(1) The number of possible basic events in the experiment is limited;
(2) The probability of each basic event in the test is equal.
There are a large number of probability models with the above two characteristics, which are called classical probability model, referred to as classical probability model for short, and also called equal possibility probability model.
Limited (only a limited number of all possible basic events)
Equal probability (the probability of each basic event is equal)
A function can be called a probability density function if it meets the following conditions:
The distribution function is the variable upper bound integral of the probability density function, which is defined as:
For any b>0
Let (x, σ, μ) be a measure space and f be a generalized real-valued measurable function defined on X, then for any real number t>0, it has:
Generally speaking, if g is a non-negative generalized real-valued measurable function and does not fall within the definition domain of f, there are:
Let x be a random variable, the expected value is u, and the standard deviation is σ. For any real number k>0
The more experiments, the more the average value of samples tends to the average value of the whole population.
With the increase of test times, the distribution of sample mean tends to normal distribution.
Parameter estimation, a statistical inference. The process of estimating unknown parameters in the population distribution according to random samples extracted from the population.
From the form of estimation, it can be divided into point estimation and interval estimation:
From the methods of constructing estimators, there are moment estimation, least square estimation, likelihood estimation, Bayesian estimation and so on.
(1) Find the estimator of unknown parameters;
(2) Point out the accuracy of the estimator under a certain reliability.
Among them, the reliability is generally expressed by probability, such as the reliability is 95%; Accuracy is measured by the proximity or error between the estimator and the estimated parameter (or the parameter to be estimated).
Sample moments are used to estimate the population moments, so as to obtain the estimation of parameters in the population distribution. Its ideological essence is to replace the distribution and moment of the population with the empirical distribution and moment of the sample. The advantage of moment estimation method is that it is simple and easy to operate, and it does not need to know what distribution the population is in advance. The disadvantage is that when the group type is known, the information provided by distribution is not fully utilized. Generally speaking, the moment estimator is not unique.
It was put forward by British statistician R.A. Fisher in 19 12, and the likelihood function was constructed by sample distribution density, and the maximum likelihood estimation of parameters was obtained.
The least square method describes the idea that the loss formed by the true value of the parameter to be estimated and the data points of the actual sample should be minimal. That is, the truth value is the least square of the total error, which is based on the fact that if the error is random, it should fluctuate around the truth value.
Refer to Zhihu user fine-tuning
According to past experience and analysis. Probability that can be obtained before experiment or sampling.
Transcendental probability refers to the probability obtained from past experience and analysis, such as the total probability formula, which often appears in the form of "cause" in the problem of "finding the result from the cause"
It means that something has happened, and you have to calculate the probability that the cause of this thing is caused by a certain factor.
Posterior probability refers to what kind of event is most likely to happen according to the obtained "result" information, just like Bayesian formula, which is the "reason" in the problem of "finding the reason"
Therefore, the accuracy of prediction is improved. Most machine learning models try to get posterior probability.
Let's give a few more examples to understand:
Recently, the weather is very hot. I went to the supermarket and bought a watermelon. I don't have much experience and don't know how to choose a mature melon. At this time, as a science student, oh, there are such considerations:
If I don't know anything about this watermelon, including color, shape, and whether the pedicle falls off. According to common sense, the probability of watermelon ripening is about 60%. Then, this probability p (ripe melon) is called prior probability.
In other words, the prior probability is the probability obtained from past experience and analysis, which does not need sample data and is not affected by any conditions. Just like Redstone only judges whether watermelon is ripe according to common sense rather than the state of watermelon, this is a priori probability.
Let's take another look. Hongshi learned a common sense to judge whether watermelon is ripe or not, that is, to see if the pedicle falls off. Generally speaking, when the pedicel falls off, the probability of watermelon ripening is high, about 75%. If pedicel abscission is regarded as a result, and then the probability of watermelon ripening is inferred, this probability P (melon ripening | pedicel abscission) is called posterior probability. Posterior probability is similar to conditional probability.
Those who play League of Legends account for 60% of China's total population, and those who don't play League of Legends account for 40%:
For the convenience of mathematical description, here we use the variable x to represent the value. According to the definition of probability and addition principle, we can write the following expression:
P(X= playing lol)= 0.6；; P(X= don't play lol)=0.4, this probability is obtained statistically, that is, the probability distribution of x is known, which we call prior probability;
In addition, 80% of people who play lol are men, 20% are little sisters, and 20% of people who don't play lol are men and 80% are little sisters. Here I use the discrete variable y to represent the gender value and write the corresponding conditional probability distribution:
P(Y= male |X= playing lol)=0.8, P(Y= little sister |X= playing lol)=0.2.
P(Y= male |X= not playing lol)=0.2, P(Y= little sister |X= not playing lol)=0.8.
Then I want to ask what is the probability that the player is a lol player when the player is known to be a male:
According to Bayesian criterion, we can get:
P(X= playing lol|Y= male) =P(Y= male |X= playing lol)*P(X= playing lol)/
[P(Y= male |X= playing lol) P(X= playing lol)+P(Y= male |X= not playing lol) P(X= not playing lol)]
The last calculated P(X= lol|Y= male) is called the posterior probability of x, which is obtained after observing the occurrence of event Y.
Lao Wang next door is going to a place 10 kilometers away on business. He can choose to walk, ride a bike or drive. It will take him some time to reach his destination.
In this incident, the mode of transportation (walking, cycling or driving) can be regarded as the cause and the time spent can be regarded as the result. If it took Lao Wang an hour to complete the journey of 10 km, he probably rode it. Of course, it is unlikely that Lao Wang was a fitness expert, or drove there but the traffic jam was serious.
If Lao Wang * * * completed the distance of 10 km in two hours, he probably walked. If Lao Wang only spent 20 minutes, it was probably driving. This probability distribution P (mode of transportation | time) is the posterior probability.
Lao Wang feels good when he gets up in the morning. He wanted to exercise and decided to run there. It is also possible that Lao Wang wants to be a literary youth, tries to enjoy the recent popular riding and decides to ride. Maybe Lao Wang wanted to show off his wealth and decided to drive. Lao Wang's choice has nothing to do with the time of arrival at his destination. Determine the probability distribution of the cause before the result, and P (mode of transportation) is the prior probability.
Posterior probability cannot be obtained directly, so we need to find a way to calculate it. The solution is to introduce Bayesian formula. The expression of posterior probability is called conditional probability, which is generally written as p(A|B), that is, the probability that A only occurs when event B occurs. We can easily get it from the formula of conditional probability.
The posterior probability can be calculated by the Bayesian formula above.
When the current state and all past states of a stochastic process are given, the conditional probability distribution of its future state only depends on the current state; In other words, when the current state is given, it is conditionally independent of the past state (that is, the historical path of the process), then this stochastic process has Markov properties. Processes with Markov properties are usually called Markov processes.
In the field of automatic control, Lyapunov stability can be used to describe the stability of a dynamic system. If the trajectory of any initial condition of this dynamic system can be kept near the equilibrium state, it can be called local Lyapunov stability.
If the trajectory of any initial condition near the equilibrium state finally approaches, then the system can be said to be asymptotically stable everywhere. Exponential stability can be used to ensure the minimum attenuation rate of the system and also to estimate the speed of trajectory convergence.
Lyapunov stability can be used for linear and nonlinear systems. However, the stability of linear systems can be obtained in other ways, so Lyapunov stability is mainly used to analyze the stability of nonlinear systems. The concept of Lyapunov stability can be extended to infinite manifolds, that is, structural stability, which is the behavior of considering a group of different but "close" solutions in differential equations. Input State Stability (ISS) is the application of Lyapunov stability to systems with inputs.
There are two hypotheses in statistics, the original hypothesis and the alternative hypothesis.
When the original hypothesis will happen, we calculate the p value ==0.05, think that this event will not happen, and reject the original hypothesis. At this time, we will make the first mistake. (Violation of the principle of small probability events)
When the original hypothesis will not happen, we calculate the value of p ==0.95, think that this event will happen, accept the original hypothesis, and then we will make the second mistake. (Violation of the principle of small probability events)
In the actual hypothesis testing, the first kind of errors should be avoided first, followed by the second kind of errors.
Because the hypothesis test is biased, the correctness of the hypothesis test is higher only when the hypothesis test rejects the original hypothesis (according to the conclusion of small probability events). When rejecting the original hypothesis, we should try to avoid false rejection, so we should avoid the first mistake.
Hypothesis test is "weak in conclusion" when accepting the original hypothesis. Under normal circumstances, we will not let this result appear and try to reject the original hypothesis.
/p/55780975
/p/26464206
/p/39 125269
/Question/24261751Answer/158547500
Probability theory and mathematical statistics, fourth edition