Pearsonschool Com Dimensionm Game
Description: We will be using A REASON FOR SCIENCE which will introduce and explore topics in science including Life Science, Earth Science and Physical Science. Fun hands on activities are used to ask and answer questions and develop a love of experimentation and learning. Class will involve discussion, problem solving and activities, home study will involve some reading and scripture connection. Small journaling may be completed in class or at home depending on time. There will be minimal homework, but you will be able to follow up with each topic as you decide. There is NO BOOK for at home use! Work pages needed will be sent home each week.
Point-of-use videos reinforce key grammar skills; ExamView Assessment Suite (diagnostic and assessment) Electronic test generator allows teachers to customize assessment/; Tabula Digita: Dimension, www.pearsonschool.com/dimension. Students hone their grammar skills with this fast-paced, multiplayer video game.
Description: This class will introduce young learners to the wonderful world of language arts. We will explore the rules and structure of our language, including capitalization, punctuation, compound words, synonyms, antonyms, plurals, contractions, the basic parts of speech, and more. Fun games and activities in class will help solidify these rules. Young students will discover tools for writing as they investigate the different types of sentences, how to form complete sentences, main ideas, and supporting details. By the end of the year students will be able to write short paragraphs.
Description: This combination class is sure to get the creative juices flowing. As students read novels from various genres, they will begin to look for common characteristics. What makes a “good story”? What literary elements does an author use? We will explore these topics and try our hand at writing our own stories, using good literature as our model. Students will become proficient in the writing process as they develop their stories from prewriting through publishing.
Grammar usage and mechanics will be incorporated as well as a study on the parts of speech. Games and mini lessons will be used to introduce concepts in bite-size pieces. The goal for this class is to encourage a love for reading and to build knowledge and confidence for budding authors! Description: A fun and thoughtful Introduction to Literature class. This course is designed to get the intermediate-age student familiar with the elements of literature, literary terms, and devices, while still perfecting their comprehension skills with increasingly difficult text.
They will read a variety of genres and learn about plot structure and character development, while practicing the art of conversation with fellow students. The following genres will be read this year: fantasy, adventure, historical fiction, realistic fiction, and drama. A short story collection will be provided by the teacher. Description: This class will introduce and encourage students to read and appreciate a variety of literary styles, and encourage an enthusiasm for reading. Students will become familiar with elements of literature such as the story plot, setting, character development, and problem and conflict. Vocabulary and comprehension of the novel will be stressed as well.
Genres will include fiction, biography, fantasy, adventure, and historical fiction. We will incorporate group activities, as well as projects to enhance the literature. Description: Designed for students who have established an elementary foundation in language arts, this course will combine writing, grammar, and literature components. As we read quality middle school literature, we will focus on different paragraph and essay types including description, example, cause and effect, and process. Research skills will be strengthened using key wording, paraphrasing, and outlining; students will practice source-citing and avoiding plagiarism. Several class presentations will be assigned.
Grammar and mechanics will be taught in mini-sessions throughout the year. This course's literature-based units will include assigned reading, literary terms, vocabulary, and book-related research portfolios.
Those books will likely include The Horse and His Boy, The Witch of Blackbird Pond, Out of the Dust, and The Giver. Description: Welcome to Part 1 of a 2 year study of American History. This study begins at 1000 and goes through 1800s. Using America the Beautiful book 1 as our text. The study is chronological and covers American history and geography from a Christian worldview. Most reading will be completed at home, but they will be required to do some writing in class. Class time will be spent with review, taking basic notes, and hands-on activities.
Students will have projects throughout the year to be completed at home. Description: This class will serve as a transition class from elementary sciences to the Apologia curriculum.
Using the BJU Curriculum as described on their website: Life Sciences is the study of the living wonders that surround us every day. Your students will learn to appreciate the wonders of Nature, as well as its Creator, in BJU Press 7th Grade Life Science Curriculum. The basics of science are covered, including classification of species, cell structure, relationships among organisms, genetics, microbiology, botany, zoology, and ecology, as well as more complicated topics such as evolution and creation, reproduction, and man's relationship to the environment. Description: In the fall semester, students will begin with civics, learning about America’s Founding Documents, our Founding Fathers, the branches of the government, voting, and elections by use of our textbook, projects, and games. In the spring semester, students will switch to world geography, learning about Virginia, the United States, and TRAVELING the 7 continents studying the physical geography and the culture, while sampling food from each continent.
There will be a minimum of one field trip during the school year. Description: This class is designed as an informative, interactive introduction to high school literature.
In this course, students will read a broad range of short stories, poetry, fiction, and nonfiction to strengthen their understanding of literature. Key literary elements such as setting, plot, theme, characterization, and figurative language will be emphasized. Through active group discussion and class participation, students will develop skills in reading comprehension and literary analysis. Additionally, we will practice media literacy by accessing and evaluating auditory, visual, and written media. Assignments will include reading, analysis essays, vocabulary, comprehension quizzes, summaries, journal entries, and oral presentations. Description: Recommended for students in the 9th grade and up; students should have completed at least one year of formal grammar and writing studies. Using techniques from the Institute for Excellence in Writing and Format Writing, students will develop an organized, creative writing style.
Grammar will be applied to writing paragraphs and essays in order to develop correct and varied sentence structure. Writing assignments include different formats such as process, cause and effect, and persuasive styles in order to prepare students for later high school and college composition classes. Many writing assignments will be research-oriented beginning with paragraphs and then essays of various lengths.
Class will also include lessons in outlining, condensing, source citing, and avoiding unintentional plagiarism. A variety of high school vocabulary words will also be studied. Description: This course is recommended for older high school students with several years of formal writing experience. Using techniques from the Institute for Excellence in Writing and Along These Lines, students will develop an organized, creative writing style. Writing assignments incorporate an abundance of research in addition to formats such as process, description, cause and effect, and persuasion.
Class will also include lessons in outlining, condensing, source citing, and avoiding plagiarism. Additionally, students will evaluate and respond to persuasive articles to help prepare for the new SAT style writing. This course will continue to review the basics of grammar and mechanics, and significant time will be spent on varied sentence structure.
Vocabulary studies will also be a major focus of the class. Description: As long as humans have lived together in communities, we have devised ideas and considered how to foster a society full of peace, prosperity, morality, and beauty. Could such a society exist? And why does this search for a perfect world typically backfire? Authors have long explored the limits of the ideal and the flip side—the dark, nightmare world of an imagined society characterized by poverty, fear, and oppression.
In this course, we will identify and analyze how dystopian authors turn an inquisitive eye on their own societies, and we will. Description: Nations define themselves by the stories they tell, and America’s story has been deeply influenced by its many great writers. This course is a chronological study of American literature which emphasizes close reading and clear writing.
Together we will examine the contexts of the development of the unique voices of American writers as they invented new genres and means of expression while this whole new kind of country was inventing itself. Readings will likely include Puritan authors, Benjamin Franklin’s Autobiography, The Scarlet Letter, Narrative of the Life of Frederick Douglass, Moby Dick, The Adventures of Huckleberry Finn, The Call of the Wild, The Great Gatsby, Our Town, and To Kill a Mockingbird, as well as short stories, essays, and poems by Irving, Bryant, Poe, Emerson, Thoreau, Dickinson, Longfellow, Whitman, Harte, Dunbar, Frost, McKay, and Hughes. Students will respond to readings through Socratic discussions, journal responses, and formal writing assignments. Description: Using James Stobaugh’s World History: Observations and Assessments from Early Cultures to Today the students will learn a comprehensive overview of world history.
Additional handouts, including primary sources, maps, and timelines will be used with this class to enhance the details of certain time periods. Also, classtime will include debates and discussions that encourage students to think about past historical events and how they can learn from them and relate them to current event.
This will be an exciting new approach for students to understand the hand of God in the shaping of history of our world!! Description: This course is a continuation of WH 1 but can be taken as a stand-alone course. Students will continue their journey through time and the study of all things “human” from a Christian perspective. We travel through the Enlightened World, the European World, and the Modern World. Students will have the opportunity to express mastery of the material through various learning styles and will include comprehension, writing, artwork, oral and visual projects, Socratic seminars, worksheets, and tests/quizzes, including both a Midterm and Final Exam. Description: This twice a week class covers basic Algebra I including integers, real numbers, the language of algebra, solving linear and quadratic equations and inequalities, graphing, solving systems of equations, operations with polynomials and radicals, factoring polynomials, and solving rational equations. Students will develop thinking processes and problem-solving abilities that are essential to success in future math courses and in everyday life.
The goal of the teacher is for students to truly understand the concepts and see that math can be fun! **Please note that this class will be using the 2nd Edition of the BJU Algebra 1 book. After using the 3rd edition for two years, the teacher has found that the students are more successful when using the 2nd edition.
Description: Geometry includes all topics in a high school geometry course, including perspective, space, and dimension associated with practical and axiomatic geometry. Students learn how to apply and calculate measurements of lengths, heights, circumference, areas, and volumes. Geometry introduces trigonometry and allows students to work with transformations. Students will use logic to create proofs and constructions and will work with key geometry theorems and proofs. The class will involve both presentation of the material, and a time in which to work through the questions which arise outside of class.
Students will be given a time to present questions from the homework which they have completed and corrected at home. Prerequisite: Successful completion of Algebra 1. Description: The class will involve both presentation of the material, and a time in which to work through the questions which arise outside of class. Students will be given a time to present questions from the homework which they have completed and corrected at home.
This course is designed to help students make the transition from intermediate algebra to calculus. This course emphasizes carefully explained concepts, procedures and worked out problems in step by step form. It contains active participation and graphing with polynomial, rational, exponential, logarithmic, and trigonometric functions.
Analytic trigonometry, systems of equations and inequalities, matrices and determinants, conic sections and more are covered. Description: This great textbook will cover function basics, polynomial functions, rational functions, exponential and logarithmic functions, radical/power functions, triangle trigonometry, trigonometric functions and identities, series and sequences, probability and statistics, matrices, and determinants. If time permits, we will also cover polar coordinates and vectors and an introduction to Calculus. The goal of the teacher is for students to truly understand the concepts and see that math can be fun!
Description: Physics is the study of matter, energy, and the interaction between them—what could be more fascinating? This college-prep lab course is suited for students who have a fairly good grasp of algebra and have a little experience working with sine cosine and tangent. Our text covers one-dimensional and two-dimensional motion, Newton’s laws and their application, gravity, work and energy, momentum, periodic motion, waves, optics, electrostatics, electrodynamics, electrical circuits, and magnetism. Description: There is an interactive textbook online that will allow for multiple level of students in this class.
Latin 1 Description: Students will interpret simple texts and phrases by learning basic vocabulary and develop awareness of Roman culture. Related subjects included mythology, geography, art, and architecture. Latin will enhance learning of English grammar and well as other Romance languages. Studying Latin aids in posting higher SAT scores. Science and Math derives about 80% of their terminology from Latin.
In addition to the sciences, many legal terms are also derived from Latin. A bonus in learning Latin is for those interested in pursuing medicine, theology, litigation or politics.
Latin 2 continues where Ecce Romani 1a ends. Students will interpret simple texts and phrases by learning basic vocabulary and develop awareness of Roman culture. Related subjects included mythology, geography, art, and architecture. Latin will enhance learning of English grammar and well as other Romance languages. Studying Latin aids in posting higher SAT scores. Science and Math derives about 80% of their terminology from Latin.
In addition to the sciences, many legal terms are also derived from Latin. A bonus in learning Latin is for those interested in pursuing medicine, theology, litigation or politics. Talent is given by God. Our gift to Him is fulfilling itThis course is designed for students who desire further study in the fine arts field. Emphasis will be made on developing old and new techniques in the areas of drawing, painting, sculpting and collaging.
Media to be covered: pencil, charcoal, watercolor, acrylic, pastel, tempera, specialty paper and clay. Students will study art movements and recreate famous works of art. Artist grade materials are used for most projects. During the school year, students work on 5 independent projects of their choice.
Curriculum includes studying and recreating famous works of art. Grades are based on classroom attendance and participation. Please feel free to email Mrs. Johnson with any questions. Description: This course will help students become well rounded in the fundamentals of digital photography and photoshop. Four areas of instruction will be emphasized: How cameras work, how composition works, how lighting works, how to use photoshop for post-processing. Students will receive instruction, demonstration, and see samples of the desired outcomes at the beginning of each class.
They will be allowed to go outside and photograph assignments, based on what they are learning. The most useful part of classroom instruction will be weekly reviews/critique of photographs students have taken the previous week. This promotes understanding of what makes a successful photograph and what does not. Description: This twelve-week course will help prepare your student for the SAT through practice tests, weekly reading, vocabulary lists, reviewing attack strategies, and weekly Q&A. It is designed to provide students with the tools to allow each student to perform at his or her best on the SAT.
Class will consist of eight classes focused on Critical Reading and Writing and four weeks focused on Math. Students will practice basic SAT testing strategies and work on the three areas of the new test—Critical Reading, Mathematics, and Writing, including the essay portion.
The course will focus on developing critical thinking skills and using alternative methods to solve problems. Subjects covered include: when (not) to use a calculator, choosing an answer by process of elimination, grammar review, and metacognition. While whole group instruction will be integral to our class sessions, students will also work in small groups and independently to ensure optimal individual progress. The math portion covers review of Algebra I, Geometry and Algebra II through sample problems.
The reading/writing portion will include great test-taking tips for understanding vocabulary context and identifying passage tone and purpose. Reading class time will also focus on the essay portion of the SAT: how to prepare with source summaries, develop an argument, and support it with strong examples. Students will have quizzes on the assigned skills as well as practice sessions in class and at home. Description: This course will be a fantastic exposure to the world of culinary arts. Students will be introduced to various kitchen tools, methods of cooking, and learn how to make a recipe work for them.
Each week the students will prepare a tasty dish in the class kitchen. The students will be encouraged to try and replicate the dishes at home. Students will learn safety measures and be encouraged to familiarize themselves with their own family kitchen and help prepare meals at home using the skills taught in class.
The idea that perceptual and cognitive systems must incorporate knowledge about the structure of the environment has become a central dogma of cognitive theory. In a Bayesian context, this idea is often realized in terms of “tuning the prior”—widely assumed to mean adjusting prior probabilities so that they match the frequencies of events in the world. This kind of “ecological” tuning has often been held up as an ideal of inference, in fact defining an “ideal observer.” But widespread as this viewpoint is, it directly contradicts Bayesian philosophy of probability, which views probabilities as degrees of belief rather than relative frequencies, and explicitly denies that they are objective characteristics of the world. Moreover, tuning the prior to observed environmental frequencies is subject to overfitting, meaning in this context overtuning to the environment, which leads (ironically) to poor performance in future encounters with the same environment.
Whenever there is uncertainty about the environment—which there almost always is—an agent's prior should be biased away from ecological relative frequencies and toward simpler and more entropic priors. The Lord's prior The conception of “tuning” that is tacitly adopted in many modern treatments is that the optimal Bayesian observer is correctly tuned when its priors match those objectively in force in the environment (the “Lord's prior”). In Bayesian probability theory, priors represent the knowledge brought to bear on a decision problem by factors other than the data at hand, that is, the state of beliefs “prior to” (really, separate from) any consideration of the evidence. (Gauss's term was ante eventum cognitum: “before the cognitive act”; see.) In the Lord's prior view, priors are said to match the world when each event class h, which objectively occurs in the environment with probability p( h), is assigned a prior of p( h).
A simple extension replaces the discrete event h with the continuous parameter x, in which case we would want the prior on x, p( x), to be equal to the objective probability density function p( x). This condition defines what is referred to in the perception literature as an ideal observer, that is, an agent that makes optimal decisions based on assumptions that are in fact true in the environment, and which thus whose decision that are optimal in that environment. This conception also underlies some of the enthusiam for natural image statistics in the literature, in which Bayesian inference is endowed with priors drawn from statistical summaries of the world, or proxies thereof such as databases of natural images. In a natural image statistics framework, the best way to set a prior is canvass the world and ask what its prior is. Indeed, in many treatments, setting priors empirically is held up as a desirable aspiration, self-evidently superior to alternatives which are derided as arbitrary or “subjective.” A recent example is, who criticize Bayesian models on a number of fronts but comment (p. 173) that “the prior can be a strong point of the model if it is derived from empirical statistics of real environments” and later (p. 180) lament that “[u]nfortunately, the majority of rational analyses do not include any measurements from actual environments.” Associated with this view is the idea that natural selection will put adaptive pressure on agents to adopt the “true” prior—that over the course of generations, it will nudge innate priors toward their true environmental values ().
Implicit in this idea is an assumption that having true probabilistic beliefs is maximally beneficial to the organism. This assumption has been criticized because the utility function may well favor something other than truth (; ).
But even if one improves this view by coupling it with a suitable loss function (), the central dictum is that inference benefits by having priors that are “empirically correct.”. Frequentist versus epistemic views of probability But this viewpoint, agreeable as it may seem to modern ears, is actually at odds with traditional Bayesian philosophy of probability. The distinction revolves around competing views of what “probability” means, generally involving the distinction between the frequentist and subjectivist (or epistemic or Bayesian) conceptions of probability. Frequentists (e.g.,;; ) define probability strictly in terms of some “infinitely repeated random experiment,” such as an infinite sequence of coin tosses. The probability of h (e.g., “heads”) is defined as the ratio of number of trials on which h occurs to the total number of trials in such a thought experiment. Most psychologists are so accustomed to this way of looking at probability that we struggle to think about it any other way.
(The common use of the term “base rate” as a synonym for “prior probability” reflects this attitude.) But the frequentist conception is extremely limiting. For example, it automatically means that probabilities can only be assigned to stochastic events that are, in principle, capable of being repeated many times with different outcomes.
For example, a frequentist cannot assign a probability to a scientific hypothesis, say the existence of gravitons, because the proposition that gravitons exist is presumably either true or false and cannot be assessed by repeated sampling (e.g., tabulating universes to assess the fraction in which gravitons exist). Historically, frequentists have been willing to accept this limitation, restricting probability calculations to properties of random samples and other plainly stochastic events.
But Bayesians, beginning with and continuing with influential twentieth-century theorists such as,,, and, wanted to use the theory to support inferences about the probability of the (fixed, not random) state of the world based on the (random) data at hand, a paradigm referred to historically as “inverse probability.” In the frequentist view, the state of the world does not have a probability, because it has a fixed value and cannot be repeatedly sampled with different outcomes. (It is not a “random variable”.) Hence, instead of thinking of p( h) as the relative frequency of h, Bayesians think of it as the degree of belief that h is true, referred to as the subjectivist, epistemic, or Bayesian view. In the epistemic view, the uncertainty expressed by a probability value relates only to the observer's state of knowledge (not randomness in the world) and changes whenever this knowledge changes. Epistemic probabilities are not limited to events that can be repeated, and thus can be extended to propositions whose truth value is fixed but unknown, like the truth of scientific hypotheses. A Bayesian would happily assign a probability to the proposition that gravitons exist (e.g., p(gravitons exist) = 0.6), reflecting a net opinion about this proposition given the ensemble of knowledge and assumptions he or she finds applicable.
To Bayesians, frequencies (counts of outcomes) arise when the world is sampled, but they do not play a foundational role in defining probability. Indeed, Bayesians have often derided the “infinitely repeated random experiment” upon which frequentism rests as a meaningless thought experiment—impossible to observe, even in principle, in reality. As a consequence of this divergence in premises, frequentists tend to view probabilities as objective characteristics of the outside world, while Bayesians regard them as strictly mental constructs.
To frequentists, probabilities are real facts about the environment, about which observers can be right or wrong. But to Bayesians, probabilities simply describe a state of belief. To put it perhaps too coarsely: To frequentists, probabilities are facts, while to Bayesians they are opinions. This point was put perhaps most strikingly by Bruno de Finetti, a key figure in the twentieth-century renaissance of Bayesian inference, who began his Theory of probability with the phrase “PROBABILITY DOES NOT EXIST,” a sentence he insisted be typeset in all capital letters (see ). Why would a probability theorist make such a peculiar remark? What De Finetti meant was simply that probability is not an objective characteristic of the world, but rather a representation of our beliefs about it. Actual events, if we record them and tabulate the proportion of the time they occur (e.g., the number of heads divided by the number of tosses), are frequencies, not probabilities, and are only related to probabilities in a more indirect way (which Bayesians then debate at great length).
The mathematical rules of probability theory are about these beliefs and how they relate to each other and to evidence, not about frequencies of events in the outside world. Ipso facto, probabilities in general, and prior probabilities in particular, cannot be assessed by tabulating events. As put it (p. 916): “the phrase ‘estimating a probability ’ is just as much a logical incongruity as ‘assigning a frequency ’ or ‘drawing a square circle.’”, who first laid out the logic of modern Bayesianism, put it bluntly: “A prior probability is not a statement about frequency of occurrence in the world or any portion of it.”. What is the true value of a probability? These sentiments are so at odds with the contemporary common wisdom in cognitive science about probabilities—that they simply represent relative frequency of occurrence in the world—that the modern reader struggles to understand what was meant. But the core of the epistemic view is simply that probabilities are not objective characteristics of the outside world that have definite values.
Consider the simple example of baseball batting averages. What is the probability a given baseball player will get a “hit” at his next at-bat? By baseball convention, this probability is approximated by a tabulation of the player's past performance: hits divided by at-bats. But now the player steps up to the plate. What is the probability he will get a hit at this at-bat? The pitcher is left-handed, so we can improve our estimate by limiting the calculation to previous encounters with left-handed pitchers. It is a home game, so we can refine the estimate still further; a runner is on base; today is Sunday; and so forth.
Every additional factor further refines the estimate to a more comparable set of circumstances but also reduces the quantity of relevant data upon which to base our estimate. In the limit, every at-bat is unique, at which point the entire notion of generalizing from past experience breaks down. But even if we had infinite data, and plenty of data for each subcondition we might imagine, which of these subconditions is the right one—which ones gives the “true” probability of a hit today? A moment's thought suggests that there is no objectively correct answer to this question.
It depends on what factors are considered causally relevant, which depends on the observer's model of the situation, as causal influences cannot be definitively determined on the basis of experience alone. More notationally careful Bayesians (e.g.,, or ) often acknowledge this point by notating the prior on h as p( h a), rather than simply p( h), with a representing the ensemble of background knowledge or assumptions believed by the agent to be relevant to the prior probability of h.
(The prior is not “unconditional” as it is often described in informal treatments.) What you think about the prior on h depends on your model. And as in any inductive situation, there is no deductively certain model, but only a (perhaps infinite) collection of models that are inductively persuasive to various degrees—each of which potentially assigns a different probability to h. There is no right answer, only a range of plausible answers. The baseball problem is a variant of a problem discussed by the early frequentist John (glossed by as the “tubercular authoress from Scotland” problem). As a frequentist, Venn's solution was to insist that probabilities were only definable with respect to large ensembles of “similar” cases, never individual events—a restriction that severely limits the scope of probability theory, and which Bayesians do not accept. For example, to meet Venn's criteria for determining the probability of hitting safely, the batter would have to be tested over a long sequence of trials under identical conditions, much as Fisher (an even more dogmatic frequentist) was to propose several decades later as a method of carrying out experiments. But such a procedure would plainly preclude determining the probability of a hit from tabulations of past performance in actual baseball games.
But the normal way of computing baseball averages is perfectly coherent in the epistemic view. Previous performance (such as the proportion of hits in previous at-bats) is simply evidence influencing the observer's degree of belief that the batter will hit safely in his next at-bat—not, as in the frequentist view, an estimate of the “true” probability of a hit, which they would regard as meaningless. The Bayesian observer is free to take more factors into account, or fewer, depending on the chosen model of the situation, which determines which factors are believed relevant. In this view, the adopted probability of a hit, whether the batting average or some more refined estimate, is simply an estimate and not the “truth.” Probabilities are not true or false but simply characteristics of models (not of reality). It is perfectly reasonable to regard a coin as having heads probability 0.5, but what this really means is that our model of the coin is as a p =.5 Bernoulli process, and we believe, but cannot be sure, that our model is right. There is no ground truth. Probabilities do not have “true” values in the environment, and the Lord's prior does not exist.
It is important to understand that the epistemic view of probability is essential to the Bayesian program, and it cannot be lightly set aside without making inverse probability effectively impossible. What else, if not the “truth?” To summarize the argument so far: Frequentists think of events in the world as having definite objective probabilities. Many contemporary researchers, adopting Bayesian techniques but frequentist attitudes, consider the observer ideally tuned when it adopts as its prior the empirically “true” prior—a concept that does not, in fact, play any role in Bayesian theory. From a Bayesian point of view, priors are simply beliefs, informed by the observer's model and assumptions along with previously observed data, and are not, in principle, subject to empirical validation. Of course, while in Bayesian theory priors cannot literally be true or false, they certainly can influence the decisions the observer makes and thus the outcomes it enjoys. So what prior should the observer adopt?
One of the benefits of viewing probabilities epistemically is that it frees us from assuming that the answer to this question is automatically “the true one.” Instead, we are at liberty to consider the choice of prior in a more openended way. From an epistemic viewpoint, there may well be choices of prior that work better than the one that matches environmental relative frequencies. In the epistemic tradition, the observer is free to adopt whatever prior he or she wants for whatever reasons he or she wants—not just as the result of tabulation or measurement—and so we can ask which choice actually works best. Thus, it is quite conceivable that an observer under adaptive pressure to be “tuned” to the environment would do well not to adopt what we normally think of as the “true” prior. In a very concrete sense, posterior beliefs may be optimized with another prior. This may sound extremely counterintuitive, because it suggests the existence of a class of observers superior to ideal observers. But the mathematical argument is extremely straightforward, and indeed all its main elements are familiar from the Bayesian literature.
In what follows I develop this argument, showing that the choice of prior should be influenced by more than just the fit between the prior and the world. Quantifying the match between the head and the world From here on we denote by p( h) the “true” prior of h in the world, bearing in mind as discussed above that this really means that p( h) is the prior on h in the model of the world that we are working with. Pres A Ply Label Template more. Expectations taken relative to this prior should be thought of as reflecting not ground truth but a particular hypothetical model that we wish evaluate. Given that, we would like to quantify the discrepancy between p (the world) and a given prior q adopted by a particular observer (the head).
A conventional quantification of this match, adopted nearly universally in information-theoretic statistics (see ), is the Kullback-Leibler distance or divergence D( p║ q), defined as. (1) which can be thought of as the expectation (under p, ranging over hypotheses h i) of the log of the ratio between p and q.
The divergence is useful measure of the discrepancy between two priors because it quantifies the inefficiency of assuming q when p is in fact true. That is, it measures the number of extra bits required to encode the world via the observers' model q compared to the Shannon optimal code under p ().
I conducted a simple Monte Carlo simulation designed to measure the performance of observers with various priors q in a world actually governed by p. In this situation, an observer that assumes prior q equal to the true prior p is an “ideal observer” and has maximum probability of classifying observations correctly. The aim of the simulation is to see how performance varies as q is varied over the space of possible priors. The simulation assumes a simple classification task with data x ∈ R 2 generated by one of two sources A and B, each of which is circular Gaussian density with distinct means μ A, μ B ∈ R 2, and common variance σ 2. All these parameters are known to the observers except the priors.
Classes A and B occur in fact with probability p( A) and p( B) = 1 − p( A), respectively. Tested priors q run the full range of possible priors in step sizes of 0.2, with each prior evaluated 10,000 times and the results averaged. Shows performance (classification proportion correct) as a function of the divergence D( p ║ q) ranging over choices of q.
As one would expect, performance decreases linearly with divergence from the true prior: The ideal observer ( D( p ║ q)=0) is best, and others degrade as their assumptions increasingly diverge from that of the ideal. In the evolutionary simulacrum imagined by, adaptive pressure would urge organisms up this slope, minimizing divergence from the environment. (2) can be thought of as a measure of the symmetry of the probabilities and is maximized when they are all equal. The plot in shows that—collapsing over divergence—more entropic priors actually perform better. This is true for all three tested true priors, that is, regardless of their entropy.
This effect is thus independent of the degree to which the prior is tuned to the environment; assuming equal degree of tuning (i.e., divergence), the more entropic the prior, the more accurate the resulting classifications. The conventional intuition is that the “true” ecological prior provides ideal performance, but this simulation shows that this is not all there is to it. Regardless of the degree of tuning—and even for ideally tuned (zero divergence) observers—more entropic priors are better.
Shows a slightly more complex simulation with four classes instead of two. (This makes the prior space three-dimensional instead of one-dimensional, cubing the number of priors tested, so in this version only 5,000 trials were run per prior.) The influence of divergence is as before, and the effect of entropy is more clear than before.
Note that the larger number of classes is inherently more confusable, meaning that absolute ideal performance is worse than before (Bayes error is greater). But again ranging over the space of priors, more entropic priors lead to objectively superior performance. Bias and variance This suggests that tuning an organism to its environment involves somewhat more than collecting statistics from the environment, interpreting them as the true priors, and endowing the organism with them. Historical Bayesians raised a philosophical objection to this idea, and the above analysis provides a more tangible one. Mere tuning does not, in fact, optimize performance. Another way of looking at this is in terms of the degree to which we “believe” the data that the environment has provided us in the past.
If we have a small amount of data, the data are likely to include a fair amount of noise along with the signal. Even with the large data sets often used in the natural image statistics literature, the wobbles and wiggles of an empirically tabulated database are plainly visible in the plots. Do we think that each of these wobbles and wiggles represents a genuine and robust elevation or depression in the probability of conditions in the world? Common sense suggests not, and in this case, common sense is backed up by standard theory in the form of what is referred to as bias/variance or complexity/data-fit trade-off (; ). The bias/variance tradeoff is a simple consequence of the fact that more complex models (e.g., with more parameters or fudge factors) can generally fit data better than simpler ones, simply because the extra parameters can always be fit so that the loss function is reduced.
In the limit, a sufficiently complex model (e.g., a high enough dimension polynomial) can fit any data, even if the model is completely wrong. Fitting the data “too well” in this sense is called overfitting. At the opposite extreme, fitting the data too coarsely, with a model that is too simple, is called underfitting. Somewhere in the middle is a perfect balance, which, unfortunately, there is no general way of finding, because there is no absolute way of deciding what is signal and what is noise.
But generally to avoid overfitting, one must be willing to allow the data to be fit imperfectly, leaving some variance unexplained. Indeed, in any realistic situation, one does not really want to fit the data perfectly, because some of the data are noise—random fluctuations unlikely to be repeated. Overfitting thus inevitably leads to poor generalization, because some aspect of the learning was predicated on data that were unrepresentative of future data. For this reason, virtually every working inference mechanism includes (implicitly or explicitly) a damping process to restrain the complexity of models, sometimes referred to as regularization (see; ).
Overtuning to the environment In the context of fitting our observer to the world, what this means is that setting priors to match observed frequencies risks overfitting the world, or what might be called “overtuning” to environment. The conventional view is that one cannot “overtune”; the optimal observer is one whose prior matches the Lord's prior exactly, and the closer one can come to it, the better.
But in view of the bias/variance tradeoff, one would be unwise to fit one's prior too closely to any finite set of observations about how the world behaves, because inevitably the observations are a mixture of reliable and ephemeral factors. One may object that with a sufficiently large quantity of prior data, perhaps on evolutionary time scales and with learning encoded genetically, the prior can be estimated with arbitrarily high precision.
But this conception assumes a fixed, repeating Bernoulli sequence with a static prior—a fishbowl with an infinitely repeated probabilistic matrix. In practice, environmental conditions are not singular, perfect, and unchanging. In reality, probabilities vary over time, space, and context, in potentially unknown and unpredictable ways.
The environment inevitably contains uncertainty, not only about the classification of items on individual trials but about the nature of the probabilistic schema itself. To fit past experience perfectly is to overtune. Uncertainty about the environment For concreteness, one can imagine that the observer believes him- or herself to be in an environment where the true prior is p( h), but that he or she might be in an alternative (counterfactually nearby) environment with a slightly different true prior, whose value is randomly distributed about p( h). Equivalently, one can simply imagine that the prior is believed to be p( h) but that this belief is tempered by some uncertainty; in this conception, the true prior is a fixed but unknown value, and the prior distribution captures the observer's uncertainty about its value. The former scenario is more frequentist in “feel,” and the latter more subjectivist, but they are mathematically equivalent: Both can be cast mathematically in terms of a distribution of environments, with the priors governing them centered on a “population mean” plus some error distribution.
In either conception, our observer must contend with a prior whose value cannot be regarded as a fixed value but rather as a probability distribution over possible values. I modeled this situation in another Monte Carlo simulation by assuming that the “true” prior is itself chosen stochastically. First, we choose an imaginary prior p 0 (the “population mean” about which environments are chosen), and a vector e of probabilistic noise (components e i chosen uniformly from (0,1), then normalized). We then create the actual environmental prior p by mixing the mean with a quantity ε of noise. (3) The noise coefficient ε modulates the magnitude of uncertainty about the true nature of the environment.
Five levels of e were used,.1,.2,.3,.4, and.5. Zero noise ε = 0 corresponds to the previous simulation, results of which are included here for comparison. All other parameters are as before. Once the true prior has been chosen, we again evaluate priors q ranging over the space of possible priors and evaluate their performance in the chosen environment. Shows the results. The decrease in performance with divergence from the true prior is again visible, as is the increase with the entropy of the subjective prior. The novel element here is the modulation of this latter effect by ε, the magnitude of noise or uncertainty about the meta-environmental mean.
The more uncertainty, the better mean performance in this new (and more uncertain) environment. More specifically, given a fixed level of divergence from the true prior, the more uncertainty the observer's prior contains, the better its performance (see inset). In this sense, having a more “random” prior works better, even at a fixed level of ostensible tuning (divergence). Results of the simulation with various levels of uncertainty added to the true prior, showing (A) decrease in performance with divergence and (B) increase in performance with entropy, separated by the level ε of noise. Inset shows mean performance.
This plot suggests performance that is, in a very literal sense, superior to that of an ideal observer of the same environment. The well-versed reader will recoil at this characterization, because by definition performance cannot exceed that of the ideal. But the classical ideal observer presumes a perfectly well-defined environment, whose governing probabilities are fixed and invariant; indeed, the entire point of the construct is to model optimal performance given such knowledge. (Nothing proposed here involves performance superior to the ideal observer in the classical situation.) But in real circumstances, there is almost always uncertainty about the environment, outside of the imaginary world of the well-defined and infinitely repeated random experiment. The Elder Scrolls V Skyrim Crack Download Tpb here. Real environments exhibit uncertainty, not only about the outcomes of individual trials but also about the underlying governing probabilistic schema. In such environments, an “overlyidealized” ideal observer lacks robustness and can perform poorly when conditions stray from the assumptions.
The regularized ideal observer This leads to the idea of the regularized ideal observer, illustrated graphically in. The regularized ideal observer is really a spectrum of possible priors, with the classical ideal observer having D( p║ q) = 0 (i.e., p = q) at one pole and a completely entropic prior at the other. (This is not Jaynes' maximum-entropy prior, which already incorporates everything that is known, but the prior with maximum entropy given the hypothesis space alone—e.g., equal priors on all hypotheses.). The class of regularized priors, depicted as a “bead on a string” connecting the ecological prior to the point of maximum entropy. Sliding the bead all the way to the maximum-entropy point ignores past experience; sliding it all the way. As a concrete example, imagine you have a coin that has been flipped some large number of times and come up heads 57% of the time.
What prior should one use in the future? The “frequentist prior” is p( h) =.57. The answer based on symmetry considerations alone (two apparently similar sides, treat them equally) is p( h) = 0.5. The regularized ideal prior is the class of priors in between, that is, 0.5.
When probability noise is added to a prior, the result is (usually) a higher-entropy prior. Illustrates the same idea in “prior space,” the space of possible priors p (here depicted two-dimensionally, but the same applies in higher dimensions). If the true prior p is perturbed, the prior tends to move in the direction of maximum entropy. Our coin with observed frequency 57% heads might actually have higher than 57% probability of heads, but it is more likely to have less (i.e., closer to 50%), giving it higher entropy. This effect becomes more extreme the higher the dimension; the more degrees of freedom in the prior, the more likely they will blur when noise is added, resulting in greater entropy.
Conclusion The idea that perceptual and cognitive mechanisms derive their success in part from a meaningful connection to the statistics of the natural world, perhaps first suggested by, and greatly extended by many more recent authors, is a profound insight. An agent's choice of priors, implicitly entailed by its decisions and behavior, has tangible implications; if they are misset, performance is materially diminished. But how exactly do you set the priors to achieve optimal performance?
The argument in this article is that the contemporary fashion of setting them from tabulations in the environment has deep conceptual problems, invoking a hodgepodge of conflicting ideas from Bayesian and frequentist camps that would be accepted by neither. As Jorma Rissanen, the founder of Minimum Description Length theory, remarked about setting the prior: “[o]ne attempt is to try to fit it to the data, but that clearly not only contradicts the very foundation of Bayesian philosophy but without restrictions on the priors disastrous outcomes can be prevented only by ad hoc means” (, p. This article develops techniques for conceptualizing the prior that (a) avoid contradictions with the foundations of Bayesian inference, and (b) suitably restrict—or, more properly, regularize—its relation to the environment.
In the Bayesian tradition, the observer's prior may legitimately be based on any kind of knowledge or beliefs, including but not limited to data about frequencies. Of course, some subjective priors are better than others. From an evolutionary point of view, the best prior is one that maximizes adaptive fitness, not one that happens to agree with a relative frequencies in the environment (cf.; ). The main point of this article is that “Bayesian frequentist” attitudes—faith in the Lord's prior—are not only epistemologically naive but, moreover, risk overtuning. Overtuning, in turn, leads to a fragility of performance in future encounters with the same class of environments, which is maladaptive. Priors must be suitably regularized to truly optimize the fit between mind and world. 1The terminology is somewhat confusing because Bayesians are further divided into subjectivists, such as De Finetti, who thought of probabilities as characteristics of individual believers (sometimes called personalism) and objectivists such as Jaynes, who assume that all rational observers given identical data should converge on identical beliefs.
Nevertheless, it is important to understand that all historical Bayesians, subjectivists and objectivists alike, conceived of probabilities epistemically; they were all “subjectivists” in the broader sense. For example, Jaynes, an influential objectivist, spent much of his treatise () criticizing, even mocking, the frequentist view. Examples include (p. 916): “In our terminology, a probability is something that we assign, in order to represent a state of knowledge, or that we calculate from previously assigned probabilities according to the rules of probability theory. A frequency is a factual property of the real world that we measure or estimate” and continues (same page) “[P]robabilities change when we change our state of knowledge; frequencies do not.” Later (p. 1001), he derides the confusion between frequencies and probabilities, arguing forcefully against the idea that probabilities are physical characteristics of the outside world, concluding: “[D]efining a probability as a frequency is not merely an excuse for ignoring the laws of physics; it is more serious than that. We want to show that maintenance of a frequency interpretation to the exclusion of all others requires one to ignore virtually all the professional knowledge that scientists have about real phenomena.
If the aim is to draw inferences about real phenomena, this is hardly the way to begin.” 2Naturally, this contentious literature contains a variety of views of probability beyond frequentist and epistemic. Some early authors (e.g., Poisson; see ) use the word chance to refer to objective probabilities of events (sometimes called physical probability, see ), reserving probability for the epistemic sense. Further distinguishes credence (how strongly one believes a proposition) from epistemic probability (how strongly evidence supports it). A number of authors, notably including Karl, have argued for a view of probability as propensity, meaning the objective (not epistemic) tendency for an event to occur, defined in a way avoids the pitfalls of frequentism (see discussion). The historical debate concerning the interpretation of probability reflects fascinating and deeply held disagreements about the nature of induction. See for clear statements of several opposing philosophies, and for an in-depth history of the debate. 3To objectivist Bayesians, such as Jaynes, they are opinions that any rational observer would agree to when faced with the same data—but they are still beliefs, not facts; see.
4The full quotation (, p. X) is: “PROBABILITY DOES NOT EXIST. The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time,, or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.” 5Personal communication, 2007. 6The divergence is not necessarily symmetric (in general D( p ║ q)≠ D( q ║ p)), and often the average ([ D( p ║ q) + D( q ║ p)] /2 is used as a symmetric measure of distance between distributions. But note that here (and in similar contexts in the information-theoretic literature), it is the form given in that we want, because it takes expectations relative to the “truth” (model of the world) p, which is what we are interested in.