this paper - William M. Briggs
... true, or M may even be complete gibberish. Then no probability at all can be assigned. If you do assign a probability it is because you are adding information that was not given to you, information you suppose that is true, but that may be false. The argument is changed and you cannot say your assig ...
... true, or M may even be complete gibberish. Then no probability at all can be assigned. If you do assign a probability it is because you are adding information that was not given to you, information you suppose that is true, but that may be false. The argument is changed and you cannot say your assig ...
On the Ordering of Probability Forecasts - Sankhya
... possibility that q A (0) > q B (0), and still A >V M (d) B. It can be shown by simple examples that well calibrated forecasters with these properties exist. Similarly, q A (1) = q B (1) = 0 rules out the possibility that q A (1) < q B (1) and still A ≥V M (nd) B. Again, it can be shown by simple exa ...
... possibility that q A (0) > q B (0), and still A >V M (d) B. It can be shown by simple examples that well calibrated forecasters with these properties exist. Similarly, q A (1) = q B (1) = 0 rules out the possibility that q A (1) < q B (1) and still A ≥V M (nd) B. Again, it can be shown by simple exa ...
pdf
... Proposition 2.1: Suppose that PrY is an arbitrary distribution on Y, L is an arbitrary loss function, P consists of all distributions on X × Y with marginal PrY , and a∗ is an optimal action for PrY (with respect to the loss function L). Then EPrY [La∗ ] = inf δ∈D(X ,A) supPr∈P EPr [Lδ ]. A standar ...
... Proposition 2.1: Suppose that PrY is an arbitrary distribution on Y, L is an arbitrary loss function, P consists of all distributions on X × Y with marginal PrY , and a∗ is an optimal action for PrY (with respect to the loss function L). Then EPrY [La∗ ] = inf δ∈D(X ,A) supPr∈P EPr [Lδ ]. A standar ...
Weighted Sets of Probabilities and Minimax Weighted Expected
... some probability over states. What measure should be used? There are two hypotheses that T-800 entertains: (1) the stairs are ten feet high and (2) the stairs are fifteen feet high. Each of these places a different probability on states. If the stairs are ten feet high, we can take all of the 1, 000 ...
... some probability over states. What measure should be used? There are two hypotheses that T-800 entertains: (1) the stairs are ten feet high and (2) the stairs are fifteen feet high. Each of these places a different probability on states. If the stairs are ten feet high, we can take all of the 1, 000 ...
Inducing Probability Distributions from Knowledge Bases with (In
... models of K). In (Knight 2002), probability theory was used to measure the degree of (in)consistency (of subsets) of a knowledge base. Assume that P is the set of all probability distributions definable on the set of possible worlds from a language (Paris 1994), then given an inconsistent KB K (or a ...
... models of K). In (Knight 2002), probability theory was used to measure the degree of (in)consistency (of subsets) of a knowledge base. Assume that P is the set of all probability distributions definable on the set of possible worlds from a language (Paris 1994), then given an inconsistent KB K (or a ...
Nash Equilibrium with Lower Probabilities
... One can think of a lower probability measure b as the vector b E EX . For any two lower probability measures b1 ; b2 2 B and 2 0; 1 , the convex combination b b1 1 b2 is defined by: b E b1 E 1 b2 E , for all E X . It is easy to verify that B, like , is a convex set. It is natural to think ...
... One can think of a lower probability measure b as the vector b E EX . For any two lower probability measures b1 ; b2 2 B and 2 0; 1 , the convex combination b b1 1 b2 is defined by: b E b1 E 1 b2 E , for all E X . It is easy to verify that B, like , is a convex set. It is natural to think ...
Relative frequencies
... relative frequency interpretation first of all that the structures it describes (long runs, with assignments of real numbers to subsets thereof through the concept of limit) be models of the theory. But that would require relative frequency to be countably additive which it is not. Secondly we would ...
... relative frequency interpretation first of all that the structures it describes (long runs, with assignments of real numbers to subsets thereof through the concept of limit) be models of the theory. But that would require relative frequency to be countably additive which it is not. Secondly we would ...
FINITE ADDITIVITY VERSUS COUNTABLE ADDITIVITY
... space have a probability, and for each set this may be calculated by summing over the probabilities of the singleton sets of the points it contains. (This is of course in the countably additive case; in the finitely additive case, one can have all finite sets of zero probability, but total mass 1.) I ...
... space have a probability, and for each set this may be calculated by summing over the probabilities of the singleton sets of the points it contains. (This is of course in the countably additive case; in the finitely additive case, one can have all finite sets of zero probability, but total mass 1.) I ...
A Simple Sequential Algorithm for Approximating Bayesian Inference
... is what people will infer about the causal properties of this block, and which class of blocks it belongs to. Given this hypothesis space, we can consider how an ideal learner should update his or her beliefs in light of the evidence provided by its interaction with the machine. Assume that the lea ...
... is what people will infer about the causal properties of this block, and which class of blocks it belongs to. Given this hypothesis space, we can consider how an ideal learner should update his or her beliefs in light of the evidence provided by its interaction with the machine. Assume that the lea ...
Week 3, Lecture 2, Conditional probabilities
... • There are several ways in which a sample space can be viewed. ...
... • There are several ways in which a sample space can be viewed. ...
The naïve see causal connections everywhere. Consider the fact... the New Jersey lottery twice. The naïve find it irresistible... Coincidences and How to Think about Them
... observations are very improbable. You can say that the hypothesis is false or that something very improbable has just occurred. Fisher was right about the disjunction. However, what does not follow is that the hypothesis is false; in fact, as just noted, it doesn’t even follow that you have obtained ...
... observations are very improbable. You can say that the hypothesis is false or that something very improbable has just occurred. Fisher was right about the disjunction. However, what does not follow is that the hypothesis is false; in fact, as just noted, it doesn’t even follow that you have obtained ...
Unreliable probabilities, risk taking, and decision making
... For each set, the subject was then asked to choose between lottery tickets involving the same events, where a ticket was to be conceived as yielding a win of 100 SwKr if the event occurred but no monetary loss if it did not occur. One hypothesis that obtained support in this experiment was that for ...
... For each set, the subject was then asked to choose between lottery tickets involving the same events, where a ticket was to be conceived as yielding a win of 100 SwKr if the event occurred but no monetary loss if it did not occur. One hypothesis that obtained support in this experiment was that for ...
Conditional probability in the light of qualitative belief change
... the west than in the east, then the figure will be higher. So the procedure provides neither a general solution nor, within its domain, a unique one. This is not to dismiss Rényi’s way around the problem out of hand. In practical situations, it may often be the best thing to do. Suppose that in an e ...
... the west than in the east, then the figure will be higher. So the procedure provides neither a general solution nor, within its domain, a unique one. This is not to dismiss Rényi’s way around the problem out of hand. In practical situations, it may often be the best thing to do. Suppose that in an e ...
Naive Bayesian Classifier
... In our example, for the attribute-value pair student = yes of X, we need to count the number of customers who are students, and for which buy = yes (which contributes to P (X|buy = yes)) and the number of customers who are students and for which buy = no (which contributes to P (X|buy = no)). But wh ...
... In our example, for the attribute-value pair student = yes of X, we need to count the number of customers who are students, and for which buy = yes (which contributes to P (X|buy = yes)) and the number of customers who are students and for which buy = no (which contributes to P (X|buy = no)). But wh ...
Study Materials
... 14.Find the probability that a number selected at random from the numbers 1 to 25 is not a prime when each of the given numbers is equally likely to be selected. 15.A bag contains 4 white and 5 blue balls .They are mixed thoroughly and one ball is drawn at random. What is the probability of getting ...
... 14.Find the probability that a number selected at random from the numbers 1 to 25 is not a prime when each of the given numbers is equally likely to be selected. 15.A bag contains 4 white and 5 blue balls .They are mixed thoroughly and one ball is drawn at random. What is the probability of getting ...
The Axioms of Subjective Probability
... as, on a set of propositionsor events. This relation, oftenreferredto as a qualitative or comparativeprobabilityrelation,can be taken either as an undefined primitive(intuitive views) or as a relation derived from a preference relation (decision-oriented approach). In the lattercase, to say that you ...
... as, on a set of propositionsor events. This relation, oftenreferredto as a qualitative or comparativeprobabilityrelation,can be taken either as an undefined primitive(intuitive views) or as a relation derived from a preference relation (decision-oriented approach). In the lattercase, to say that you ...
Risk, Uncertainty, and Profit
... cannot help attributing to other creatures similarly constituted and behaving in the same way with himself "insides," to use Descartes' picturesque term, like his own. We perceive the world before we react to it, and we react not to what we perceive, but always to what we infer. The universal form o ...
... cannot help attributing to other creatures similarly constituted and behaving in the same way with himself "insides," to use Descartes' picturesque term, like his own. We perceive the world before we react to it, and we react not to what we perceive, but always to what we infer. The universal form o ...
Dismissal of the illusion of uncertainty in the assessment of a
... been found at the crime scene and that DNA analyses performed by a forensic laboratory have led it to report a match E between the genetic profiles characterizing the recovered material, y, and the control material x found on a suspect, respectively. The competing propositions of interest to the Cou ...
... been found at the crime scene and that DNA analyses performed by a forensic laboratory have led it to report a match E between the genetic profiles characterizing the recovered material, y, and the control material x found on a suspect, respectively. The competing propositions of interest to the Cou ...
• Elementary propositions can be combined to form complex
... • The joint probability distribution of two random variables is the product of their domains • E.g., P(Weather, Cavity) is a 4 × 2 table of probabilities • Full joint probability distribution covers the complete set of random variables used to describe the world • For continuous variables it is not ...
... • The joint probability distribution of two random variables is the product of their domains • E.g., P(Weather, Cavity) is a 4 × 2 table of probabilities • Full joint probability distribution covers the complete set of random variables used to describe the world • For continuous variables it is not ...
Pseudo-Bayesian Updating
... Axiom 2. (Conservatism) µ(A2 ) ≥ µ(A1 ) =⇒ µα (A1 ) = µα (A2 ). Axiom 2 says that the DM, upon receiving information that contradicts her prior, makes minimal adjustments to her beliefs. The prior µ encapsulates all the credible information that the DM received in the past and, therefore, stands on ...
... Axiom 2. (Conservatism) µ(A2 ) ≥ µ(A1 ) =⇒ µα (A1 ) = µα (A2 ). Axiom 2 says that the DM, upon receiving information that contradicts her prior, makes minimal adjustments to her beliefs. The prior µ encapsulates all the credible information that the DM received in the past and, therefore, stands on ...
Tutorial: Defining Probability for Science.
... probability of the same outcome poses a problem for scientific uses, and can be seen to be one of the motivations behind alternative definitions of probability. However, appropriate use of frequentist probability requires some care. We need to first consider if this is a valid motivation, or just a ...
... probability of the same outcome poses a problem for scientific uses, and can be seen to be one of the motivations behind alternative definitions of probability. However, appropriate use of frequentist probability requires some care. We need to first consider if this is a valid motivation, or just a ...
Reducing belief simpliciter to degrees of belief
... But there are further ontological possibilities: A cognitive agent might actually instantiate both qualitative and quantitative mental states of belief, without either of the two being ontologically reducible to the other one. For example, imagine one part of the brain to determine one’s qualitative ...
... But there are further ontological possibilities: A cognitive agent might actually instantiate both qualitative and quantitative mental states of belief, without either of the two being ontologically reducible to the other one. For example, imagine one part of the brain to determine one’s qualitative ...
Advanced probability: notes 1. History 1.1. Introduction. Kolmogorov
... randomly chosen point falls on a particular meridian). We should pass to the limit only after absorbing the new information’ 1.3. Measure-theoretic probability before the Grundbegriffe. 1.3.1. The invention of measure theory by Borel and Lebesgue. Borel is considered the father of measure theory. Hi ...
... randomly chosen point falls on a particular meridian). We should pass to the limit only after absorbing the new information’ 1.3. Measure-theoretic probability before the Grundbegriffe. 1.3.1. The invention of measure theory by Borel and Lebesgue. Borel is considered the father of measure theory. Hi ...
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.