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.
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
- You enter the room and the thermostat reads hot. what does this tell you about the variables?
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
- If I were to close the window, and turn on the AC would that make it hotter or colder"
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 │
└────────────────────────────────────────┘
- In what scenarios would it still be hotter after turning on the AC and closing the window?
rand((timeofday, outsidetemp, insidetemp, thermostat), thermostatnew - thermostat > 0.0, 10, alg = RejectionSample)