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bkamins/BKAMINS_DSS2018.ipynb

Created June 7, 2018 18:32
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Support material for presentation at DSS2018
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BKAMINS_DSS2018.ipynb
{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Data Science Summit 2018\n",
"### Bogumił Kamński\n",
"\n",
"## Testy A/B: od teorii do praktyki\n",
"\n",
"Take a survey: https://bit.ly/2HsfruG"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# Julia v0.6.3, DataFrames v0.11.6, DataFramesMeta v0.3.0, StatPlots v0.7.2\n",
"using DataFrames, DataFramesMeta, StatPlots"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"samplemax (generic function with 1 method)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# return an index of a random element of x that is eaual to maximum(x)\n",
"function samplemax(x)\n",
" m, ind, c = x[1], 1, 1\n",
" for i in 2:length(x)\n",
" v = x[i]\n",
" v > m && ((m, ind, c) = (v, i, 1))\n",
" if v == m\n",
" c += 1\n",
" c * rand() < 1 && (ind = i)\n",
" end\n",
" end\n",
" ind\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"failure (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# structure holding beliefs about Beta distribution of alternatives probability of success\n",
"struct Belief\n",
" α::Float64\n",
" β::Float64\n",
"end\n",
"\n",
"prob(b::Belief) = b.α / (b.α + b.β)\n",
"success(b::Belief) = Belief(b.α + 1, b.β)\n",
"failure(b::Belief) = Belief(b.α, b.β + 1)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"run_experiment (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# runs simulation experiments; stores simulation results in a DataFrame\n",
"function run_experiment(run_id, reps, ground_truth, α₀, β₀, choice_rule)\n",
" decision = fill(-1, reps)\n",
" cum_payoff = fill(-1, reps)\n",
" options = length(ground_truth)\n",
" beliefs = [Belief(α₀, β₀) for i in 1:options]\n",
" for i in 1:reps\n",
" option = choice_rule(beliefs)\n",
" result = rand() < ground_truth[option]\n",
" cum_payoff[i] = i == 1 ? result : cum_payoff[i-1] + result\n",
" beliefs[option] = result ? success(beliefs[option]) : failure(beliefs[option])\n",
" decision[i] = samplemax(prob.(beliefs))\n",
" end\n",
" best = maximum(ground_truth)\n",
" DataFrame(run_id = run_id,\n",
" method = string(choice_rule),\n",
" iter = 1:reps,\n",
" loss = best .- ground_truth[decision],\n",
" correct_selection = ground_truth[decision] .== best,\n",
" payoff = cum_payoff)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"balanced_rule (generic function with 1 method)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# rule randomly choosing an alternative to measure\n",
"balanced_rule(beliefs) = rand(1:length(beliefs))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"adaptive_rule (generic function with 1 method)"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# creates a vector of expected probability of success of best alternative\n",
"function genc(p)\n",
" len = length(p)\n",
" c = fill(-Inf, len)\n",
" for (i, v) in enumerate(p), j in 1:len\n",
" i != j && (c[j] = max(c[j], v))\n",
" end\n",
" c\n",
"end\n",
"\n",
"# rule adaptively choosing what to measure to maximize probability of correct beliefs at the end of the process\n",
"function adaptive_rule(beliefs)\n",
" p = prob.(beliefs)\n",
" c = genc(p)\n",
" for (i,b) in enumerate(beliefs)\n",
" n = b.α + b.β\n",
" t = n * (c[i]<p[i] ? (p[i]-c[i])/c[i] : (c[i]-p[i])/(1-c[i]))\n",
" c[i] = (c[i] < p[i] ? (1-p[i]) : p[i]) ^ t / t\n",
" end\n",
" samplemax(c)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"gittins_rule (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# helper function for Gittins index approximation\n",
"function ψ(s)\n",
" if s ≤ 0.2\n",
" sqrt(s/2)\n",
" elseif s ≤ 1\n",
" 0.49 - 0.11/sqrt(s)\n",
" elseif s ≤ 5\n",
" 0.63 - 0.26/sqrt(s)\n",
" elseif s ≤ 15\n",
" 0.77 - 0.58/sqrt(s)\n",
" else\n",
" ls = log(s)\n",
" sqrt(2ls-log(ls)-3.9173186)\n",
" end\n",
"end\n",
"\n",
"# rule based on approximation of Gittins index that tries to maximize cumulative payoff\n",
"function gittins_rule(beliefs)\n",
" p = prob.(beliefs)\n",
" n = [b.α+b.β for b in beliefs]\n",
" v = p .+ sqrt.(p.*(1.-p)./(1.+n)).*ψ.(1000./(1.+n))\n",
" samplemax(v)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"experiment (generic function with 1 method)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# executes a set of experiments for assumed ground truth probabilities of success\n",
"function experiment(ground_truth)\n",
" res = DataFrame(run_id = Int[], method = String[], iter = Int[],\n",
" loss = Float64[], correct_selection = Bool[], payoff = Int[])\n",
" α = mean(ground_truth)\n",
" for rule in [balanced_rule, adaptive_rule, gittins_rule], i in 1:10000\n",
" if i % 1000 == 0\n",
" println((rule, i))\n",
" end\n",
" append!(res, run_experiment(i, 1000, ground_truth, α, 1-α, rule))\n",
" end\n",
"\n",
" agg = @by(res, [:method, :iter], loss=mean(:loss),\n",
" pcs=mean(:correct_selection),\n",
" payoff=mean(:payoff))\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(balanced_rule, 1000)\n",
"(balanced_rule, 2000)\n",
"(balanced_rule, 3000)\n",
"(balanced_rule, 4000)\n",
"(balanced_rule, 5000)\n",
"(balanced_rule, 6000)\n",
"(balanced_rule, 7000)\n",
"(balanced_rule, 8000)\n",
"(balanced_rule, 9000)\n",
"(balanced_rule, 10000)\n",
"(adaptive_rule, 1000)\n",
"(adaptive_rule, 2000)\n",
"(adaptive_rule, 3000)\n",
"(adaptive_rule, 4000)\n",
"(adaptive_rule, 5000)\n",
"(adaptive_rule, 6000)\n",
"(adaptive_rule, 7000)\n",
"(adaptive_rule, 8000)\n",
"(adaptive_rule, 9000)\n",
"(adaptive_rule, 10000)\n",
"(gittins_rule, 1000)\n",
"(gittins_rule, 2000)\n",
"(gittins_rule, 3000)\n",
"(gittins_rule, 4000)\n",
"(gittins_rule, 5000)\n",
"(gittins_rule, 6000)\n",
"(gittins_rule, 7000)\n",
"(gittins_rule, 8000)\n",
"(gittins_rule, 9000)\n",
"(gittins_rule, 10000)\n"
]
}
],
"source": [
"# Experiment 1: 1 good alternative, and 10 bad ones\n",
"ex1 = experiment([fill(0.1, 10); 0.15]);"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
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"@df ex1 plot(:iter, :loss, group = :method, title=\"Expected loss, one better\")"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
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"source": [
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},
{
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},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@df ex1 plot(:iter, :payoff, group = :method, legend=:topleft, title=\"Cumulative payoff, one better\")"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(balanced_rule, 1000)\n",
"(balanced_rule, 2000)\n",
"(balanced_rule, 3000)\n",
"(balanced_rule, 4000)\n",
"(balanced_rule, 5000)\n",
"(balanced_rule, 6000)\n",
"(balanced_rule, 7000)\n",
"(balanced_rule, 8000)\n",
"(balanced_rule, 9000)\n",
"(balanced_rule, 10000)\n",
"(adaptive_rule, 1000)\n",
"(adaptive_rule, 2000)\n",
"(adaptive_rule, 3000)\n",
"(adaptive_rule, 4000)\n",
"(adaptive_rule, 5000)\n",
"(adaptive_rule, 6000)\n",
"(adaptive_rule, 7000)\n",
"(adaptive_rule, 8000)\n",
"(adaptive_rule, 9000)\n",
"(adaptive_rule, 10000)\n",
"(gittins_rule, 1000)\n",
"(gittins_rule, 2000)\n",
"(gittins_rule, 3000)\n",
"(gittins_rule, 4000)\n",
"(gittins_rule, 5000)\n",
"(gittins_rule, 6000)\n",
"(gittins_rule, 7000)\n",
"(gittins_rule, 8000)\n",
"(gittins_rule, 9000)\n",
"(gittins_rule, 10000)\n"
]
}
],
"source": [
"# Experiment 2: 11 alternatives evenly spaced between 0.1 and 0.2\n",
"ex2 = experiment(0.1:0.01:0.2);"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
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],
"source": [
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]
},
{
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],
"source": [
"@df ex2 plot(:iter, :pcs, group = :method, legend=:topleft, title=\"Probability of correct selection, linspace\")"
]
},
{
"cell_type": "code",
"execution_count": 16,
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" 221.133,265.453 221.678,265.109 222.224,264.771 222.769,264.434 223.314,264.087 223.859,263.736 224.404,263.393 224.949,263.044 225.494,262.697 226.039,262.357 \n",
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" 237.485,255.121 238.03,254.788 238.575,254.438 239.12,254.098 239.665,253.746 240.21,253.388 240.755,253.043 241.3,252.694 241.845,252.347 242.39,252.007 \n",
" 242.935,251.675 243.48,251.324 244.025,250.983 244.57,250.636 245.115,250.308 245.66,249.965 246.205,249.626 246.75,249.268 247.295,248.931 247.84,248.591 \n",
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" 264.737,237.86 265.282,237.512 265.827,237.158 266.372,236.824 266.917,236.473 267.462,236.12 268.007,235.788 268.552,235.443 269.097,235.089 269.642,234.738 \n",
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" 379.195,164.016 379.74,163.672 380.285,163.305 380.83,162.956 381.375,162.58 381.92,162.228 382.465,161.883 383.01,161.525 383.555,161.177 384.1,160.827 \n",
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" 400.997,149.8 401.542,149.454 402.087,149.096 402.632,148.73 403.177,148.372 403.722,148.008 404.267,147.658 404.812,147.295 405.357,146.949 405.902,146.596 \n",
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" 460.951,110.467 461.496,110.103 462.041,109.746 462.586,109.384 463.131,109.016 463.676,108.661 464.221,108.298 464.766,107.945 465.311,107.587 465.857,107.232 \n",
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" 471.852,103.278 472.397,102.908 472.942,102.552 473.487,102.181 474.032,101.822 474.577,101.46 475.122,101.101 475.667,100.736 476.212,100.382 476.757,100.024 \n",
" 477.302,99.6661 477.847,99.303 478.392,98.9436 478.937,98.5834 479.483,98.2232 480.028,97.8752 480.573,97.5295 481.118,97.1723 481.663,96.8055 482.208,96.4439 \n",
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},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@df ex2 plot(:iter, :payoff, group = :method, legend=:topleft, title=\"Cumulative payoff, linspace\")"
]
}
],
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