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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The document outlines a probabilistic framework involving various statistical models, including Poisson and Dirichlet processes, for data generation and inference. It details categorical distributions and MCMC sampling methods for estimating parameters like mean (μ) and weight variables (w). Additionally, it presents the usage of normal distributions for observed data and integrates deterministic functions for specific outcomes based on conditions.
![4
Pr[✓|x] =
Pr[x|✓] Pr[✓]
P
✓ Pr[x|✓] Pr[✓]](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-4-320.jpg)
![8
Pr[ θ, Z, D ]
Pr[ θ | D ]
Z D](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-8-320.jpg)
![9
X
y2Dom(Y )
Pr[X, y] = Pr[X]
X
y2Dom(Y )
Pr[y|X] = Pr[X] ⇥ 1
+ =](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-9-320.jpg)
![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[✓]](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-10-320.jpg)
![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 ]](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-14-320.jpg)
![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}]](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-17-320.jpg)
![21
M = MCMC(input=[mu, x])
M.sample(iter=10000)
✤](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-21-320.jpg)
![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](https://image.slidesharecdn.com/2014-m-tokyoscipy6-140801092607-phpapp02/85/PyMC-24-320.jpg)
PyMCがあれば，ベイズ推定でもう泣いたりなんかしない
- 1. 1
- 3. 3
- 4. 4 Pr[✓|x] = Pr[x|✓] Pr[✓] P ✓ Pr[x|✓] Pr[✓]
- 5. 5
- 6. 6
- 7. 7 • • •
- 8. 8 Pr[ θ, Z, D ] Pr[ θ | D ] Z D
- 9. 9 X y2Dom(Y ) Pr[X, y] = Pr[X] X y2Dom(Y ) Pr[y|X] = Pr[X] ⇥ 1 + =
- 10. 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[✓]
- 11. 11 • • • • • •
- 12. 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. 13 z, z=1, . . . , K M N ✓ t ↵ w
- 14. 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 ]
- 15. 15
- 17. 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}]
- 18. 18 • • • • •
- 19. 19 mu = Normal('mu', 0, 1 / (0.1 ** 2)) ✤
- 20. x 20 x = Normal('x', mu=mu, tau=1/(1.0**2), value=x_sample, observed=True) x
- 21. 21 M = MCMC(input=[mu, x]) M.sample(iter=10000) ✤
- 23. 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. 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
- 25. 25