In statistical modelling, a type of model is called a (finite) mixture distribution. This states that we have a random variable $X_M$ with distribution function $P_M$ satisfying a finite mixture distribution if and only if there is some finite sequence of probability density functions $P_1, P_2, ... , P_n$ and some set of weights $p_1, p_2,...,p_n \geq 0$ such that $\sum_{i} p_i$=1 that give:

$P_M = \sum_{i}^{n} p_i P_i$

For example, if $P_1 \sim N(-1,1)$ and $P_2 \sim N(5,1)$ and weights $p_1 = p_2 = 0.5$, this looks like...

SRAP UPLOADED THE FILE SINCE I COULDN'T GET IT WORKING.

You can see the peaks at the means for $P_1$ (-1) and $P_2$ (5), this kind of clustering behaviour is typical of mixtures. You can calculate the mean (here denoted $E$ and variance (here denoted $V$ for mixture distributions as the weighted mean of the two variables. So in this case, we have:

$E(X_M)=0.5 \int X_1 P(X_1) + 0.5 \int X_2 P(X_2) = 0.5E(X_1)+0.5E(X_2)$

which is $0.5*-1+0.5*5=2$.

Where $X_1$ satisfies $P=N(-1,1)$ and $X_2$ satisfied $P=N(5,1)$. Generally, this implies that the mean is the sum of the two means. Also, the variance is the sum of the two variances (if the variables are independent).

Now consider we have some data which are estimating Newton's second law $F=ma$, measuring the impact force $F$ and acceleration $a$ to estimate the mass $m$. This is estimated very precisely, and the measurement errors are harmlessly assumed to be Gaussian/Normal. So, this would give:

$F=ma \rightarrow F \sim m*N(a,\sigma^2)$ where $\sigma^2$, the measurement error, is very close to 0 since the measurements are precise (this is a high school physics experiment and analysis rendered in terms of the statistics involved. You can now imagine an alternative construction, a mixture distribution model of $F$ such as:

$F\sim 0.999*m*N(a,\sigma^2)+0.001*m*t_1$

This is saying that most of the time we observe faithful measurements of Newton's second law, but 1 in 1000 times we observe something completely nuts and unexpected - the t distribution on 1 degree of freedom. By shrinking the probability multiplying the t distribution term this model can be made consistent with any set of observations - it will look just like Newton's second law but with a hidden pathological component.

What does this pathological component do? Well, it induces two properties in the observations.

(1)The measurements of force have infinite variance.
(2)The measurements of force have infinite mean.

This means the estimate of the measurement error are entirely pointless, and that it is impossible to obtain an estimate of the acceleration through the mean of repeated concordant measurements. As a recent thread suggested: if we have no evidence by which to distinguish a proposition p from not-p, we can't learn anything else. For a sufficiently small p, we can render this model consistent with every set of observations of Newton's second law. But since we cannot distinguish the behaviour of Newton's law from the pathological law, we cannot learn anything about Newton's second law from any data.

Thus there is no evidence for Newton's second law.