By Christophe Dabancourt

ISBN-10: 2212123507

ISBN-13: 9782212123500

Apprendre à programmer : Algorithmes et belief objet

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**Example text**

With common mean μ and finite variance A random variable X is said to have a Gaussian or normal distribution, with mean parameter μ and variance parameter σ 2 if 10 fX (x) = (2πσ 2 )−1/2 exp{−(x − μ )2 /(2σ 2 )} . If μ = 0 and σ = 1, Z is said to be a standard normal random variable. 12) where Φ (·) is the CDF for a Gaussian random variable with mean zero and unit variance. 13) for some function g of x (or sums of the analogous form, in the discrete case). 13) includes, for example, the expectation of X, its variance, and the probability that X ∈ A, for sets A ⊂ IR.

Xn or a new response value Y∗ corresponding to an observed predictor X∗ = x∗ . This form of estimation is typically called prediction. , Y = ±1), the predic- 13 The distinction between parametric and nonparametric models can be made more rigorous than this, but the definitions given here will suffice. 32 2 Preliminaries tion problem is called classification. Predictions too may take either point or set forms. Hypothesis testing involves weighing the evidence contained in observations, say x1 , .

Note that for n large enough, we will have the data-based estimator X) ¯ That is, with enough observations, the information in w ≈ 0, in which case μ¯ ≈ x. the data will overwhelm that in the prior; for lesser amounts of data, however, the prior information plays an increasingly greater role. Producing a Bayesian interval estimate of μ under our model, in analogy to a frequentist confidence interval, amounts to finding a set C such that P(μ ∈ C|X = x) = 1 − α . The probability underlying the definition of C here is with respect to the posterior distribution,19 and that, as we have already remarked, is Gaussian.

### Apprendre a programmer. Algorithmes et conception objet by Christophe Dabancourt

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