统计代写 | Statistical Learning and Decision Making代考|Conditional Distributions

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Conditional Distributions

The previous section introduced the idea of independence, which can help reduce the number of parameters used to define a joint distribution. However, as was mentioned, independence can be too strong of an assumption. This section will introduce the idea of conditional independence, which can help reduce the number of independent parameters without making assumptions that are as strong. Before discussing conditional independence, we will first introduce the notion of a conditional distribution, which is a distribution over a variable given the value of one or more others.

The definition of conditional probability states that
$$P(x \mid y)=\frac{P(x, y)}{P(y)}$$
where $P(x \mid y)$ is read as “probability of $x$ given $y$.” In some contexts, it is common to refer to $y$ as evidence.

Since a conditional probability distribution is a probability distribution over one or more variables given some evidence, we know that
$$\sum_{x} P(x \mid y)=1$$
for a discrete $X$. If $X$ is continuous, it integrates to 1 .
We can incorporate the definition of conditional probability into equation (2.18) to obtain a slightly different form of the law of total probability:
$$P(x)=\sum_{y} P(x \mid y) P(y)$$
for a discrete distribution.
Another useful relationship that follows from the definition of conditional probability is Bayes’ rule: ${ }^{12}$
$$P(x \mid y)=\frac{P(y \mid x) P(x)}{P(y)}$$
If we have a representation of a conditional distribution $P(y \mid x)$, we can apply Bayes’s rule to swap the $y$ and $x$ to obtain the conditional distribution $P(x \mid y)$.
We will now discuss a variety of ways to represent conditional probability distributions over discrete and continuous variables.

统计代写 | Statistical Learning and Decision Making代考|Discrete Conditional Models

A conditional probability distribution over discrete variables can be represented using a table. In fact, we can use the same discrete factor representation that we used in section $2.3 .1$ for joint distributions. Table $2.3$ shows an example of a table representing $P(X \mid Y, Z)$ with all binary variables. In contrast with a joint table (e.g, table $2.1$ ), the column containing the probabilities need not sum to 1. However, if we sum over the probabilities that are consistent with what we are conditioning on, we must get 1 . For example, conditioning on $y^{0}$ and $z^{0}$ (the evidence), we have
$$P\left(x^{0} \mid y^{0}, z^{0}\right)+P\left(x^{1} \mid y^{0}, z^{0}\right)=0.08+0.92=1$$ Conditional probability tables can become quite large. If we were to create a table like table $2.3$ where all variables can take on $m$ values and we are conditioning on $n$ variables, there would be $m^{n+1}$ rows. However, since the $m$ values of the variable we are not conditioning on must sum to 1 , there are only $(m-1) m^{n}$ independent parameters. There is still an exponential growth in the number of variables on which we condition. When there are many repeated values in the conditional probability table, a decision tree (introduced in section 2.3.1) may be a more efficient representation.

统计代写 | Statistical Learning and Decision Making代考|Conditional Linear Gaussian Models

The conditional linear Gaussian model combines the ideas of conditional Gaussian and linear Gaussian models to be able to handle conditioning a continuous variable on both discrete and continuous variables. Suppose we want to represent $p(X \mid Y, Z)$, where $X$ and $Y$ are continuous and $Z$ is discrete with values $1: n$. The conditional density function is then
$$p(x \mid y, z)= \begin{cases}\mathcal{N}\left(x \mid m_{1} y+b_{1}, \sigma_{1}^{2}\right) & \text { if } z^{1} \ \vdots \ \mathcal{N}\left(x \mid m_{n} y+b_{n}, \sigma_{n}^{2}\right) & \text { if } z^{n}\end{cases}$$
Above, the parameter vector $\theta=\left[m_{1: n}, b_{1: n}, \sigma_{1: n}\right]$ has $3 n$ components.

We can use a sigmoid ${ }^{13}$ model to represent a distribution over a binary variable conditioned on a continuous variable. For example, we may want to represent $P\left(x^{1} \mid y\right)$, where $x$ is binary and $y$ is continuous. Of course, we could just set a threshold $\theta$ and say $P\left(x^{1} \mid y\right)=0$ if $y<\theta$ and $P\left(x^{1} \mid y\right)=1$ otherwise. However, in many applications, we may not want to have such a hard threshold that results in assigning zero probability to $x^{1}$ for certain values of $y$.

Instead of a hard threshold, we could use a soft threshold that assigns low probabilities when below a threshold and high probabilities when above a threshold. One way to represent a soft threshold is to use a logit model, which produces a sigmoid curve:
$$P\left(x^{1} \mid y\right)=\frac{1}{1+\exp \left(-2 \frac{y-\theta_{1}}{\theta_{2}}\right)}$$
The parameter $\theta_{1}$ governs the location of the threshold, and $\theta_{2}$ controls the “softness” or spread of the probabilities. Figure $2.10$ shows an example plot of $P\left(x^{1} \mid y\right)$ with a logit model.

∑X磷(X∣是)=1

统计代写 | Statistical Learning and Decision Making代考|Conditional Linear Gaussian Models

p(X∣是,和)={ñ(X∣米1是+b1,σ12) 如果 和1 ⋮ ñ(X∣米n是+bn,σn2) 如果 和n

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