# A Tutorial in Data Science: Lecture 4 – Statistical Inference via Systems of Hypothesis-Trees

by | Jan 12, 2021 | Math Lecture

As from Lecture 1, letting $$X^n$$ be a random variable representing the n qualities that can be measured for the thing under investigation, $$\Omega$$, itself the collected gathering of all its possible appearances, $$\omega \in \Omega$$ such that each manifestation can be measured as a real number, i.e. (X^n:\omega \rightarrow {\mathbb{R}}^n\). Each sampled measurement of $$X^n$$ through interaction with $$\omega$$ is given as an $$\hat{X}^n(t_i)$$, each one constituting a unit of indexable time in the catalogable measurement process. Thus, the set of sampled measurements, a sample space, is a partition of ‘internally orderable’ test times within the measurement action, $$\{ \hat{X}^n(t): t \in \pi \}$$.

$$\Omega$$ is a state-system, i.e the spatio-temporality of the thing in question, in that it has specific space-states $$\omega$$ at different times $$\Omega(t)=\omega$$. $$X$$ is the function that measures $$\omega$$. What if the measurement is not Real, but Complex: $$X: \Omega \rightarrow \mathbb{C}$$? While a real number results from a finite, approximate, or open-ended process of objective empirical measurement, an imaginary number results from a subjective intuition or presupposition to measurement. Every interaction with $$\Omega$$ lets it appear as $$\omega$$, which is quantified by $$X$$. From these interactions, we seek to establish truths about $$\Omega$$ as quantifying the probability that the Claim $$C$$ is correct, which is itself a quantifiable statement about $$\Omega$$.

Ultimately, we seek the nature of how $$\Omega$$ appears differently depending on one’s interactions with it (i.e. samplings), as thus the actual distribution ($$\mathcal{D}$$) of the observed measurements, using our measurement apparatus $X$, that is, we ask about $$\mathcal{D}X(\Omega)=f{X(\Omega)}$$. The assumptions will describe the class $$\mathcal{C}$$ of the family $$\mathcal{F}$$ of distribution functions which $$f_X$$ belongs to, i.e. $$f_X \in \mathcal{F}{\mathcal{C}}$$, for the $$\hat{X}$$ measurements of the appearances of $$\Omega$$, while the sampling will give the parameter $$\theta$$, such that $$f_X =f{\mathcal{C}}(\theta)$$. The hypothesis distribution-parameter ($$\theta^*$$) may be either established by prior knowledge ($$\theta_0$$) or some the present n-sampling of the state-system ($$\theta_1$$). Thus, the parameter obtained from the present sampling $$\hat{\theta}=\Theta(\hat{X_1}, \cdots \hat{X_n})$$ is either used to judge the validity of a prior parameter estimation ($$\theta^=\theta_0$$) or is assessed in its own right (i.e. $$\theta^*=\theta_1=\hat{\theta}$$) as representative of the actual object’s state-system distribution, the difference between the two hypothesis set-ups, \textit{a priori vs. a posteriori}, being whether the present experiment is seen has having a bias or not. In either the prior or posteriori cases, $$H_{-}:\theta_0=\theta|\hat{\theta}$$ or $$H_{+}:\hat{\theta}=\theta$$, one uses the present sampling to establish the validity of a certain parameter value. If $$\hat{\Delta} \theta =\theta_0-\hat{\theta}$$ is the expected bias of the experiment, then $$H_{-}:\hat{\theta}+\hat{\Delta}\theta=\theta|\hat{\theta}$$ \& $$H_{+}:\hat{\theta}=\theta|\hat{\theta}$$. Thus, in all experiments, the statistical question is primarily that of the bias of the experiment that samples a parameter, whether it is 0 or not, i.e. $$H_{-}:|\hat{\Delta}\theta|>0$$ or $$H_{+}:\hat{\Delta}\theta=0$$.

The truth of the bias of the experiment, i.e. how representative it is, can only be given by our prior assumptions, $$A$$, such as to know the validity of our claim about the state-system’s distributional parameter, $$P(C|A)=P(\theta=\theta^*|\hat{\theta})=P(\Delta \theta=\hat{\Delta}\theta)$$, as the probability our expectation of bias is correct. Our prior assumption, $$A: f_X \in \mathcal{F}_{\mathcal{C}}$$ is about the distribution of the $$k$$-parameters in the class-family of distributions, where $$\mathcal{F}_{\mathcal{C}}={f(k)}, \ s.t. \ f_X=f(\theta)$$, that is about $$\mathcal{D}K(\mathcal{F}_{\mathcal{C}})$$. Here, $$K$$ is a random variable that samples state-systems in the wider class of generally known objects, or equivalently their distributions (i.e. functional representations), measuring the $$k$$-parameter of their distribution, such that $$f_K(\mathcal{F}_{\mathcal{C}})=\mathcal{D}_K(\mathcal{F}_{\mathcal{C}})$$. The distributed-objects in $$\mathcal{F}_{\mathcal{C}}$$ are themselves relative to the measurement system $$X$$ although they may be transformed into other measurement units, in that this distribution class is of all possible state-systems which $$X$$ might measure sample-wise, for which we seek to know specifically about the $$\Omega$$ in question to obtain its distributional $$k$$-parameter value of $$\theta$$. Essentially, the assumption $$A$$ is about a meta-state-system as the set of all objects $$X$$ can measure, and thus has more to do with $$X$$, the subject’s method of measurement, and $$\Theta$$, the parametrical aggregation of interest, than with $$\Omega$$, the specific object of measurement.

$$\theta \in \Theta$$, the set of all the parameters to the family $$\mathcal{F}$$ of relevant distributions, in that $$\Theta$$ uniquely determines $$f$$, in that $$\exists M: \Theta \rightarrow f \in \mathcal{F}$$, or $$f=\mathcal{F}(\Theta)$$.

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