nLab
stochastic order

Contents

Contents

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

Note that, if one does not require the measures to be normalized, one has for example p2pp\le 2p whenever the measure pp 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 XX, 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 pqp\le q if and only if p(U)q(U)p(U)\le q(U) for every upper set UU.

  • In the latter case, we say that pqp\le q if and only if p(U)q(U)p(U)\le q(U) for every Scott-open set UU, 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 pqp\le q if and only if p(U)q(U)p(U)\le q(U) for every open set UU. 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 XX be an ordered topological space, and let f,g:ΩXf,g:\Omega\to X be random elements? with values in XX on the probability space (Ω,μ)(\Omega,\mu). We say that fgf\le g in the stochastic order if and only if f *μg *μf_*\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 XX and YY and monotone maps f,g:XYf,g:X\to Y, we say that fgf\le g, or we draw a 2-cell fgf\Rightarrow g, if and only if for every xXx\in X, f(x)g(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 PP 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:XYf,g:X\to Y are such that fgf\le g, then also PfPgP f\le P g as maps PXPYP X\to P Y. The latter condition means that for all pPXp\in P X, (Pf)(p)(Pg)(p)(P f)(p) \le (P g)(p) in PYP Y, so we need a preorder relation on PYP Y that is suitably compatible with the order on YY, in such a way that PP 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:XYf,g:X\to Y as above, PfPgP f\le P g if and only if fgf\le g.

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

See also

References

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Last revised on February 26, 2020 at 09:36:43. See the history of this page for a list of all contributions to it.