Parametric and Interval Models

In this notebook, we will show how to create and work with parametric and interval models. ## Parametric Models

Stormvogel represents parametric transition values as sympy expressions. Any polynomial or rational function over sympy.Symbol parameters is a valid transition value, which means you can write them using ordinary Python arithmetic.

import sympy as sp

For example, a polynomial in two variables:

x, y, z = sp.symbols("x y z")

polynomial1 = x**2 + y**2
print(polynomial1)

polynomial2 = 6 * z**3 + z + 2
print(polynomial2)

rational_function = polynomial1 / polynomial2
print(rational_function)

x**2 + y**2
6*z**3 + z + 2
(x**2 + y**2)/(6*z**3 + z + 2)

To create a parametric model (e.g. pMC or pMDP) we simply use these sympy expressions as transition probabilities. As an example, we build a parametric Knuth–Yao dice: a coin with success probability x is flipped repeatedly to simulate a six-sided die.

from stormvogel import model, bird
from stormvogel.show import show

# A single symbolic parameter for the coin bias.
x = sp.Symbol("x")


# we build the Knuth–Yao dice using the bird model builder
def delta(
    s: bird.State,
) -> list[tuple[float | sp.Expr, bird.State]] | None:
    match s.s:
        case 0:
            return [(x, bird.State(s=1)), (1 - x, bird.State(s=2))]
        case 1:
            return [(x, bird.State(s=3)), (1 - x, bird.State(s=4))]
        case 2:
            return [(x, bird.State(s=5)), (1 - x, bird.State(s=6))]
        case 3:
            return [(x, bird.State(s=1)), (1 - x, bird.State(s=7, d=1))]
        case 4:
            return [
                (x, bird.State(s=7, d=2)),
                (1 - x, bird.State(s=7, d=3)),
            ]
        case 5:
            return [
                (x, bird.State(s=7, d=4)),
                (1 - x, bird.State(s=7, d=5)),
            ]
        case 6:
            return [(x, bird.State(s=2)), (1 - x, bird.State(s=7, d=6))]
        case 7:
            return [(1, s)]


def labels(s: bird.State):
    if s.s == 7:
        return f"rolled{str(s.d)}"


knuth_yao_pmc = bird.build_bird(
    delta=delta,
    init=bird.State(s=0),
    labels=labels,
    modeltype=model.ModelType.DTMC,
)

show(knuth_yao_pmc)

/tmp/ipykernel_1255/1846297749.py:10: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  s: bird.State,
/tmp/ipykernel_1255/1846297749.py:11: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  ) -> list[tuple[float | sp.Expr, bird.State]] | None:
/tmp/ipykernel_1255/1846297749.py:37: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  def labels(s: bird.State):
/tmp/ipykernel_1255/1846297749.py:44: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  init=bird.State(s=0),
/tmp/ipykernel_1255/1846297749.py:14: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(x, bird.State(s=1)), (1 - x, bird.State(s=2))]
/tmp/ipykernel_1255/1846297749.py:16: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(x, bird.State(s=3)), (1 - x, bird.State(s=4))]
/tmp/ipykernel_1255/1846297749.py:18: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(x, bird.State(s=5)), (1 - x, bird.State(s=6))]
/tmp/ipykernel_1255/1846297749.py:20: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(x, bird.State(s=1)), (1 - x, bird.State(s=7, d=1))]
/tmp/ipykernel_1255/1846297749.py:23: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (x, bird.State(s=7, d=2)),
/tmp/ipykernel_1255/1846297749.py:24: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (1 - x, bird.State(s=7, d=3)),
/tmp/ipykernel_1255/1846297749.py:28: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (x, bird.State(s=7, d=4)),
/tmp/ipykernel_1255/1846297749.py:29: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (1 - x, bird.State(s=7, d=5)),
/tmp/ipykernel_1255/1846297749.py:32: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(x, bird.State(s=2)), (1 - x, bird.State(s=7, d=6))]

<stormvogel.visualization.JSVisualization at 0x7f581019a120>

The symbol x was auto-declared on the model as soon as it appeared in a transition. You can inspect the parameters any time:

print("Parameters:", knuth_yao_pmc.parameters)

Parameters: ('x',)

We can now evaluate the model by assigning the parameter x to any concrete value. This induces a regular DTMC with fixed probabilities.

p = 1 / 2

eval_knuth_yao_pmc = knuth_yao_pmc.get_instantiated_model({"x": p})
show(eval_knuth_yao_pmc)

<stormvogel.visualization.JSVisualization at 0x7f580fcc3b10>

Interval Models

We can also set an interval between two values x and y as transition value, meaning that we don’t know the probability precisely, but we know it is between x and y. We represent intervals using model.Interval, where we have two attributes: lower and upper. Both of these should be an element of type Number, i.e., int, float or Fraction.

interval = model.Interval(1 / 3, 2 / 3)
print(interval)

[0.3333333333333333,0.6666666666666666]

Similar to parametric models, creating an interval model is as straightforward as just setting some interval objects as transition values.

from stormvogel import bird
from stormvogel.show import show

# We create our interval values
interval = model.Interval(2 / 7, 6 / 7)
inv_interval = model.Interval(1 / 7, 5 / 7)


# we build the knuth yao dice using the bird model builder
def delta(s: bird.State) -> list[tuple[float | model.Interval, bird.State]] | None:
    match s.s:
        case 0:
            return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=2))]
        case 1:
            return [(interval, bird.State(s=3)), (inv_interval, bird.State(s=4))]
        case 2:
            return [(interval, bird.State(s=5)), (inv_interval, bird.State(s=6))]
        case 3:
            return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=7, d=1))]
        case 4:
            return [
                (interval, bird.State(s=7, d=2)),
                (inv_interval, bird.State(s=7, d=3)),
            ]
        case 5:
            return [
                (interval, bird.State(s=7, d=4)),
                (inv_interval, bird.State(s=7, d=5)),
            ]
        case 6:
            return [(interval, bird.State(s=2)), (inv_interval, bird.State(s=7, d=6))]
        case 7:
            return [(1, s)]


def labels(s: bird.State):
    if s.s == 7:
        return f"rolled{str(s.d)}"


knuth_yao_imc = bird.build_bird(
    delta=delta,
    init=bird.State(s=0),
    labels=labels,
    modeltype=model.ModelType.DTMC,
)

show(knuth_yao_imc)

/tmp/ipykernel_1255/1271625296.py:10: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  def delta(s: bird.State) -> list[tuple[float | model.Interval, bird.State]] | None:
/tmp/ipykernel_1255/1271625296.py:36: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  def labels(s: bird.State):
/tmp/ipykernel_1255/1271625296.py:43: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  init=bird.State(s=0),
/tmp/ipykernel_1255/1271625296.py:13: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=2))]
/tmp/ipykernel_1255/1271625296.py:15: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(interval, bird.State(s=3)), (inv_interval, bird.State(s=4))]
/tmp/ipykernel_1255/1271625296.py:17: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(interval, bird.State(s=5)), (inv_interval, bird.State(s=6))]
/tmp/ipykernel_1255/1271625296.py:19: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=7, d=1))]
/tmp/ipykernel_1255/1271625296.py:22: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (interval, bird.State(s=7, d=2)),
/tmp/ipykernel_1255/1271625296.py:23: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (inv_interval, bird.State(s=7, d=3)),
/tmp/ipykernel_1255/1271625296.py:27: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (interval, bird.State(s=7, d=4)),
/tmp/ipykernel_1255/1271625296.py:28: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  (inv_interval, bird.State(s=7, d=5)),
/tmp/ipykernel_1255/1271625296.py:31: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.
  return [(interval, bird.State(s=2)), (inv_interval, bird.State(s=7, d=6))]

<stormvogel.visualization.JSVisualization at 0x7f580fbf8050>