转发：概率（probability)和似然（likelihood)的共同点和区别

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概率（probability)和似然（likelihood)，都是指可能性，都可以被称为概率，但在统计应用中有所区别。    概率是给定某一参数值，求某一结果的可能性的函数。
    例如，抛一枚匀质硬币，抛10次，6次正面向上的可能性多大？
    解读：“匀质硬币”，表明参数值是0.5，“抛10次，六次正面向上”这是一个结果，概率（probability)是求这一结果的可能性。
    似然是给定某一结果，求某一参数值的可能性的函数。
    例如，抛一枚硬币，抛10次，结果是6次正面向上，其是匀质的可能性多大？
    解读：“抛10次，结果是6次正面向上”，这是一个给定的结果，问“匀质”的可能性，即求参数值=0.5的可能性。
   出处：http://blog.sina.com.cn/s/blog_e8ef033d0101oa4k.html

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关键词：Probability Likelihood bability Ability like 共同点

去学测度，观点层面马上提高！

iharpst 发表于 2014-2-10 23:54
去学测度，观点层面马上提高！

弱弱地问一句：“测度”是一门学科吗？

耕耘使者 发表于 2014-2-11 00:26
弱弱地问一句：“测度”是一门学科吗？

至少是一本书，当然可以开相关的课

耕耘使者 发表于 2014-2-11 00:26
弱弱地问一句：“测度”是一门学科吗？

也有的翻译为“测量” measurement。。都可以的

多谢两位朋友！

iharpst 发表于 2014-2-10 23:54
去学测度，观点层面马上提高！

测度论~~~~好难得

The answer depends on whether you are dealing with discrete or continuous random variables.

Discrete Random Variables

Suppose that you have a stochastic process that takes discrete values (e.g., outcomes of tossing a coin 10 times, number of customers who arrive at a store in 10 minutes etc). In such cases, we can calculate the probability of observing a particular set of outcomes by making suitable assumptions about the underlying stochastic process (e.g., probability of coin landing heads is p and that coin tosses are independent).

Denote the observed outcomes by O and the set of parameters that describe the stochastic process as θ. Thus, when we speak of probability we want to calculate P(O|θ). In other words, given specific values for θ, P(O|θ) is the probability that we would observe the outcomes represented by O.

However, when we model a real life stochastic process, we often do not know θ. We simply observe O and the goal then is to arrive at an estimate for θ that would be a plausible choice given the observed outcomes O. We know that given a value of θ the probability of observing O is P(O|θ). Thus, a 'natural' estimation process is to choose that value of θ that would maximize the probability that we would actually observe O. In other words, we find the parameter values θ that maximize the following function:

L(θ|O)=P(O|θ)

L(θ|O) is called as the likelihood function. Notice that by definition the likelihood function is conditioned on the observed O and that it is a function of the unknown parameters θ.

Continuous Random Variables

In the continuous case the situation is similar with one important difference. We can no longer talk about the probability that we observed O given θ as in the continuous case P(O|θ)=0. Without getting into technicalities, the basic idea is as follows:

Denote the probability density function (pdf) associated with the outcomes O as: f(O|θ). Thus, in the continuous case we estimate θ given observed outcomes O by maximizing the following function:

L(θ|O)=f(O|θ)

In this situation, we cannot technically assert that we are finding the parameter value that maximizes the probability that we observe O as we maximize the pdf associated with the observed outcomes O.

Lisrelchen 发表于 2014-2-23 22:44
The answer depends on whether you are dealing with discrete or continuous random variables. Discrete ...

非常感谢！

你搞贝叶斯的话，会对这个东西理解的更好。 prob, likelihood, prior之类的。