### 统计代写 | Statistical Learning and Decision Making代考|probabilistic reasoning

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写 | Statistical Learning and Decision Making代考|Degrees of Belief and Probability

In problems involving uncertainty, it is essential to be able to compare the plausibility of different statements. We would like to be able to represent, for example, that proposition $A$ is more plausible than proposition $B$. If A represents “my actuator failed”, and $B$ represents “my sensor failed”, then we would write $A \succ B$. Using this basic relation $\succ$, we can define several other relations:
\begin{aligned} &A \prec B \text { if and only if } B \succ A \ &A \sim B \text { if and only if neither } A \succ B \text { nor } B \succ A \ &A \succeq B \text { if and only if } A \succ B \text { or } A \sim B \ &A \preceq B \text { if and only if } B \succ A \text { or } A \sim B \end{aligned}
We want to make certain assumptions about the relationships induced by the operators $\succ, \sim$, and $\prec$. The assumption of universal comparability requires exactly one of the following to hold: $A \succ B, A \sim B$, or $A \prec B$. The assumption of transitivity requires that if $A \succeq B$ and $B \succeq C$ then $A \succeq C$. Universal comparability

and transitivity assumptions lead to an ability to represent plausibility by a realvalued function $P$ that has the following two properties: 3
\begin{aligned} &P(A)>P(B) \text { if and only if } A \succ B \ &P(A)=P(B) \text { if and only if } A \sim B \end{aligned}
If we make a set of additional assumptions 4 about the form of $P$, then we can show that $P$ must satisfy the basic axioms of probability (appendix A.2). If we are certain of $A$, then $P(A)=1$. If we believe $A$ is impossible, then $P(A)=0$. Uncertainty in the truth of $A$ is represented by values in between the two extrema. Hence, probability masses must lie between 0 and 1 with $0 \leq P(A) \leq 1$.

## 统计代写 | Statistical Learning and Decision Making代考|Discrete Probability Distributions

A discrete probability distribution is a distribution over a discrete set of values. We can represent such a distribution as a probability mass function, which assigns a probability to every possible assignment of its input variable to a value. For example, suppose we have a variable $X$ that can take on one of $n$ different values: $1, \ldots, n$, or, using colon notation, $1: n{ }^{6}$ A distribution associated with $X$ specifies the $n$ probabilities of the various assignments of values to that variable, in particular $P(X=1), \ldots, P(X=n)$. Figure $2.1$ shows an example of a discrete distribution.

Ihere are constraints on the probability masses associated with discrete distributions. The masses must sum to one:
$$\sum_{i=1}^{n} P(X=i)=1$$
and $0 \leq P(X=i) \leq 1$ for all $i$

For notational convenience, we will use lowercase letters and superscripts as shorthand when discussing the assignment of values to variables. For example, $P\left(x^{3}\right)$ is shorthand for $P(X=3)$. If $X$ is a binary variable, it can take on the value true or false. 7 We will use 0 to represent false and 1 to represent true. For example, we use $P\left(x^{0}\right)$ to represent the probability that $X$ is false.

The parameters of a distribution govern the probabilities associated with different assignments. For example, if we use $X$ to represent the outcome of a roll of a six-sided die, then we would have $P\left(x^{1}\right)=\theta_{1}, \ldots, P\left(x^{6}\right)=\theta_{6}$, with $\theta_{1: 6}$ being the six parameters of the distribution. However, we need only five independent parameters to uniquely specify the distribution over the outcomes of the roll because we know that the distribution must sum to 1 .

## 统计代写 | Statistical Learning and Decision Making代考|Continuous Probability Distributions

A continuous probability distribution is a distribution over a continuous set of values. Representing a distribution over a continuous variable is a little less straightforward than for a discrete variable. For instance, in many continuous distributions, the probability that a variable takes on a particular value is infinitesimally small. One way to represent a continuous probability distribution is to use a probability density function (see figure $2.2$ ), represented with lowercase letters. If $p(x)$ is a probability density function over $X$, then $p(x) d x$ is the probability $X$ falls within the interval $(x, x+d x)$ as $d x \rightarrow 0$. Similarly to how the probability masses associated with a discrete distribution must sum to 1 , a probability density function $p(x)$ must integrate to 1 :
$$\int_{-\infty}^{\infty} p(x) \mathrm{d} x=1$$
Another way to represent a continuous distribution is with a cumulative distribution function (see figure $2.3$ ), which specifies the probability mass associated with values below some threshold. If we have a cumulative distribution function $P$ associated with variable $X$, then $P(x)$ represents the probability mass associated with $X$ taking on a value less than or equal to $x . \Lambda$ cumulative distribution function can be defined in terms of a probability density function $p$ as follows:
$$\operatorname{cdf}{X}(x)=P(X \leq x)=\int{-\infty}^{x} p\left(x^{\prime}\right) \mathrm{d} x^{\prime}$$

Related to the cumulative distribution function is the quantile function, also called the inverse cumulative distribution function (see figure 2.4). The value of quantile ${ }_{X}(\alpha)$ is the value $x$ such that $P(X \leq x)=\alpha$. In other words, the quantile function returns the minimum value of $x$ whose cumulative distribution value exceeds $\alpha$. Of course, we have $0 \leq \alpha \leq 1$.

There are many different parameterized families of distributions. We outline several in appendix B. A simple distribution family is the uniform distribution $\mathcal{U}(a, b)$, which assigns probability density uniformly between $a$ and $b$, and zero elsewhere. Hence, the probability density function is $p(x)=1 /(b-a)$ for $x$ in the interval $[a, b]$. We can use $\mathcal{U}(x \mid a, b)$ to represent the density at $x .^{8}$ The support of a distribution is the set of values that are assigned non-zero density. In the case of $\mathcal{U}(a, b)$, the support is the interval $[a, b]$. See example 2.1.

∑一世=1n磷(X=一世)=1

## 统计代写 | Statistical Learning and Decision Making代考|Continuous Probability Distributions

∫−∞∞p(X)dX=1

cdf⁡X(X)=磷(X≤X)=∫−∞Xp(X′)dX′

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