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Stream: learning: questions

Topic: Measuring and sampling

Christoph Thies (Jan 16 2021 at 05:43):

In Schmid et al. (2020) the authors define measurement of a deterministic variable, that is a variable $Z$ in a Kleisli category $\mathsf{C}$ as in Fritz et al. (2020), in an experiment as operation on string diagrams like this (adapted from diagram (86) on page 15 in Schmid et al. (2020)):
StringMeas.png

This operation follows rules with respect to $\mathsf{del}$ and $\mathsf{copy}$: (1) measurements do not affect the variables in the diagram (equation (88)), and (2) multiple measurements of the same variable are equivalent to multiple copies of one of these measurements (equation (89)).

I would like to use a nondeterministic version of this measurement in order to model sampling from populations. In a Kleisli category $\mathsf{C}$ with monad $P$ consider a population to be a distribution over a variable $Z$, that is an element of $PZ.$ Inpired by comments by @Nathaniel Virgo, let's draw variables in the first power of the monad like this
StringTube.png

Christoph Thies (Jan 16 2021 at 05:44):

As above, we have the operation of measuring the deterministic observable $PZ$:
StringTubeMeas.png

I would like to define a nondeterministic measurement of the string within the tube by sampling from this observation like this:
MonadSample.png

Does this seem like a sensible thing to do?

Tobias Fritz (Jan 16 2021 at 06:27):

The motivation behind Schmid et al's work is to have a formalism for distinguishing inference and causation. Is this sort of thing important for you? Do you need to distinguish between the biological processes and the measurement in the sitring diagrams? Because if you don't, then it's simpler: you just copy on $PZ$ and applying the sampling map to one of the copies (which corresponds to extracting a random member of the population and looking at it).

What Schmid et al do largely seems to be to take the distinction between inference and causation into account by drawing the inference wires horizontally and the causation wires vertically, so that the whole string diagram is supposed to be read diagonally from the bottom left to the top right.

Christoph Thies (Jan 16 2021 at 06:45):

I am interested in Schmid et al.'s work because I would like to talk clearly about processes, the formalism that represents them, and the inferential structures used to find them in nature.

The reason I would like to treat these monad variables explicitly is that I consider populations of populations (metapopulations), that is elements of $PPZ$. To these, $\mathsf{samp}$ can be applied once to get an element of $PZ$ (a population) and again to get an element of $Z$ (a sample). A metapopulation in $PPZ$ therefore yields two random variables, one over $Z$ and one over $PZ$. Since the relation between these two is crucial, I would like to represent them.

Tobias Fritz (Jan 16 2021 at 07:31):

Right, I understand that you need the monad around, that makes perfect sense. But I was actually meaning to ask about the measurement: it would be simpler to represent it just as copy on $PZ$, composed with $samp : PZ \to Z$ on the right copy. The price for pay for this simpler description is that the string diagram then has no formal distinction between the "real" population and the copy population to which the measurement samp is applied. So my question is, do you actually need such a formal distinction? Because if not, then this simpler description is the way to go.

Christoph Thies (Jan 16 2021 at 09:01):

I see what you mean now. Yes, I think that'd be the same.

A string diagram as in Schmid et al. represents an experiment. I would like to use the measurement operation in order to explain how measurements at different stages of the experiment are related.