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I have a question about the philosophy using probability/statistic theory to solve some real world problems that we don't have a fully understanding of its deterministic mechanism yet.

For example, to simulate some nature problems, as we don't know the full scientific mechanism behind it, we may try to borrow stochastic theories, which means we assume the nature of this real problem is random. However, it is quite possible that this real problem is governed by some deterministic way that we don't know yet. So is there any contradiction between these two. It seems we are admitting both deterministic (mechanism we don't know yet) and stochastic (use stochastic model to solve it). And how could we judge out stochastic model's results in this way?

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    $\begingroup$ Coin tossing is also a deterministic process that can be describes in terms of physics. We always use probability and statistics to approximate the real world problems. $\endgroup$
    – Tim ♦
    Nov 5, 2016 at 23:09
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    $\begingroup$ People use Newtonian mechanics all the time to solve difficult and important problems in physics, such as landing probes on Mars, even though we know Newtonian mechanics is not a correct description of the universe. This fact might help you appreciate the difference between using a model and believing something about the "real world." $\endgroup$
    – whuber ♦
    Nov 5, 2016 at 23:42
  • $\begingroup$ @Tim what if the coin toss of 100 times and determine its sequence. I don't see it is tractable in physics $\endgroup$
    – rifle123
    Nov 5, 2016 at 23:42
  • $\begingroup$ @rifle123 people have build coin-tossing machines, that will always throw the same side (stats.stackexchange.com/questions/153076/…), in terms of physics it is perfectly deterministic, while in real-life may be complicated. $\endgroup$
    – Tim ♦
    Nov 6, 2016 at 9:22

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Let's say the true data generating process is $$y_i = f(x_i, z_i)$$ Let's say we instead model it as a function $g$ of some subset of the world $x_i$ plus some error term $\epsilon_i$. $$y_i = g(x_i) + \epsilon_i$$ Hence $\epsilon_i = f(x_i, z_i) - g(x_i)$. The error term may be a garbage collector which accumulates everything you don't observe, don't model, don't understand etc... And you can treat $\epsilon_i$ as a random variable.

Small postscript... when you get down to the lowest level of physics, the world doesn't become deterministic either.

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