By understanding the real world with the concept of the network, we can know the validity of all theories and the convergence attenuation as a hierarchy. Pen @ fun @ pavilion wWw. ｂｉｑUｇE。 The info network cannot be like a clock, which can have time inversion and comparative certainty, because the network is a multi-dimensional coupling, and when the number of iterations exceeds a certain threshold, it will decay and converge, which is chaos. At the same time, it is also an error. So we end up using statistical models to understand the network and, by extension, the reality.

The statistical nature of the data is based on large-scale, random data

It is almost impossible to explore the role of a single variable because the role of the network is coupled

Regression to the mean is the achievement of another dimension of dynamic equilibrium, and the average is an intrinsic characteristic, which maintains the overall homeostasis and avoids polarization

We substitute the formula by substituting the original solution of a set of distributed data: the observations are always random, but the overall observations appear in a high-dimensional structure

A phenomenon is the result of an observation, and its essence is the distribution function (probability distribution) behind it: the Poisson distribution

Goodness-of-fit test: Determines whether a given set of observations is appropriate for a particular mathematical distribution function

Monte Carlo technique: a mathematical model that is repeatedly simulated to determine the probability distribution of relevant data (a means of traversing in which the proportions of various partial derivatives conform to a certain distribution function)

High-dimensional data: The sample is large enough to determine the parameters without error----- the treatment of small sample random errors: the ratio of the mean and standard deviation estimates, Pearson's four parameters (mean and standard deviation, symmetry, and kurtosis) are matched to one of the distributions in K. Pearson's skewed distribution series. The ratio of the first two parameter estimates has a tabulable probability distribution, and the ratio of the two sample estimates is calculated to give a known distribution.

The basic assumption is that the raw measurements follow a normal distribution.

Complex iterative formulas are transformed into multi-dimensional geometric space forms

The statistical distribution of various parameters is a high-dimensional structure, such as the continuous change of distribution parameters is the true essence of evolution

It is hypothesized that these phenotypes are the result of interactions between genes, which in turn have different probabilities

The multivariate influence of the network, through a certain constraint to divide the module, we make the relationship of a certain path appear through random adjustment, and establish a large number of correlation effects of interrelated causes

Breaking down the effects of different treatments: Fischer's ANOVA, an analysis of interactions

Degrees of freedom reconcile the varied and anomalous results observed by different authors

The distribution of extreme values determines the convergence range of the hierarchy, and knowing the relationship between the distribution of extreme values and the distribution of normal values makes it possible to predict the occurrence of extreme cases: extreme statistics

The distribution is probabilistic, and its error from reality is also probabilistic

The maximum likelihood estimators are always consistent, and if one agrees with several assumptions that are considered "regularityconditions", then the MLE is the most valid of all statistics. In addition, Fischer demonstrated that even if the MLE is biased, the magnitude of the deviation can be calculated and then subtracted from the MLE estimate to obtain a consistent, valid, and unbiased correction statistic (sequence match similarity)

Iterative algorithms, constantly approaching the intrinsic. The Bayesian formula is a treatment of probability and conforms to the hierarchy of the network. By repeating Bayes' theorem, we are able to determine the distribution of these parameters, and then the distribution of these hyperparameters. In principle, we can use super-hyper-hyperparameters to find the distribution of super-hyperparameters, and then lead this analytic hierarchy process to a deeper level, and so on. This is convergent, just as the higher-order derivatives of Taylor series decomposition are of little use in the simulation

Insects were divided into groups, kept in jars, and then experimented with different ingredients and different doses of insecticides. In the course of these experiments, he discovered a remarkable phenomenon: no matter how high the insecticide he tried to prepare, one or two insects would always be alive after the application, and no matter how much he diluted the insecticide, even if he only used a container containing the insecticide, a few insects would always die. (Expression of Probabilistic Networks, Stability)

Probability Unit Analysis: The relationship between "the dose of insecticide" and "the probability that a bug will die when the dose is used" is established. A similar concept of half-life can only be used: "50 percentlethaldoes", often denoted as "LD-50", refers to the dose at which the insecticide kills the bug with a 50% probability. At the same time: it is impossible to determine the dose required to kill a particular bug used as an experimental specimen. (This is the nature of the network, and we can only find a more certain relationship from the statistics of the whole)

The stochastic process theorem is an operation at the sequence level

Probability is the partial derivative, i.e., the relative proportion, between the different levels of the high-dimensional structure of the network

Central Limit Theorem: Everything has a distribution. The sums and differences of the various types of normal variables also obey a normal distribution. As a result, many of the statistics derived from normal random variables (variates) themselves obey normal distributions.

Operations research, the optimal allocation of resources, which is a discrete state of equilibrium (eigen) reached by hierarchical competition, uses a normal distribution to deal with the problem.

What appears to be a purely random measurement is actually generated by a deterministic system of equations, i.e., a selective representation of the network. The problem is that the coupling of multiple differential equations makes it impossible to solve them accurately, and we still need to understand them statistically

Constructing a statistic that tests goodness-of-fit obeys a probability distribution, K. Pearson proves that the x2 goodness-of-fit test obeys the same distribution regardless of the type of data used

Hypothesis testing (or significance testing) is a formal statistical method used to calculate the probability of previously observed outcomes under the assumption that the hypothesis to be tested is true

The significance test is used to draw one of three possible conclusions:

If the p-value is small (usually less than 0. 01), he asserts that some kind of result has already been manifested, if the p-value is large (usually greater than 0. 2) He claims that even if there is a result, it is too unlikely that there will be any large-scale experiments that show the result, and if the p-value is in between, he discusses how to design the next experiment to get a better result.

Interval estimates, the probability of believing that the true value of a population parameter will fall within the estimated interval, i.e., the confidence interval

The power-law distribution of the network makes the probability of extreme values relatively large, which significantly affects the results, resulting in a smaller value of the "student" t-test statistic than it would normally be (in general, a large t-test statistic corresponds to a small p-value).

The scatterplot of the observed data needs to be compared to what would be expected from a purely random distribution – a non-parametric test that eliminates noise

Eigen, a small sample is collected that is sufficiently representative and can be used to estimate the characteristics of the population

The network as a whole can be divided into several relatively independent parts (there is still a certain similarity between these levels, that is, a coupling relationship), and its further division may be somewhat duplicated. Mathematically, input-output analysis requires that the matrix describing network activity must have a unique inverse matrix, which means that once the matrix is obtained, it must be used as a mathematical "inverse matrix". The more refined the classification, the higher the probability of the existence of a unique inverse matrix because the degree of simulation of reality is constantly enhanced

The impact of a single variable is unreliable, and a more certain correlation can only be constructed at the network level. The language of the network is probability, and a certain path requires the accumulation of probability in a sequence, which fundamentally denies causality. The influence of multiple variables, i.e., the probability of Bayesian formula calculations, can only be observed at the macroscopic scale, i.e. the frequency. The many parameters of the network can never be observed exactly, but they interact with each other and influence each other

Everything we can see and touch is in fact only a shadow of the real world, and the real things that can really be found in this universe can only be obtained through pure reason. The selective representation of a probabilistic network is a real thing

In this 5000-dimensional space, these real data are not scattered, but actually tend to be a lower-dimensional space. Suppose that these points scattered in three-dimensional space all fall on the same plane or even the same line (Riemann hypothesis?), this is exactly what the real data is like. The 5,000 observations for each patient in a clinical study are not uncorrelated and scattered, because many of these measurements are correlated with each other.

In medical research, the true "dimensions" of data usually do not exceed 5. (Six degrees of separation of the network, average distance)

Correlation between power-law distributions and hidden Markov models: Determining the independent hierarchy (robustness) by finding a way to estimate the central trend of this distribution: An experiment conducted at Yale University in the 50s of the 20th century estimated the earnings of its graduates in 10 years. If they use the average, then the income is very high, because a few were multi-millionaires at the time, but, in fact, more than 80% of graduates earn less than this average on average

Network of dialectical treatment, the systematic expression of the disease (congestive heart failure is not an ordinary disease. The cause is not a simple source of infection, nor can it be alleviated by blocking the pathway of a biochemical enzyme. Hormones in the human body delicately control the heart, regulating its beating speed and contraction ability to adapt to the changing needs of the body, but the heart of patients with congestive heart failure is becoming less and less responsive to this regulation, and the main symptom of the patient is that the heart muscle is gradually weakening, and the muscles of the heart are becoming more and more hypertrophied and relaxed. As a result, patients develop edema in the lungs and ankles, and the slightest exercise can make it difficult for them to breathe. Patients may also feel drowsy and confused due to insufficient blood supply to the brain due to the blood supply to the stomach during meals. To maintain homeostasis, the affected body automatically adjusts to the reduced energy output of the heart. For many people, hormones that regulate changes in the heart muscle and other muscles reach a balance in some steady state. Although for the average person, such hormone levels are not normal. If doctors use β epinephrine contractionists or calcium-ionizing isolators during treatment, the results can complicate the patient's situation. Pulmonary edema is an important cause of death in patients with congestive heart failure. Modern medicine relies on diuretics, which can relieve edema. However, after the use of diuretics, the hormonal changes caused by the regulation of kidney function and heart function will cause new problems due to the interaction of patients.)

When designing a study, the first question that comes to mind is what to measure. The measurements in this experiment are multi-layered, and therefore their distribution functions, the parameters of which must be estimable, and their composition must be multidimensional.

Levy's proof of the central limit theorem establishes a more general set of necessary conditions, which are equivalent to having a set of randomly generated sequences of numbers one after the other: 1. Variation is bounded, so individual values cannot be infinitely large or infinitesimal. 2. The best estimate of the next number must be its previous value. Levy called such columns martingale, a convergence of the hidden Markov model and a manifestation of energy minimization

The patient's response is a martingale. The difference between the two martingale is still a martingale-linear system

Abraham Deimover introduced calculus into probability calculations

Glivenco-Cantelli Lemma: It is possible to make the less beautiful empirical distribution function (Fourier series) closer and closer to the real distribution function (Fourier series) by increasing the number of observations

More accurate measurements in turn make the difference between the predicted value of the model and the actual observed value larger, as in the uncertainty principle of quantum physics

A probability distribution is a low-dimensional projection of the network structure

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