def model(self):
"""
**Generative Model**
"""
# global parameters
gain = pyro.sample("gain", dist.HalfNormal(self.priors["gain_std"]))
init = pyro.sample(
"init",
dist.Dirichlet(torch.ones(self.Q, self.S + 1) /
(self.S + 1)).to_event(1),
)
init = expand_offtarget(init)
trans = pyro.sample(
"trans",
dist.Dirichlet(
torch.ones(self.Q, self.S + 1, self.S + 1) /
(self.S + 1)).to_event(2),
)
trans = expand_offtarget(trans)
lamda = pyro.sample(
"lamda",
dist.Exponential(torch.full(
(self.Q, ), self.priors["lamda_rate"])).to_event(1),
)
proximity = pyro.sample(
"proximity", dist.Exponential(self.priors["proximity_rate"]))
size = torch.stack(
(
torch.full_like(proximity, 2.0),
(((self.data.P + 1) / (2 * proximity))**2 - 1),
),
dim=-1,
)
# spots
spots = pyro.plate("spots", self.K)
# aoi sites
aois = pyro.plate(
"aois",
self.data.Nt,
subsample=self.n,
subsample_size=self.nbatch_size,
dim=-3,
)
# time frames
frames = (pyro.vectorized_markov(
name="frames", size=self.data.F, dim=-2)
if self.vectorized else pyro.markov(range(self.data.F)))
# color channels
channels = pyro.plate(
"channels",
self.data.C,
dim=-1,
)
with channels as cdx, aois as ndx:
ndx = ndx[:, None, None]
mask = Vindex(self.data.mask)[ndx].to(self.device)
with handlers.mask(mask=mask):
# background mean and std
background_mean = pyro.sample(
"background_mean",
dist.HalfNormal(self.priors["background_mean_std"]),
)
background_std = pyro.sample(
"background_std",
dist.HalfNormal(self.priors["background_std_std"]))
z_prev = None
for fdx in frames:
if self.vectorized:
fsx, fdx = fdx
fdx = torch.as_tensor(fdx)
fdx = fdx.unsqueeze(-1)
else:
fsx = fdx
# fetch data
obs, target_locs, is_ontarget = self.data.fetch(
ndx, fdx, cdx)
# sample background intensity
background = pyro.sample(
f"background_f{fsx}",
dist.Gamma(
(background_mean / background_std)**2,
background_mean / background_std**2,
),
)
# sample hidden model state (1+S,)
z_probs = (Vindex(init)[..., cdx, :,
is_ontarget.long()]
if z_prev is None else
Vindex(trans)[..., cdx, z_prev, :,
is_ontarget.long()])
z_curr = pyro.sample(f"z_f{fsx}",
dist.Categorical(z_probs))
theta = pyro.sample(
f"theta_f{fsx}",
dist.Categorical(
Vindex(probs_theta(
self.K, self.device))[torch.clamp(z_curr,
min=0,
max=1)]),
infer={"enumerate": "parallel"},
)
onehot_theta = one_hot(theta, num_classes=1 + self.K)
ms, heights, widths, xs, ys = [], [], [], [], []
for kdx in spots:
specific = onehot_theta[..., 1 + kdx]
# spot presence
m_probs = Vindex(probs_m(lamda, self.K))[..., cdx,
theta, kdx]
m = pyro.sample(
f"m_k{kdx}_f{fsx}",
dist.Categorical(
torch.stack((1 - m_probs, m_probs), -1)),
)
with handlers.mask(mask=m > 0):
# sample spot variables
height = pyro.sample(
f"height_k{kdx}_f{fsx}",
dist.HalfNormal(self.priors["height_std"]),
)
width = pyro.sample(
f"width_k{kdx}_f{fsx}",
AffineBeta(
1.5,
2,
self.priors["width_min"],
self.priors["width_max"],
),
)
x = pyro.sample(
f"x_k{kdx}_f{fsx}",
AffineBeta(
0,
Vindex(size)[..., specific],
-(self.data.P + 1) / 2,
(self.data.P + 1) / 2,
),
)
y = pyro.sample(
f"y_k{kdx}_f{fsx}",
AffineBeta(
0,
Vindex(size)[..., specific],
-(self.data.P + 1) / 2,
(self.data.P + 1) / 2,
),
)
# append
ms.append(m)
heights.append(height)
widths.append(width)
xs.append(x)
ys.append(y)
# observed data
pyro.sample(
f"data_f{fsx}",
KSMOGN(
torch.stack(heights, -1),
torch.stack(widths, -1),
torch.stack(xs, -1),
torch.stack(ys, -1),
target_locs,
background,
gain,
self.data.offset.samples,
self.data.offset.logits.to(self.dtype),
self.data.P,
torch.stack(torch.broadcast_tensors(*ms), -1),
use_pykeops=self.use_pykeops,
),
obs=obs,
)
z_prev = z_curr
def model(self):
r"""
Generative Model
"""
# global parameters
gain = pyro.sample("gain", dist.HalfNormal(self.priors["gain_std"]))
alpha = pyro.sample(
"alpha",
dist.Dirichlet(
torch.ones((self.Q, self.data.C)) +
torch.eye(self.Q) * 9).to_event(1),
)
pi = pyro.sample(
"pi",
dist.Dirichlet(torch.ones(
(self.Q, self.S + 1)) / (self.S + 1)).to_event(1),
)
pi = expand_offtarget(pi)
lamda = pyro.sample(
"lamda",
dist.Exponential(torch.full(
(self.Q, ), self.priors["lamda_rate"])).to_event(1),
)
proximity = pyro.sample(
"proximity", dist.Exponential(self.priors["proximity_rate"]))
size = torch.stack(
(
torch.full_like(proximity, 2.0),
(((self.data.P + 1) / (2 * proximity))**2 - 1),
),
dim=-1,
)
# aoi sites
aois = pyro.plate(
"aois",
self.data.Nt,
subsample=self.n,
subsample_size=self.nbatch_size,
dim=-2,
)
# time frames
frames = pyro.plate(
"frames",
self.data.F,
subsample=self.f,
subsample_size=self.fbatch_size,
dim=-1,
)
with aois as ndx:
ndx = ndx[:, None]
mask = Vindex(self.data.mask)[ndx].to(self.device)
with handlers.mask(mask=mask):
# background mean and std
background_mean = pyro.sample(
"background_mean",
dist.HalfNormal(self.priors["background_mean_std"]).expand(
(self.data.C, )).to_event(1),
)
background_std = pyro.sample(
"background_std",
dist.HalfNormal(self.priors["background_std_std"]).expand(
(self.data.C, )).to_event(1),
)
with frames as fdx:
# fetch data
obs, target_locs, is_ontarget = self.data.fetch(
ndx.unsqueeze(-1), fdx.unsqueeze(-1),
torch.arange(self.data.C))
# sample background intensity
background = pyro.sample(
"background",
dist.Gamma(
(background_mean / background_std)**2,
background_mean / background_std**2,
).to_event(1),
)
ms, heights, widths, xs, ys = [], [], [], [], []
is_ontarget = is_ontarget.squeeze(-1)
for qdx in range(self.Q):
# sample hidden model state (1+S,)
z_probs = Vindex(pi)[..., qdx, :, is_ontarget.long()]
z = pyro.sample(
f"z_q{qdx}",
dist.Categorical(z_probs),
infer={"enumerate": "parallel"},
)
theta = pyro.sample(
f"theta_q{qdx}",
dist.Categorical(
Vindex(probs_theta(
self.K, self.device))[torch.clamp(z,
min=0,
max=1)]),
infer={"enumerate": "parallel"},
)
onehot_theta = one_hot(theta, num_classes=1 + self.K)
for kdx in range(self.K):
specific = onehot_theta[..., 1 + kdx]
# spot presence
m = pyro.sample(
f"m_k{kdx}_q{qdx}",
dist.Bernoulli(
Vindex(probs_m(lamda,
self.K))[..., qdx, theta,
kdx]),
)
with handlers.mask(mask=m > 0):
# sample spot variables
height = pyro.sample(
f"height_k{kdx}_q{qdx}",
dist.HalfNormal(self.priors["height_std"]),
)
width = pyro.sample(
f"width_k{kdx}_q{qdx}",
AffineBeta(
1.5,
2,
self.priors["width_min"],
self.priors["width_max"],
),
)
x = pyro.sample(
f"x_k{kdx}_q{qdx}",
AffineBeta(
0,
Vindex(size)[..., specific],
-(self.data.P + 1) / 2,
(self.data.P + 1) / 2,
),
)
y = pyro.sample(
f"y_k{kdx}_q{qdx}",
AffineBeta(
0,
Vindex(size)[..., specific],
-(self.data.P + 1) / 2,
(self.data.P + 1) / 2,
),
)
# append
ms.append(m)
heights.append(height)
widths.append(width)
xs.append(x)
ys.append(y)
heights = torch.stack(
[
torch.stack(heights[q * self.K:(1 + q) * self.K],
-1) for q in range(self.Q)
],
-2,
)
widths = torch.stack(
[
torch.stack(widths[q * self.K:(1 + q) * self.K],
-1) for q in range(self.Q)
],
-2,
)
xs = torch.stack(
[
torch.stack(xs[q * self.K:(1 + q) * self.K], -1)
for q in range(self.Q)
],
-2,
)
ys = torch.stack(
[
torch.stack(ys[q * self.Q:(1 + q) * self.K], -1)
for q in range(self.Q)
],
-2,
)
ms = torch.broadcast_tensors(*ms)
ms = torch.stack(
[
torch.stack(ms[q * self.Q:(1 + q) * self.K], -1)
for q in range(self.Q)
],
-2,
)
# observed data
pyro.sample(
"data",
KSMOGN(
heights,
widths,
xs,
ys,
target_locs,
background,
gain,
self.data.offset.samples,
self.data.offset.logits.to(self.dtype),
self.data.P,
ms,
alpha,
use_pykeops=self.use_pykeops,
),
obs=obs,
)
def model(self):
r"""
**Generative Model**
Model parameters:
+-----------------+-----------+-------------------------------------+
| Parameter | Shape | Description |
+=================+===========+=====================================+
| |g| - :math:`g` | (1,) | camera gain |
+-----------------+-----------+-------------------------------------+
| |sigma| - |prox|| (1,) | proximity |
+-----------------+-----------+-------------------------------------+
| ``lamda`` - |ld|| (1,) | average rate of target-nonspecific |
| | | binding |
+-----------------+-----------+-------------------------------------+
| ``pi`` - |pi| | (1,) | average binding probability of |
| | | target-specific binding |
+-----------------+-----------+-------------------------------------+
| |bg| - |b| | (N, F) | background intensity |
+-----------------+-----------+-------------------------------------+
| |z| - :math:`z` | (N, F) | target-specific spot presence |
+-----------------+-----------+-------------------------------------+
| |t| - |theta| | (N, F) | target-specific spot index |
+-----------------+-----------+-------------------------------------+
| |m| - :math:`m` | (K, N, F) | spot presence indicator |
+-----------------+-----------+-------------------------------------+
| |h| - :math:`h` | (K, N, F) | spot intensity |
+-----------------+-----------+-------------------------------------+
| |w| - :math:`w` | (K, N, F) | spot width |
+-----------------+-----------+-------------------------------------+
| |x| - :math:`x` | (K, N, F) | spot position on x-axis |
+-----------------+-----------+-------------------------------------+
| |y| - :math:`y` | (K, N, F) | spot position on y-axis |
+-----------------+-----------+-------------------------------------+
| |D| - :math:`D` | |shape| | observed images |
+-----------------+-----------+-------------------------------------+
.. |ps| replace:: :math:`p(\mathsf{specific})`
.. |theta| replace:: :math:`\theta`
.. |prox| replace:: :math:`\sigma^{xy}`
.. |ld| replace:: :math:`\lambda`
.. |b| replace:: :math:`b`
.. |shape| replace:: (N, F, P, P)
.. |sigma| replace:: ``proximity``
.. |bg| replace:: ``background``
.. |h| replace:: ``height``
.. |w| replace:: ``width``
.. |D| replace:: ``data``
.. |m| replace:: ``m``
.. |z| replace:: ``z``
.. |t| replace:: ``theta``
.. |x| replace:: ``x``
.. |y| replace:: ``y``
.. |pi| replace:: :math:`\pi`
.. |g| replace:: ``gain``
Full joint distribution:
.. math::
\begin{aligned}
p(D, \phi) =~&p(g) p(\sigma^{xy}) p(\pi) p(\lambda)
\prod_{\mathsf{AOI}} \left[ p(\mu^b) p(\sigma^b) \prod_{\mathsf{frame}}
\left[ \vphantom{\prod_{F}} p(b | \mu^b, \sigma^b) p(z | \pi) p(\theta | z)
\vphantom{\prod_{\substack{\mathsf{pixelX} \\ \mathsf{pixelY}}}} \cdot \right. \right. \\
&\prod_{\mathsf{spot}} \left[ \vphantom{\prod_{F}} p(m | \theta, \lambda)
p(h) p(w) p(x | \sigma^{xy}, \theta) p(y | \sigma^{xy}, \theta) \right] \left. \left.
\prod_{\substack{\mathsf{pixelX} \\ \mathsf{pixelY}}} \sum_{\delta} p(\delta)
p(D | \mu^I, g, \delta) \right] \right]
\end{aligned}
:math:`z` and :math:`\theta` marginalized joint distribution:
.. math::
\begin{aligned}
\sum_{z, \theta} p(D, \phi) =~&p(g) p(\sigma^{xy}) p(\pi) p(\lambda)
\prod_{\mathsf{AOI}} \left[ p(\mu^b) p(\sigma^b) \prod_{\mathsf{frame}}
\left[ \vphantom{\prod_{F}} p(b | \mu^b, \sigma^b) \sum_{z} p(z | \pi) \sum_{\theta} p(\theta | z)
\vphantom{\prod_{\substack{\mathsf{pixelX} \\ \mathsf{pixelY}}}} \cdot \right. \right. \\
&\prod_{\mathsf{spot}} \left[ \vphantom{\prod_{F}} p(m | \theta, \lambda)
p(h) p(w) p(x | \sigma^{xy}, \theta) p(y | \sigma^{xy}, \theta) \right] \left. \left.
\prod_{\substack{\mathsf{pixelX} \\ \mathsf{pixelY}}} \sum_{\delta} p(\delta)
p(D | \mu^I, g, \delta) \right] \right]
\end{aligned}
"""
# global parameters
gain = pyro.sample("gain", dist.HalfNormal(self.gain_std))
pi = pyro.sample("pi",
dist.Dirichlet(torch.ones(self.S + 1) / (self.S + 1)))
pi = expand_offtarget(pi)
lamda = pyro.sample("lamda", dist.Exponential(self.lamda_rate))
proximity = pyro.sample("proximity",
dist.Exponential(self.proximity_rate))
size = torch.stack(
(
torch.full_like(proximity, 2.0),
(((self.data.P + 1) / (2 * proximity))**2 - 1),
),
dim=-1,
)
# spots
spots = pyro.plate("spots", self.K)
# aoi sites
aois = pyro.plate(
"aois",
self.data.Nt,
subsample=self.n,
subsample_size=self.nbatch_size,
dim=-2,
)
# time frames
frames = pyro.plate(
"frames",
self.data.F,
subsample=self.f,
subsample_size=self.fbatch_size,
dim=-1,
)
with aois as ndx:
ndx = ndx[:, None]
# background mean and std
background_mean = pyro.sample(
"background_mean", dist.HalfNormal(self.background_mean_std))
background_std = pyro.sample(
"background_std", dist.HalfNormal(self.background_std_std))
with frames as fdx:
# fetch data
obs, target_locs, is_ontarget = self.data.fetch(
ndx, fdx, self.cdx)
# sample background intensity
background = pyro.sample(
"background",
dist.Gamma(
(background_mean / background_std)**2,
background_mean / background_std**2,
),
)
# sample hidden model state (1+S,)
z = pyro.sample(
"z",
dist.Categorical(Vindex(pi)[..., :,
is_ontarget.long()]),
infer={"enumerate": "parallel"},
)
theta = pyro.sample(
"theta",
dist.Categorical(
Vindex(probs_theta(self.K,
self.device))[torch.clamp(z,
min=0,
max=1)]),
infer={"enumerate": "parallel"},
)
onehot_theta = one_hot(theta, num_classes=1 + self.K)
ms, heights, widths, xs, ys = [], [], [], [], []
for kdx in spots:
specific = onehot_theta[..., 1 + kdx]
# spot presence
m = pyro.sample(
f"m_{kdx}",
dist.Bernoulli(
Vindex(probs_m(lamda, self.K))[..., theta, kdx]),
)
with handlers.mask(mask=m > 0):
# sample spot variables
height = pyro.sample(
f"height_{kdx}",
dist.HalfNormal(self.height_std),
)
width = pyro.sample(
f"width_{kdx}",
AffineBeta(
1.5,
2,
self.width_min,
self.width_max,
),
)
x = pyro.sample(
f"x_{kdx}",
AffineBeta(
0,
Vindex(size)[..., specific],
-(self.data.P + 1) / 2,
(self.data.P + 1) / 2,
),
)
y = pyro.sample(
f"y_{kdx}",
AffineBeta(
0,
Vindex(size)[..., specific],
-(self.data.P + 1) / 2,
(self.data.P + 1) / 2,
),
)
# append
ms.append(m)
heights.append(height)
widths.append(width)
xs.append(x)
ys.append(y)
# observed data
pyro.sample(
"data",
KSMOGN(
torch.stack(heights, -1),
torch.stack(widths, -1),
torch.stack(xs, -1),
torch.stack(ys, -1),
target_locs,
background,
gain,
self.data.offset.samples,
self.data.offset.logits.to(self.dtype),
self.data.P,
torch.stack(torch.broadcast_tensors(*ms), -1),
self.use_pykeops,
),
obs=obs,
)