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PyMCがあれば，ベイズ推定でもう泣いたりなんかしない

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Toshihiro Kamishima
Toshihiro Kamishima

ベイズ推定の基本とPyMCによる簡単な実装例です． 関連資料： https://github.com/scipy-japan/tokyo-scipy/tree/master/006/shima__shima

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4 Pr[✓|x] = Pr[x|✓] Pr[✓] P ✓ Pr[x|✓] Pr[✓]
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8 Pr[ θ, Z, D ] Pr[ θ | D ] Z D
9 X y2Dom(Y ) Pr[X, y] = Pr[X] X y2Dom(Y ) Pr[y|X] = Pr[X] ⇥ 1 + =
10 Pr[✓, D] = Pr[✓|D] Pr[D] Pr[✓|D] = Pr[✓, D] Pr[D] D Z D D Pr[✓|D] = Pr[✓, D] Pr[D] = Pr[✓, D] P ✓ Pr[✓, D] = Pr[D|✓] Pr[✓] P ✓ Pr[D|✓] Pr[✓]
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12 1. N Poisson(ξ) 2. θ Dirichlet(θ ; α) 3. For n in 1, …, N (a) zn Categorical(zn | θ) (b) wn Categorical(wn | zn; β)
13 z, z=1, . . . , K M N ✓ t ↵ w
14 1. N Poisson(ξ) 2. θ Dirichlet(θ ; α) 3. For n in 1 … N (a) zn Categorical(zn | θ) (b) wn Categorical(wn | zn; β) ✤ Pr[w, z, ✓; ↵, ] = Pr[✓|↵] NY n=1 Pr[zn|✓] Pr[wn|zn, zn ]
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17 1. μ Normal(0.0, 0.12) 2. For i in 1 … N (a) xi Normal(μ, 1.02) μ x N • x • μ Pr[µ|x1, . . . , xN ] = Pr[µ|{x}]
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19 mu = Normal('mu', 0, 1 / (0.1 ** 2)) ✤
x 20 x = Normal('x', mu=mu, tau=1/(1.0**2), value=x_sample, observed=True) x
21 M = MCMC(input=[mu, x]) M.sample(iter=10000) ✤
22 Matplot.plot(mu)
23 1. p Beta(1.0, 1.0) 2. μ0 Normal(-1, 1.0) 3. μ1 Normal(1, 1.0) 4. For i in 1 … N (a) yi Bernoulli(p) (b) μ = μ0 if yi = 0; μ1 if yi = 1 (c) xi Normal(μ, 1.02) y x N ✤ xi p μ0 μ1
24 @deterministic(plot=False) def mu(y=y, mu0=mu0, mu1=mu1): out = np.empty_like(y, dtype=np.float) out[y == 0] = mu0 out[y == 1] = mu1 return out
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PyMCがあれば，ベイズ推定でもう泣いたりなんかしない