nLab stochastic order

Idea

The stochastic order is partial order or preorder between probability measures or valuations over an ordered space, which compares them in terms of “how higher up in the order the mass is”.

Definition

Let $X$ be a topological space equipped with a preorder relation, and let $p$ and $q$ be Borel measures or valuations on $X$. We say that $p \le q$ in the stochastic order if and only if for every upper open set $U\subseteq X$, we have that $p(U)\le q(U)$.

Note that, if one does not require the measures to be normalized, one has for example $p\le 2p$ whenever the measure $p$ is non-signed. However, usually one is interested in comparing probability measures (or normalized valuations). In this case, assigning a higher mass to upper sets can be indeed interpreted as “having the mass placed at higher places”.

Often one does not start with both an order and a topology on $X$, but obtains one structure from the other. If one starts with a partial order or a preorder, one can take either the Alexandrov topology or the Scott topology:

• In the former case, we say that $p\le q$ if and only if $p(U)\le q(U)$ for every upper set $U$.

• In the latter case, we say that $p\le q$ if and only if $p(U)\le q(U)$ for every Scott-open set $U$, i.e. upper and inaccessible by directed suprema. This is how one construct the probabilistic powerdomain?.

If one starts with a topological space, one can take as preorder the specialization preorder relative to the given topology. This way, we say that $p\le q$ if and only if $p(U)\le q(U)$ for every open set $U$. Equivalently, this is the pointwise order of valuations as functions on the frame of open sets. This is how one constructs the extended probabilistic powerdomain.

For random variables

The stochastic order can be extended from probability measures to random variables by comparing their laws.

Let $X$ be an ordered topological space, and let $f,g:\Omega\to X$ be random elements with values in $X$ on the probability space $(\Omega,\mu)$. We say that $f\le g$ in the stochastic order if and only if $f_*\mu\le g_*\mu$ in the stochastic order of measures defined above.

In decision theory, economics and related fields, this order is also known as first-order stochastic dominance, especially on the real line.

As a 2-functor

Categories of preorders (possibly with extra structure) and monotone maps (possibly preserving the extra structure) can be canonically equipped with the structure of a locally posetal 2-category. Explicitly this is given by the pointwise order: given preorders $X$ and $Y$ and monotone maps $f,g:X\to Y$, we say that $f\le g$, or we draw a 2-cell $f\Rightarrow g$, if and only if for every $x\in X$, $f(x)\le g(x)$.

The stochastic order can be seen as a way to turn a probability monad on a category of preorders into a 2-monad by equipping it with an action on the 2-cells. In particular, let $P$ be a probability monad on a category of preorders. In order to make it a 2-functor, and so a 2-monad, we want to say that if $f,g:X\to Y$ are such that $f\le g$, then also $P f\le P g$ as maps $P X\to P Y$. The latter condition means that for all $p\in P X$, $(P f)(p) \le (P g)(p)$ in $P Y$, so we need a preorder relation on $P Y$ that is suitably compatible with the order on $Y$, in such a way that $P$ preserves these 2-cells (inequalities). The stochastic order is the canonical choice of such a preorder.

One can actually define the stochastic order directly in terms of 2-cells as follows: given $f,g:X\to Y$ as above, $P f\le P g$ if and only if $f\le g$.

Similarly, the stochastic order can be seen as a way of making the functor $P$ enriched in truth values.

See also

• monads of probability, measures, and valuations
• probabilistic powerdomain?, extended probabilistic powerdomain
• valuation (measure theory)

References

• Tobias Fritz and Paolo Perrone, Stochastic order on metric spaces and the ordered Kantorovich monad, Advances in Mathematics 366, 2020. (arXiv:1808.09898)

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