Classical statistical procedures are typically defined by some function over sample space, where this function depends, often exclusively, on the distributions that the hypotheses under consideration assign to the sample space.
Hypothesis tests and estimations are sometimes criticised because their results generally depend on the probability functions over the entire sample space, and not exclusively on the probabilities of the observed sample.
Although both the theta model and the control model are intrinsically stochastic, the sample size was large enough for the objective function over the entire sample to be nearly deterministic, allowing for convergence of the gradient descent algorithm.
Six voice-over samples are being played for Solid.
To integrate all above functions, a sample manipulator was constructed featuring control of temperature over a wide range -- 20 K to 1000 K, six degrees of freedom for sample positioning, sample transfer with exchangeable sample-mounting modules to fulfill varied experimental needs.
This is the parameter value (hat {boldsymbol {theta }}) which maximizes (L {boldsymbol {theta }})= frac {1}{n}sum ^{n}_{i=1}log p({mathbf {Z}}_{i}; {boldsymbol {theta }})) of the log likelihood functions of the sample over the parameter space Θ, that is, L hat{{boldsymbol{theta}}}) =max_{{boldsymbol{theta}} in Theta} frac{1}{n}sum^{n}_{i=1} log p({mathbf{Z}}_{i}; {boldsymbol{theta}}). (4.32).
The choice of covariance function determines an a priori distribution over functions so that samples from the Gaussian process plotted over many input points have characteristic behaviours.
Analogously, the function ς replaces a sum over samples in the Fourier domain with an integral.
The main enhancement is obtained by analyzing the variation of the Lyapunov function over more than one sample.
We can compare these hypotheses by the following test function $F$ over the sample space: \[ F s) = \begin{cases} 1 \quad \text{if } \frac{P_{h'}(s)}{P_{h}(s)} > r,\\ 0 \quad \text{otherwise,} \end{cases} \] where $P_{h}$ and $P_{h'}$ are the probability distributions over the sample space determined by the statistical hypotheses $h$ and $h'$ respectively.
