亚洲免费在线-亚洲免费在线播放-亚洲免费在线观看-亚洲免费在线观看视频-亚洲免费在线看-亚洲免费在线视频

ML| EM

系統(tǒng) 2240 0

What's xxx

The EM algorithm is used to find the maximum likelihood parameters of a statistical model in cases where the equations cannot be solved directly. Typically these models involve latent variables in addition to unknown parameters and known data observations. That is, either there are missing values among the data, or the model can be formulated more simply by assuming the existence of additional unobserved data points.?

The motivation is as follows. If we know the value of the parameters $\boldsymbol\theta$, we can usually find the value of the latent variables $\mathbf{Z}$ by maximizing the log-likelihood over all possible values of $\mathbf{Z}$, either simply by iterating over $\mathbf{Z}$ or through an algorithm such as the Viterbi algorithm for hidden Markov models. Conversely, if we know the value of the latent variables $\mathbf{Z}$, we can find an estimate of the parameters $\boldsymbol\theta$ fairly easily, typically by simply grouping the observed data points according to the value of the associated latent variable and averaging the values, or some function of the values, of the points in each group. This suggests an iterative algorithm, in the case where both $\boldsymbol\theta$ and $\mathbf{Z}$ are unknown:

  1. First, initialize the parameters $\boldsymbol\theta$ to some random values.
  2. Compute the best value for $\mathbf{Z}$ given these parameter values.
  3. Then, use the just-computed values of $\mathbf{Z}$ to compute a better estimate for the parameters $\boldsymbol\theta$. Parameters associated with a particular value of $\mathbf{Z}$ will use only those data points whose associated latent variable has that value.
  4. Iterate steps 2 and 3 until convergence.

The algorithm as just described monotonically approaches a local minimum of the cost function, and is commonly called hard EM . The k-means algorithm is an example of this class of algorithms.

However, we can do somewhat better by, rather than making a hard choice for $\mathbf{Z}$ given the current parameter values and averaging only over the set of data points associated with a particular value of $\mathbf{Z}$, instead determining the probability of each possible value of $\mathbf{Z}$ for each data point, and then using the probabilities associated with a particular value of $\mathbf{Z}$ to compute a weighted average over the entire set of data points. The resulting algorithm is commonly called soft EM, and is the type of algorithm normally associated with EM.?

With the ability to deal with missing data and observe unidentified variables, EM is becoming a useful tool to price and manage risk of a portfolio.

Algorithm

Given a statistical model consisting of a set $\mathbf{X}$ of observed data, a set of unobserved latent data or missing values $\mathbf{Z}$, and a vector of unknown parameters $\boldsymbol\theta$, along with a likelihood function $L(\boldsymbol\theta; \mathbf{X}, \mathbf{Z}) = p(\mathbf{X}, \mathbf{Z}|\boldsymbol\theta)$, the maximum likelihood estimate (MLE) of the unknown parameters is determined by the marginal likelihood of the observed data

$L(\boldsymbol\theta; \mathbf{X}) = p(\mathbf{X}|\boldsymbol\theta) = \sum_{\mathbf{Z}} p(\mathbf{X},\mathbf{Z}|\boldsymbol\theta) $
However, this quantity is often intractable (e.g. if $\mathbf{Z}$ is a sequence of events, so that the number of values grows exponentially with the sequence length, making the exact calculation of the sum extremely difficult).

The EM algorithm seeks to find the MLE of the marginal likelihood by iteratively applying the following two steps:

1. Expectation step (E step) : Calculate the expected value of the log likelihood function, with respect to the conditional distribution of $\mathbf{Z}$ given $\mathbf{X}$ under the current estimate of the parameters $\boldsymbol\theta^{(t)}$:
$Q(\boldsymbol\theta|\boldsymbol\theta^{(t)}) = \operatorname{E}_{\mathbf{Z}|\mathbf{X},\boldsymbol\theta^{(t)}}\left[ \log L (\boldsymbol\theta;\mathbf{X},\mathbf{Z}) \right] \,$
2. Maximization step (M step): Find the parameter that maximizes this quantity:
$\boldsymbol\theta^{(t+1)} = \underset{\boldsymbol\theta}{\operatorname{arg\,max}} \ Q(\boldsymbol\theta|\boldsymbol\theta^{(t)}) \, $
Note that in typical models to which EM is applied:

  • The observed data points $\mathbf{X}$ may be discrete (taking values in a finite or countably infinite set) or continuous (taking values in an uncountably infinite set). There may in fact be a vector of observations associated with each data point.
  • The missing values (aka latent variables) $\mathbf{Z}$ are discrete, drawn from a fixed number of values, and there is one latent variable per observed data point.
  • The parameters are continuous, and are of two kinds: Parameters that are associated with all data points, and parameters associated with a particular value of a latent variable (i.e. associated with all data points whose corresponding latent variable has a particular value).

ML| EM


更多文章、技術(shù)交流、商務(wù)合作、聯(lián)系博主

微信掃碼或搜索:z360901061

微信掃一掃加我為好友

QQ號聯(lián)系: 360901061

您的支持是博主寫作最大的動力,如果您喜歡我的文章,感覺我的文章對您有幫助,請用微信掃描下面二維碼支持博主2元、5元、10元、20元等您想捐的金額吧,狠狠點擊下面給點支持吧,站長非常感激您!手機微信長按不能支付解決辦法:請將微信支付二維碼保存到相冊,切換到微信,然后點擊微信右上角掃一掃功能,選擇支付二維碼完成支付。

【本文對您有幫助就好】

您的支持是博主寫作最大的動力,如果您喜歡我的文章,感覺我的文章對您有幫助,請用微信掃描上面二維碼支持博主2元、5元、10元、自定義金額等您想捐的金額吧,站長會非常 感謝您的哦!!!

發(fā)表我的評論
最新評論 總共0條評論
主站蜘蛛池模板: 五月激情六月婷婷 | 五月婷婷激情网 | 欧美日韩国产三级 | 国产成人乱码一区二区三区在线 | 在线看的成人性视频 | 99re热这里只有精品视频 | 亚洲国产欧洲综合997久久 | se色成人亚洲综合 | 香蕉超级碰碰碰97视频蜜芽 | 99热视热频这里只有精品 | 久久女人天堂 | 国产精品女仆装在线播放 | 91精品一区二区三区久久久久 | 一级日本特黄毛片视频 | 国产区精品一区二区不卡中文 | 免费中文字幕 | 26uuu最新地址| 久久99精品久久久久久黑人 | 久久亚洲私人国产精品va | 国产女人天堂 | 一级爱爱片一级毛片-一毛 一级白嫩美女毛片免费 | 老司机福利在线播放 | 18视频在线观看 | 26uuu最新地址 | 激情综合五月亚洲婷婷 | 成人a毛片视频免费看 | 91最新视频在线观看 | 久久99热这里只有精品7 | 婷婷 综合| 成人亚洲视频 | 久久亚洲国产精品一区二区 | 免费视频精品一区二区三区 | 国产精品一区视频 | 国产无卡一级毛片aaa | 青娱乐伊人 | 天天干天天干 | 久久综合一本 | 国产精品亚洲第五区在线 | 在线播放人成午夜免费视频 | 高清国产美女一级毛片 | 亚洲综合色婷婷 |