true that all proof involves probabilities, there are different types of probabilities, only one of which – which is not statistical in nature – is sufficie ...
Dempster–Shafer theory
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty - a mathematical theory of evidence. The theory allows one to combine evidence from different sources and arrive at a degree of belief (represented by a mathematical object called belief function) that takes into account all the available evidence.In a narrow sense, the term Dempster–Shafer theory refers to the original conception of the theory by Dempster and Shafer. However, it is more common to use the term in the wider sense of the same general approach, as adapted to specific kinds of situations. In particular, many authors have proposed different rules for combining evidence, often with a view to handling conflicts in evidence better. The early contributions have also been the starting points of many important developments, including the Transferable Belief Model and the Theory of Hints.