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naive-bayes.jl
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naive-bayes.jl
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### A Pluto.jl notebook ###
# v0.16.1
using Markdown
using InteractiveUtils
# ╔═╡ 2c755fd0-216a-11ec-074d-f95187c6b596
using Combinatorics, DataFrames, Plots, PlutoUI, Random
# ╔═╡ 5191288a-d46c-44a6-b1d3-a4abc7604cb7
md"""
# Naive Bayes
## Teorema de Bayes
O Teorema de Bayes relaciona probabilidades condicionais:
$$P(A|B) = \frac{P(B|A) P(A)}{P(B)}.$$
**Exemplo:**
- Um teste aplicado em pessoas com Covid reporta Positivo para 95% das pessoas e Negativo para 5%.
- O mesmo teste em pessoas sem Covid reporta Negativo 99% das vezes e Positivo 1% das vezes.
- A taxa de incidência de COVID é 1%.
- Você faz o teste e dá Positivo. Qual a probabilidade de você ter COVID?
Vamos definir como $C$ o evento "estar com Covid" e $\oplus$ como "teste Positivo".
Usando o Teorema de Bayes, temos
$$P(C|\oplus) = \frac{P(\oplus|C) P(C)}{P(\oplus)}.$$
Pelo texto, temos
$$P(C) = 0.01, \quad P(\bar{C}) = 0.99$$
$$P(\oplus|C) = 0.95, \quad P(\oplus|\bar{C}) = 0.01$$
Podemos calcular $P(\oplus)$ pela lei de probabilidade total:
$P(\oplus) = P(\oplus|C)P(C) + P(\oplus|\bar{C})P(\bar{C}) = 0.95\times 0.01 + 0.01\times 0.99 = 0.0194$
Portanto,
$$P(C|\oplus) = \frac{0.95 \times 0.01}{0.0194} \approx 0.48969.$$
"""
# ╔═╡ c91b5d5a-d684-40bf-9843-27e65e455775
md"""
## Classificador de Bayes Ingênuo (Naive)
A ideia do classificador NB é tentar calcular a probabilidade de cada possível classe condicionada aos dados de entrada.Comparamos todas as probablidades e a probabilidade maior dita qual a classe escolhida para o indivíduo.
Isto é, para valores $x = (x_1,x_2,\dots,x_p)$, calculamos a probabilidade do indivíduo ser da classe $C_k$, dada por $P(C_k|x)$, $k = 1,\dots,N_c$.
Então, comparamos os valores de $P(C_k|x)$ e o que for maior dita qual classe é escolhida para $x$.
Para calcular $P(C_k|x)$, usamos o Teorema de Bayes, obtendo
$$P(C_k|x) = \frac{P(x|C_k) P(C_k)}{P(x)}.$$
O valor de $P(x)$ é o mesmo para todas as classes, então podemos ignorá-lo.
Basta comparar $P(x|C_k) P(C_k)$ para todos os valores de $k$.
Aqui que entra a parte da ingenuidade. Vamos assumir que as features são independentes, e que portanto podemos calcular esse valor como
$$P(C_k) P(x_1|C_k) P(x_2|C_k) \dots P(x_p|C_k).$$
Todos esses valores são estimados usando os dados de treino.
O valor de $P(C_k)$ é apenas a proporção de dados da classe.
O cálculo de $P(x_j|C_k)$ varia de acordo com o tipo dos dados.
Vamos ver apenas o mais básico.
Chega de conversa, vamos ao exemplo.
"""
# ╔═╡ 8bd6584e-9a32-4fc1-9242-12a41570e0e3
begin
n = 100
Random.seed!(0)
df = DataFrame(
tosse = rand(0:1, n),
coriza = rand(0:1, n),
olfato = rand(0:1, n),
gosto = rand(0:1, n),
)
sigmoid(t) = 1 / (1 + exp(0.3-t))
df.covid = round.(Int, sigmoid.(0.3 * randn(n) + Matrix(df) * [0.15; 0.2; 0.25; 0.25] .- 0.14))
df
end
# ╔═╡ f432aa4f-5695-42ac-85a8-e2e1b5bb0331
begin
PCc = [count(df.covid .== c) / n for c = 0:1]
PxjispgivenCc = [
begin
idx = findall(df.covid .== c)
round(count(df[idx,j] .== p) / length(idx), digits=2)
end
for j = 1:4, p = 0:1, c = 0:1
]
histogram(df.covid, leg=false)
title!("Proporção de classes")
xticks!([0.25; 1.25], ["0", "1"])
end
# ╔═╡ 7853c4cc-b45d-4b19-8935-ba58a209627e
Markdown.parse("""
\$\$P(C_0) = $(PCc[1]). \\qquad P(C_1) = $(PCc[2])\$\$
""")
# ╔═╡ 4ebf2f1b-a354-4852-8d89-762f234f6c0d
let
plt = plot(layout=grid(2, 4), size=(600, 300), leg=false)
ylabel!(plt[1,1], "Covid = 0")
ylabel!(plt[2,1], "Covid = 1")
for c = 0:1
idx = findall(df.covid .== c)
for j = 1:4
C = [count(df[idx,j] .== p) for p = 0:1]
histogram!(plt[c+1,j], df[idx,j])
annotate!(plt[c+1,j], [
(0.25, C[1] + 7, text(PxjispgivenCc[j,1,c+1], 10)),
(1.25, C[2] + 7, text(PxjispgivenCc[j,2,c+1], 10)),
])
title!(plt[c+1,j], "P(x$j|C$c)")
end
end
xticks!([0.25; 1.25], ["0", "1"])
yticks!(0:10:50)
xlims!(-0.3, 1.8)
ylims!(0, 50)
plot!()
end
# ╔═╡ 0bc84bfd-8f85-47d1-ba44-c82caf36cf47
let
x = [0; 1; 1; 0]
px0 = PCc[1] * prod(PxjispgivenCc[i,x[i]+1,1] for i = 1:4)
md"""
Exemplo:
- Tenho _coriza_ e perdi o _olfato_, mas sem _tosse_ e sentindo _gosto_ normal.
- Dados = $x = (0, 1, 1, 0)$.
- Probabilidades:
$$\begin{align*}
P(\bar{C}|x)P(x) & = P(\bar{C}) P(x_1=0|\bar{C}) P(x_2=1|\bar{C}) P(x_3=1|\bar{C}) P(x_4=0|\bar{C}) \\
& = 0.46 \times 0.59 \times 0.35 \times 0.39 \times 0.67 \\
& \approx 0.0248
\end{align*}$$
$$\begin{align*}
P(C|x)P(x) & = P(C) P(x_1=0|C) P(x_2=1|C) P(x_3=1|C) P(x_4=0|C) \\
& = 0.54 \times 0.39 \times 0.63 \times 0.76 \times 0.37 \\
& \approx 0.0373
\end{align*}$$
- Portanto, provavelmente é Covid. Melhoras.
"""
end
# ╔═╡ 27ca7c6a-b379-487c-a43d-5065f455bfdf
md"""
Como temos apenas 16 opções, calculei e coloquei abaixo:
"""
# ╔═╡ bb71281f-1e53-4770-a226-fb43827f9a54
begin
df_all = DataFrame(
hcat([[i1, i2, i3, i4] for i1=0:1, i2=0:1, i3=0:1, i4=0:1][:]...)',
names(df)[1:4],
)
df_all.pred = zeros(Int, 16)
for i = 1:16
pred = zeros(2)
for c = 0:1
pred[c+1] = PCc[c+1]
for j = 1:4
p = df_all[i,j] + 1
pred[c+1] *= PxjispgivenCc[j,p,c+1]
end
end
df_all.pred[i] = argmax(pred) - 1
end
df_all[1:8,:]
end
# ╔═╡ ebc116a0-3133-4367-a946-205025cc8cc9
df_all[9:16,:]
# ╔═╡ 5bcaab4f-10b1-4f5c-9362-68e256e9213c
let
df.pred = zeros(Int, n)
pred = zeros(n, 2)
for i = 1:n
for c = 0:1
pred[i,c+1] = PCc[c+1]
for j = 1:4
p = df[i,j]+1
pred[i,c+1] *= PxjispgivenCc[j,p,c+1]
end
end
df.pred[i] = argmax(pred[i,:]) - 1
end
p = count(df.covid .== df.pred) / n
"Também calculamos a acurácia do conjunto original: $(100p)% "
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Combinatorics = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
[compat]
Combinatorics = "~1.0.2"
DataFrames = "~1.2.2"
Plots = "~1.22.3"
PlutoUI = "~0.7.12"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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