Tutorial

Counterfactuals

The utility of replace may not be obvious at first glance. We can use replace and cond separately and in combination to ask lots of different kinds of questions. In this example, we model the relationship betwee the weather outside and teh thermostat reading inside a house. Broadly, the model says that the weather outside is dictataed by the time of day, while the temperature inside is determined by whether the air conditioning is on, and whether the window is open.

Notebook

First, setup simple priors over the time of day, and variables to determine whether the air conditioning is on and whether hte iwndow is open:

timeofday = uniform([:morning, :afternoon, :evening])
is_window_open = bernoulli(0.5)
is_ac_on = bernoulli(0.3)

Second, assume that the outside temperature depends on the time of day, being hottest in the afternoon, but cold at night:

function outside_temp_(rng)
  if timeofday(rng) == :morning
    normal(rng, 20.0, 1.0)
  elseif timeofday(rng) == :afternoon
    normal(rng, 32.0, 1.0)
  else
    normal(rng, 10.0, 1.0)
  end
end

Remember, in this style we have to use ciid to convert a function into a RandVar

outside_temp = ciid(outside_temp_, T=Float64)

The inside_temp before considering the effects of the window is room temperature, unless the ac is on, which makes it colder.

function inside_temp_(rng)
  if Bool(is_ac_on(rng))
    normal(rng, 20.0, 1.0)
  else
    normal(rng, 25.0, 1.0)
  end
end

inside_temp = ciid(inside_temp_, T=Float64)

47:Omega.normal(100.0, 1.0)::Float64

Finally, the thermostat reading is inside_temp if the window is closed (we have perfect insulation), otherwise it's just the average of the outside and inside temperature

function thermostat_(rng)
  if Bool(is_window_open(rng))
    (outside_temp(rng) + inside_temp(rng)) / 2.0
  else
    inside_temp(rng)
  end
end

thermostat = ciid(thermostat_, T=Float64)

Now with the model built, we can ask some questions:

Samples from the prior

The simplest task is to sample from the prior:

julia> rand((timeofday, is_window_open, is_ac_on, outside_temp, inside_temp, thermostat), 5, alg = RejectionSample)
5-element Array{Any,1}:
 (:afternoon, 0.0, 0.0, 32.349, 26.441, 26.441)   
 (:afternoon, 1.0, 0.0, 30.751, 25.143, 27.947)
 (:morning, 1.0, 0.0, 16.928, 24.146, 20.537)     
 (:afternoon, 1.0, 0.0, 30.521, 25.370, 27.946)
 (:morning, 1.0, 1.0, 16.495, 20.203, 18.349) 

Conditional Inference

samples = rand((timeofday, iswindowopen, isacon, outsidetemp, insidetemp, thermostat), thermostat > 30.0, 5, alg = RejectionSample)


julia> samples = rand((timeofday, is_window_open, is_ac_on, outside_temp, inside_temp, thermostat),
                       thermostat > 30.0, 5, alg = RejectionSample)
5-element Array{Any,1}:
 (:evening, 1.0, 0.0, 33.64609872046609, 26.822449458789542, 30.234274089627817) 
 (:afternoon, 1.0, 0.0, 34.37763909867243, 26.16221853550574, 30.269928817089088)
 (:evening, 1.0, 0.0, 34.32198183192978, 26.6773921624331, 30.499686997181442)   
 (:afternoon, 1.0, 0.0, 34.05126597960254, 26.51833791813246, 30.2848019488675)  
 (:afternoon, 1.0, 0.0, 32.92982568498735, 27.56800059609554, 30.248913140541447)

Counter Factual

thermostatnew = replace(thermostat, is_ac_on => 1.0, is_window_open => 0.0)
diffsamples = rand(thermostatnew - thermostat, 10000, alg = RejectionSample)
julia> mean(diffsamples)
-4.246869797640215

So in expectation, that intervention will make the thermostat colder. But we can look more closely at the distribution:

julia> UnicodePlots.histogram([diffsamples...])

                 ┌────────────────────────────────────────┐ 
   (-11.0,-10.0] │ 37                                     │ 
    (-10.0,-9.0] │▇▇▇▇ 502                                │ 
     (-9.0,-8.0] │▇▇▇▇▇▇▇▇▇▇▇ 1269                        │ 
     (-8.0,-7.0] │▇▇▇▇▇ 581                               │ 
     (-7.0,-6.0] │▇▇▇▇ 497                                │ 
     (-6.0,-5.0] │▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇ 3926 │ 
     (-5.0,-4.0] │▇ 65                                    │ 
     (-4.0,-3.0] │ 5                                      │ 
     (-3.0,-2.0] │ 3                                      │ 
     (-2.0,-1.0] │▇ 97                                    │ 
      (-1.0,0.0] │▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇ 1960                  │ 
       (0.0,1.0] │▇▇▇▇ 494                                │ 
       (1.0,2.0] │▇▇ 197                                  │ 
       (2.0,3.0] │▇▇ 237                                  │ 
       (3.0,4.0] │▇ 118                                   │ 
       (4.0,5.0] │ 12                                     │ 
                 └────────────────────────────────────────┘ 

rand((timeofday, outsidetemp, insidetemp, thermostat), thermostatnew - thermostat > 0.0, 10, alg = RejectionSample)