### 金融代写|金融计量经济学Financial Econometrics代考|Invariance-Based Explanation

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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|金融计量经济学Financial Econometrics代考|Formulation of the Problem

Need for solving the inverse problem. Once we have a model of a system, we can use this model to predict the system’s behavior, in particular, to predict the results of future measurements and observations of this system. The problem of estimating future measurement results based on the model is known as the forward problem.

In many practical situations, we do not know the exact model. To be more precise, we know the general form of a dependence between physical quantities, but the parameters of this dependence need to be determined from the observations and from the results of the experiment. For example, often, we have a linear model $y=$ $a_{0}+\sum_{i=1}^{n} a_{i} \cdot x_{i}$, in which the parameters $a_{i}$ need to be experimentally determined. The problem of determining the parameters of the model based on the measurement results is known as the inverse problem.

To actually find the parameters, we can use, e.g., the Maximum Likelihood method. For example, when the errors are normally distributed, then the Maximum Likelihood procedure results in the usual Least Squares estimates; see, e.g., Sheskin $\mathrm{~ ( 2 0 1 1 ) . ~ F o ̄ ́ ~ e x a ̄ m p ̄ l e ́ , ~ f o ̄ r ~ a ̄ ~ g e n e n e ́ r a ̆ l ~ l i n ̃ e a r r ~ m o}$ several tuples of corresponding values $\left(x_{1}^{(k)}, \ldots, x_{n}^{(k)}, y^{(k)}\right), 1 \leq k \leq K$, then we can find the parameters from the condition that
$$\sum_{k=1}^{K}\left(y^{(k)}-\left(a_{0}+\sum_{i=1}^{n} a_{i} \cdot x_{i}^{(k)}\right)\right)^{2} \rightarrow \min {a{0}, \ldots, a_{n}}$$
Need for regularization. In some practical situations, based on the measurement results, we can determine all the model’s parameters with reasonably accuracy. However, in many other situations, the inverse problem is ill-defined in the sense that several very different combinations of parameters are consistent with all the measurement results.

This happens, e.g., in dynamical systems, when the observations provide a smoothed picture of the system’s dynamics. For example, if we are tracing the motion of a mechanical system caused by an external force, then a strong but short-time force in one direction followed by a similar strong and short-time force in the opposite direction will (almost) cancel each other, so the same almost-unchanging behavior is consistent both with the absence of forces and with the above wildly-oscillating force. A similar phenomenon occurs when, based on the observed economic behavior, we try to reconstruct the external forces affecting the economic system.

In such situations, the only way to narrow down the set of possible solutions is to take into account some general a priori information. For example, for forces, we may know-e.g., from experts-the upper bound on each individual force, or the upper bound on the overall force. The use of such a priori information is known as regularization; see, e.g., Tikhonov and Arsenin (1977).

## 金融代写|金融计量经济学Financial Econometrics代考|General and Probabilistic Regularizations

General idea of regularization and its possible probabilistic background. In general, regularization means that we dismiss values $a_{i}$ which are too large or too small. In some cases, this dismissal is based on subjective estimations of what is large and what is small. In other cases, the conclusion about what is large and what is not large is based on past experience of solving similar problem-i.e., on our estimate of the frequencies (= probabilities) with which different values have been observed in the past. In this paper, we consider both types of regularization.

Probabilistic regularization: towards a precise definition. There is no a priori reason to believe that different parameters have different distributions. So, in the first approximation, it makes sense to assume that they have the same probability distribution. Let us denote the probability density function of this common distribution by $\rho(a)$.

In more precise terms, the original information is invariant with respect to all possible permutations of the parameters; thus, it makes sense to conclude that the resulting joint distribution is also invariant with respect to all these permutationswhich implies, in particular, that all the marginal distributions are the same.

Similarly, in general, we do not have a priori reasons to prefer positive or negative values of each the coefficients, i.e., the a priori information is invariant with respect to changing the sign of each of the variables: $a_{i} \rightarrow-a_{i}$. It is therefore reasonable to conclude that the marginal distribution should also be invariant, i.e., that we should have $\rho(-a)=\rho(a)$, and thus, $\rho(a)=\rho(|a|)$.

Also, there is no reason to believe that different parameters are positively or negatively correlated, so it makes sense to assume that their distributions are statistically independent. This is in line with the general Maximum Entropy (=Laplace Indeterminacy Principle) ideas Jaynes and Bretthorst (2003), according to which we should not pretend to be certain – to be more precise, if several different probability distributions are consistent with our knowledge:

• we should not select distributions with small entropy (measure of uncertainty),
• we should select the one for which the entropy is the largest.
If all we know are marginal distributions, then this principle leads to the conclusion that the corresponding variables are independent; see, e.g., Jaynes and Bretthorst $(2003) .$

Due to the independence assumption, the joint distribution of $n$ variables $a_{i}$ take the form $\rho\left(a_{0}, a_{1}, \ldots, a_{n}\right)=\prod_{i=0}^{n} \rho\left(\left|a_{i}\right|\right)$. In applications of probability and statistics, it is usually assumed, crudely speaking, that events with very small probability are not expected to happen. This is the basis for all statistical tests-e.g., if we assume that the distribution is normal with given mean and standard deviation, and the probability that this distribution will lead to the observed data is very small (e.g., if we observe a 5 -sigma deviation from the mean), then we can conclude, with high confidence, that experiments disprove our assumption. In other words, we take some threshold $t_{0}$, and we consider only the tuples $a=\left(a_{0}, a_{1}, \ldots, a_{n}\right)$ for which $\rho\left(a_{0}, a_{1}, \ldots, a_{n}\right)=$ $\prod_{i=0}^{n} \rho\left(\left|a_{i}\right|\right) \geq t_{0}$. By taking logarithms of both sides and changing signs, we get an equivalent inequality
$$\sum_{i=0}^{n} \psi\left(\left|a_{i}\right|\right) \leq p_{0}$$

## 金融代写|金融计量经济学Financial Econometrics代考|Natural Invariances

Scale-invariance: general idea. The numerical values of physical quantities depend on the selection of a measuring unit. For example, if we previously used meters and now start using centimeters, all the physical quantities will remain the same, but the numerical values will change-they will all get multiplied by 100 .

In general, if we replace the original measuring unit with a new measuring unit which is $\lambda$ times smaller, then all the numerical values get multiplied by $\lambda$ :
$$x \rightarrow x^{\prime}=\lambda \cdot x$$
Similarly, if we change the original measuring units for the quantity $y$ to a new unit which is $\lambda$ times smaller, then all the coefficients $a_{i}$ in the corresponding dependence $y=a_{0}+\cdots+a_{i} \cdot x_{i}+\cdots$ will also be multiplied by the same factor: $a_{i} \rightarrow \lambda \cdot a_{i}$.
Scale-invariance: case of probabilistic constraints. It is reasonable to require that the corresponding constraints should not depend on the choice of a measuring unit. Of course, if we change $a_{i}$ to $\lambda \cdot a_{i}$, then the value $p_{0}$ may also need to be accordingly changed, but overall, the constraint should remain the same. Thus, we arrive at the following definition.

Definition 4 We say that probability constraints corresponding to the function $\psi(z)$ are scale-invariant if for every $p_{0}$ and for every $\lambda>0$, there exists a value $p_{0}^{\prime}$ such that
$$\sum_{i=0}^{n} \psi\left(\left|a_{i}\right|\right)=p_{0} \Leftrightarrow \sum_{i=0}^{n} \psi\left(\lambda \cdot\left|a_{i}\right|\right)=p_{0}^{\prime}$$
Scale-invariance: case of general constraints. In general, the degree of impossibility is described in the same units as the coefficients themselves. Thus, invariance would mean that if replace $a$ and $b$ with $\lambda \cdot a$ and $\lambda \cdot b$, then the combined value $a * b$ will be replaced by a similarly re-scaled value $\lambda \cdot(a * b)$. Thus, we arrive at the following definition.

## 金融代写|金融计量经济学Financial Econometrics代考|Formulation of the Problem

$$\sum_{k=1}^{K}\left(y^{(k)}-\left(a_{0}+\sum_{i=1} ^{n} a_{i} \cdot x_{i}^{(k)}\right)\right)^{2} \rightarrow \min {a {0}, \ldots, a_{n}}$$

## 金融代写|金融计量经济学Financial Econometrics代考|General and Probabilistic Regularizations

• 我们不应该选择熵小的分布（不确定性的度量），
• 我们应该选择熵最大的那个。
如果我们所知道的只是边际分布，那么这个原理就会得出相应变量是独立的结论；例如，参见 Jaynes 和 Bretthorst(2003).

∑一世=0nψ(|一个一世|)≤p0

## 金融代写|金融计量经济学Financial Econometrics代考|Natural Invariances

X→X′=λ⋅X

∑一世=0nψ(|一个一世|)=p0⇔∑一世=0nψ(λ⋅|一个一世|)=p0′

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