15 - 2 - Gaussian Distribution (10 min).srt

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In this video, I'd like to
在这个视频中
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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talk about the Gaussian distribution, which
我将介绍高斯分布

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is also called the normal distribution.
也称为正态分布

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In case you're already intimately
如果你已经

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familiar with the Gaussian distribution, it is
对高斯分布非常熟悉了

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probably okay to skip this video.
那么也许你可以直接跳过这段视频

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But if you're not sure or
但是 如果你不确定

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if it's been a while since you've
或者你已经有段时间

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worked with a Gaussian distribution or the normal
没有接触高斯分布

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distribution then please do
或者正态分布了

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watch this video all the way to the end.
那么 请从头到尾看完这段视频

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And in the video after this,
在下一个视频中

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we'll start applying the Gaussian
我们将应用高斯分布

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distribution to developing an anomaly detection algorithm.
来推导一套异常检测算法

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Let's say x is a
假设x是一个

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real value random variable, so x is a real number.
实数随机变量 因此x是一个实数

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If the probability distribution of
如果x的概率分布

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x is Gaussian, it
服从高斯分布

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would mean Mu and variant
其中均值为μ

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sigma squared, then we'll
方差为σ平方

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write this as x the
那么将它记作

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random variable tilde.
随机变量x 波浪号

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That's this little tilde
这个小小的波浪号

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as distributed as and then
读作 服从...分布

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to denote the Gaussian Distribution, sometimes
为了表示高斯分布

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you're going to write script n, parentheses
有时你将使用大写字母N

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Mu, sigma squared.
括号μ σ平方

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So, this script's end stands for
因此这个大写字母N表示

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normal, since Gaussian and normal
Normal (正态)

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distribution, they mean the same
因为高斯分布就是正态分布

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phase of synonymous and a
他们是同义词

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Gussian distribution is parameterized
然后 高斯分布

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by 2 parameters, by a
有两个参数

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mean parameter which we
一个是均值

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denote Mu, and a variance
我们记作μ

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parameter, which we denote by sigma squared.
另一个是方差 我们记作σ平方

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If we pluck the Gaussian distribution
如果我们将高斯分布

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or Gaussian probability density, it
的概率密度函数绘制出来

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will look like the bell shaped
它看起来将是这样一个钟形的曲线

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curve, which you may have seen before.
大家之前可能就见过

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And so, this bell-shaped curve
这个钟形曲线

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is parameterized by those 2 parameters Mu and sigma.
有两个参数 分别是μ和σ

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And the location of the
其中μ控制

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center of this bell-shaped curve
这个钟形曲线

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is the mean Mu, and the
的中心位置

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width of this bell-shaped curve,
σ控制这个钟形曲线的宽度

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so it's roughly that, is
因此 参数σ

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the, this parameter
有时也称作

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sigma, is also called one standard deviation.
一个标准差

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And so, this specifies the
这条钟形曲线决定了

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probability of x taking
x取不同数值

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on different values, so x
的概率密度分布

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taking on values, you know
因此 x取中心这些值

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in the middle here is pretty high
的概率相当大

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since the Gaussian density here
因为高斯分布的概率密度

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is pretty high whereas
在这里很大

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x taking on values further and
而x取远处和更远处数值

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further away will be diminishing
的概率将逐渐降低

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in probability. Finally, just
直至消失

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for completeness, let me write
最后 为了讲述的完整性

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out the formula for the Gaussian
让我写下高斯分布的数学公式

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distribution so the property
x的概率分布

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of x and I'll
我有时不写p(x)

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sometimes write this instead of
我会用这个代替

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p of x, I'm going
我会写成

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to write this as p of
p 括号 x

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x semicolon Mu comma sigma squared.
分号 μ 逗号 σ 平方

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And so this denotes that the probability of
这个表示

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x is parametrized by
x的概率分布

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the two parameters Mu and sigma squared.
由两个参数控制μ和σ平方

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And the formula for the
高斯分布的概率密度公式

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Gaussian density is this,
是这样的

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1 over 2pi, sigma e
2π开方 乘以 σ 分之 1

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to the negative x minus Mu squared over 2 sigma squared.
乘以一个e的指数函数 其中指数项为 负的 x减μ的平方除以2倍的σ平方

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So there's no need to memorize this
其实我们并不需要

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formula, you know, this
记住这个公式

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is just the formula for the
它只是左边这条钟形曲线

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bell-shaped curve over here on the left.
对应的公式

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There's no need to memorize it and
我们没有必要记住它

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if you ever need to use this,
当我们真的需要用到它时

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you can always look this up.
我们总可以查资料找到它

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And so that figure on the
如果你选定

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left, that is what you get
μ值

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if you take a fixed
以及σ值

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value of Mu and a
然后绘制p(x)曲线

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fixed value of sigma and
那么你将得到

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you plot p of x. So this
左边这幅图

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curve here, this is really
因此这条曲线

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p of x plotted as a
其实就是

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function of x, you know,
给定μ值

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for a fixed value of Mu
以及σ平方 也就是方差值时

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and of sigma squared sigma squared, that's called the variance.
p(x)的函数图像

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And sometimes it's easier to think in terms of sigma.
也许有些时候我们使用σ会更方便

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So sigma is called the
而σ被称作

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standard deviation and it,
标准差

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so it specifies the
它确定了

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width of this Gaussian probability
高斯分布概率密度函数

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density whereas the square
的宽度

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of sigma, so sigma squared, is
而σ平方

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called the variance. Let's look
则称作方差

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at some examples of what the Gaussian distribution looks like.
让我们看几个高斯分布的图像

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If Mu equals zero, sigma equals 1.
如果μ取0 σ取1

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Then we have a Gaussian distribution
那么我们的高斯分布

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that is centered around zero, because that's Mu.
将以0为中心 因为μ等于0

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And the width of this Gaussian, so
而高斯分布的宽度

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that's one standard deviation is sigma over there.
将是一个标准差 也就是σ

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Let's look at some examples of
让我们看几个高斯分布的图像

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Gaussians. If Mu
如果μ取0

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is equal to zero it equals 1.
σ取1

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Then that corresponds to a
那么这将对应

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Gaussian distribution that is centered
一个以0为中心

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at zero since Mu is zero.
的高斯分布

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And the width of this Gaussian
而高斯分布的宽度

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is Gaussian thus controlled
高斯分布的宽度

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by sigma by that variance parameter sigma.
由标准差σ决定

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Here's another example.
来看另一个例子

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Let's say Mu is equal to
如果μ取0

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zero and sigma is equal to one-half.
σ取0.5

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So the standard deviation is
也就是说标准差

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one-half and the variance
是0.5

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sigma squared would therefore be
方差σ平方

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the square of 0.5 would be 0.25.
是0.5的平方 也就是0.25

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And in that case the Gaussian distribution,
这时候 高斯分布

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the Gaussian probability density looks
高斯分布的概率密度函数曲线

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like this, is also centered at zero.
会是这样的 以0为中心

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But now the width of
然而 现在它的宽度

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this is much smaller because
小了许多

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the smaller variance, the
因为方差变小了

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width of this Gaussian density
高斯密度函数的宽度

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is roughly half as wide.
大约是之前的一半

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But because this is a
但是 因为这是

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probability distribution, the area under
一个概率分布

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the curve, that is the shaded
因此曲线下的面积

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area there, that area
这些阴影区域的积分

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must integrate to 1.
一定是1

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This is a property of probability distributions.
这是概率分布的一个特性

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And so, you know, this
因此

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is a much taller Gaussian density because
这个高斯密度曲线更高

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it's half as wide, with
因为它只有一半宽

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half the standard deviation, but it's twice as tall.
只有一半的标准差 但是它有两倍高

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Another example, if sigma is
再看一个例子

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equal to 2, then you
如果σ等于2

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get a much fatter, or much wider Gaussian density.
那么你将得到一个更胖更宽的高斯密度曲线

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And so here, the sigma
在这里

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parameter controls that this
σ参数决定了

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Gaussian density has a wider width.
曲线会更宽

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And once again, the area under
同样的

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the curve, that is this shaded
曲线下方的面积 这快阴影区域

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area, you know, it always integrates to 1.
的积分一定是1

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That's a property of probability
这是概率分布的一个特性

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distributions, and because it's
因为它更宽

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wider, it's also half as
因此它只有一半高

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tall, in order to just integrate to the same thing.
这样积分才能保持不变

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And finally, one last example would be,
最后一个例子

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if we now changed the Mu
如果我们也改变参数μ

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parameters as well, then instead
那么曲线

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of being centered at zero, we
将不再以0为中心

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now we have a Gaussian distribution
现在我们的高斯分布

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that is centered at three, because
以3为中心

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this shifts over the entire Gaussian distribution.
因为整个高斯分布被平移了

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Next, lets take about the parameter estimation problem.
接下来 让我们来看参数估计问题

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So what is the parameter estimation problem?
那么 什么是参数估计问题？

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Let's say we have a data set
假设我们有一个数据集

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of m examples, so x1
其中有m个样本

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through x(m), and let's say
从x(1)到x(m)

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each of these examples is a real number.
假设他们都是实数

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Here in the figure, I've plotted an
在这幅图里

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example of a data set,
我画出了整个数据集

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so the horizontal axis is the
图中的横轴

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x axis and, you know, I
是x轴

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have a range of examples of x and I've just plotted them
我的样本x取值分布广泛

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on this figure here.
我就将它们画在这里

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And the parameter estimation problem is, let's
而参数估计问题就是

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say I suspect that these examples
假设我猜测这些样本

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came from a Gaussian distribution so
来自一个高斯分布的总体

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let's say I suspect that each of my example x(i) was distributed.
假设我猜测每一个样本xi服从某个分布

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That's what this tilde thing means.
这里的波浪号表示 服从...分布

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Thus, I suspect that each of
因此 我猜测

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these examples was distributed according
这里的每个样本

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to a normal distribution or
服从正态分布

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Gaussian distribution with some
或者高斯分布

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parameter Mu and some parameter sigma squared.
它有两个参数 μ和σ平方

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But I don't know what the values of these parameters are.
然而 我不知道这些参数的值是多少

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The problem with parameter estimation is,
参数估计问题就是

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given my data set I want
给定数据集

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to figure out, I want to
我希望能找到

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estimate, what are the
能够估算出

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values of Mu and sigma squared.
μ和σ平方的值

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So, if you're given a
因此 如果你有

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data set like this, you know,
这样一个数据

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it looks like maybe, if I
它看起来好像

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estimate what Gaussian distribution the
如果我试图找到

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data came from, maybe that
它来自哪个高斯分布

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might be roughly the Gaussian distribution
也许这个就是

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it came from, with Mu
它对应的高斯分布

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being the center of the distribution and
其中μ对应分布函数的中心

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sigma the standard deviation controlling the width of this Gaussian distribution.
而标准差σ控制高斯分布的宽度

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It seems like a reasonable
这条曲线似乎

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fit to the data, because, you know, it
很好的拟合了数据

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looks the data has
因为看起来

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a very high probability of being
这个数据集

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in the central region, low probability of
在中心区域的概率比较大

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being further out, low probability of being further out, and so on.
而在外围 在边缘的概率越来越小

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And so, maybe this is
因此 也许这是

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a reasonable estimate of
对μ和σ平方

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Mu and of sigma squared,
的一个不错的估计

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that is, if it corresponds to
也就是说

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a Gaussian distribution, that then looks like this.
我们的数据对应这样一个高斯分布

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So, what I'm going to
那么 接下来

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do is write out the
我将写下

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formulas, the standard formulas
对μ和σ平方

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for estimating the parameters from
进行参数估计的

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Mu and sigma squared.
标准公式

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The way we are going to
我们估计μ的方法是

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estimate Mu is going to
对我的

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be just the average
所有样本

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of my example.
求平均值

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So Mu is the mean parameter,
μ就是平均值参数

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so I'm going to take my
因此 我将

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training set, take my m
使用我的训练集

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examples and average them.
使用我的m个样本 对它们取平均

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And that just gives me the center of this distribution.
这样我就得到了高斯分布的中心位置

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How about sigma squared?
那么如何估计σ平方呢？

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Well the variance, I'll just
σ平方表示方差

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write out the standard formula again,
我们也来写下方差的标准计算公式

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I'm going to estimate as sum
我先将所有的样本

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of 1 through m of x(i),
从x(1)到x(m)

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minus Mu squared,
减去平均值μ

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and so this Mu here is
平方再求和

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actually the Mu that I compute
实际上μ是用

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over here using this formula,
之前的这个公式计算出来的

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and what the variance is, or
而方差的含义

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one interpretation of the variance,
至少一种方差的定义

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is that, if you look at the
是将这一项

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this term, that's the square
所有样本

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difference between the value
的差值平方和

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I've got in my example minus
我已经将样本减去了平均值

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the mean, minus the center, minus the mean of distribution.
减去了中心位置 分布的平均值

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And so, you know, the
因此 你看

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variance, I'm gonna estimate
我会将方差估计为

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as just the average of
样本减去平均值

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the square differences, between my examples, minus the mean.
差值取平方 再求平均

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And as a side comment,
这里顺便提一下

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only for those of you that are experts
你们有些人

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in statistics, if you're
可能精通统计学

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an expert in statistics and if
如果你是统计方面的专家

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you've heard of maximum likelihood estimation,
听说过极大似然估计

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then these estimates are actually the
那么这里的估计

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maximum likelihood estimates of the parameters
实际就是对μ和σ平方

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of Mu and sigma squared.
的极大似然估计

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But if you haven't heard of that before, don't worry about it.
不过如果你没听说过这些 没关系

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All you need to know is that
你只需要知道

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these are the two standard formulas
这里有两个标准公式

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for how you try to
给定数据集 你可以利用它们

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figure out what our Mu and sigma squared given the dataset.
估算出μ和σ平方的值

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Finally one last side comment.
最后 再提一句

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Again only for those of
同样是针对那些

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you that has maybe taken a statistics class before.
学习过统计课程的人

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But if you have taken a statistics
如果你以前上过统计课程

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class before, some of you
你们可能会见过

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may have seen the formula here
这么一个公式

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where, you know, this is m minus
在这里不是m

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1, instead of m. So
而是m减1

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this first term becomes 1
第一项写成

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over m minus 1, instead
m减1分之1

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of 1 over m. In machine
而不是m分之1

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learning, people tend to use this 1 over m formula.
在机器学习领域 大家喜欢用m分之1

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But in practice, whether it
然而在实际使用中

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is 1 over m or 1
到底是选择使用

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over m minus one, makes essentially
m分之1还是m减1分之1

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no difference, assuming, you know, m is
其实区别很小

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reasonably large, it's a large training set size.
只要你有一个还算大的训练集

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So, just in case you've seen
因此 如果你见过m减1分之1

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this other version before, in either
这个版本的公式

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version it works just equally
它同样可以很好的估计出参数

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well, but in machine
只是在机器学习领域

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learning most people tend to
大部分人更习惯使用

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use 1 over m in this formula.
m分之1这个版本的公式

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And the two versions have slightly
这两个版本的公式

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different theoretical properties, slightly different
在理论特性和数学特性上稍有不同

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mathematical properties, but in
但是在实际使用中

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practice it really makes very little difference, if any.
他们的区别甚小 几乎可以忽略不计

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So, hopefully, you now have
现在 我想

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a good sense of what the
你大概已经对高斯分布的样子

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Gaussian distribution looks like,
有些感觉了

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as well as how to estimate
你也知道

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00:10:02,270 --> 00:10:03,730
the parameters, mu and sigma
如何估计高斯分布中的

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00:10:04,010 --> 00:10:05,770
squared, of the Gaussian distribution, and
参数μ和σ平方

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00:10:05,910 --> 00:10:07,510
if you're given a training set,
只要你有一个训练集

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00:10:07,850 --> 00:10:08,940
that is if you're given a
如果你猜测它

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00:10:09,240 --> 00:10:10,220
set of data that you suspect
来自一个高斯分布

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comes from a Gaussian
你就可以估计出它的参数值

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00:10:12,410 --> 00:10:15,190
distribution with unknown parameters using sigma squared.
μ和σ平方

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00:10:16,190 --> 00:10:17,510
In the next video, we'll start
在下一个视频中

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to take this and apply it
我们将利用这些知识

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to develop the anomaly detection algorithm.
推导出异常检测算法