probability-daf-hl

# probability-daf-hl - CHAPTER 1 An Introduction to...

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C H A P T E R 1 An Introduction to Probability As the previous chapters have illustrated, it is often quite easy to come up with physical models that determine the effects that result from various causes — we know how image intensity is determined, for example. The diﬃculty is that effects could have come from various causes and we would like to know which — for example, is the image dark because the light level is low, or because the surface has low albedo? Ideally, we should like to take our measurements and determine a reasonable description of the world that generated them. Accounting for uncertainty is a crucial component of this process, because of the ambiguity of our measurements. Our process of accounting needs to take into account reasonable preferences about the state of the world — for example, it is less common to see very dark surfaces under very bright lights than it is to see a range of albedoes under a reasonably bright light. Probability is the proper mechanism for accounting for uncertainty. Axiomatic probability theory is gloriously complicated, and we don’t attempt to derive the ideas in detail. Instead, this chapter will first review the basic ideas of probability. We then describe techniques for building probabilistic models and for extracting information from a probabilistic model, all in the context of quite simple examples. In chapters ?? , 2, ?? and ?? , we show some substantial examples of probabilistic methods; there are other examples scattered about the text by topic. Discussions of probability are often bogged down with waffle about what prob- ability means , a topic that has attracted a spectacular quantity of text. Instead, we will discuss probability as a modelling technique with certain formal, abstract properties — this means we can dodge the question of what the ideas mean and concentrate on the far more interesting question of what they can do for us. We will develop probability theory in discrete spaces first, because it is possible to demonstrate the underpinning notions without much notation (section 1.1). We then pass to continuous spaces (section 1.2). Section 1.3 describes the important notion of a random variable, and section 1.4 describes some common probability models. Finally, in section 1.5, we get to probabilistic inference, which is the main reason to study probability. 1.1 PROBABILITY IN DISCRETE SPACES Probability models compare the outcomes of various experiments. These outcomes are represented by a collection of subsets of some space; the collection must have special properties. Once we have defined such a collection, we can define a proba- bility function. The interesting part is how we choose a probability function for a particular application, and there are a series of methods for doing this.

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