Conditional probability table

Abstract: Kolmogorov's axiomatization of probability includes the familiarratio formula for conditional probability: $$({\text{RATIO}}) P(A|B) = \frac{{P(A \cap B)}}{{P(B)}}{\text{ }}(P(B) >0).$$ Call this the ratio analysis of conditional probability. It has become so entrenched that it is often referred to as the definition of conditional probability.I argue that it is not even an adequate analysis of that concept. I prove what I call the Four Horn theorem, concluding that every probability assignment has uncountably many ‘trouble spots’. Trouble spots come in four varieties: assignments of zero togenuine possibilities; assignments of infinitesimals to such possibilities; vague assignments to such possibilities; and no assignment whatsoever to such possibilities. Each sort of trouble spot can create serious problems for the ratio analysis. I marshal manyexamples from scientific and philosophical practice against the ratio analysis. I conclude more positively: we should reverse the traditional direction of analysis. Conditional probability should be taken as the primitive notion, and unconditional probability should be analyzed in terms of it. “I'd probably be famous now If I wasn't such a good waitress.” Jane Siberry, “Waitress”

Abstract: Studied whether subjects understand conditional statements ("if p then q") as indicating high conditional probability. In a series of 5 experiments (paper-and-pencil as well as Internet-based), a total of 4,376 participants estimated the probability that a given conditional is true or judged whether a conditional was true or false, after being provided with information about the frequencies of the relevant truth table cases. Judgments were influenced by the ratio of cases in which p is q and p is not q, the frequency of cases in which p is q, and the pragmatic utility associated with believing a thematic conditional statement. Results are concluded to support the conditional probability account, mental model theory, and the influence of personal utilities in the cognitive processing of conditional statements.

Abstract: This paper presents a semi-supervised training method for linear-chain conditional random fields that makes use of labeled features rather than labeled instances. This is accomplished by using generalized expectation criteria to express a preference for parameter settings in which the model’s distribution on unlabeled data matches a target distribution. We induce target conditional probability distributions of labels given features from both annotated feature occurrences in context and adhoc feature majority label assignment. The use of generalized expectation criteria allows for a dramatic reduction in annotation time by shifting from traditional instance-labeling to feature-labeling, and the methods presented outperform traditional CRF training and other semi-supervised methods when limited human effort is available.