Hidden Markov Models
Contents
Hidden Markov Models¶
In this section, we introduce Hidden Markov Models (HMMs).
Boilerplate¶
# Install necessary libraries
try:
import jax
except:
# For cuda version, see https://github.com/google/jax#installation
%pip install --upgrade "jax[cpu]"
import jax
try:
import jsl
except:
%pip install git+https://github.com/probml/jsl
import jsl
try:
import rich
except:
%pip install rich
import rich
# Import standard libraries
import abc
from dataclasses import dataclass
import functools
import itertools
from typing import Any, Callable, NamedTuple, Optional, Union, Tuple
import matplotlib.pyplot as plt
import numpy as np
import jax
import jax.numpy as jnp
from jax import lax, vmap, jit, grad
from jax.scipy.special import logit
from jax.nn import softmax
from functools import partial
from jax.random import PRNGKey, split
import inspect
import inspect as py_inspect
from rich import inspect as r_inspect
from rich import print as r_print
def print_source(fname):
r_print(py_inspect.getsource(fname))
Utility code¶
def normalize(u, axis=0, eps=1e-15):
'''
Normalizes the values within the axis in a way that they sum up to 1.
Parameters
----------
u : array
axis : int
eps : float
Threshold for the alpha values
Returns
-------
* array
Normalized version of the given matrix
* array(seq_len, n_hidden) :
The values of the normalizer
'''
u = jnp.where(u == 0, 0, jnp.where(u < eps, eps, u))
c = u.sum(axis=axis)
c = jnp.where(c == 0, 1, c)
return u / c, c
Example: Casino HMM¶
We first create the “Ocassionally dishonest casino” model from [DEKM98].
Fig. 3 Illustration of the casino HMM.¶
There are 2 hidden states, each of which emit 6 possible observations.
# state transition matrix
A = np.array([
[0.95, 0.05],
[0.10, 0.90]
])
# observation matrix
B = np.array([
[1/6, 1/6, 1/6, 1/6, 1/6, 1/6], # fair die
[1/10, 1/10, 1/10, 1/10, 1/10, 5/10] # loaded die
])
pi, _ = normalize(np.array([1, 1]))
pi = np.array(pi)
(nstates, nobs) = np.shape(B)
WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)
Let’s make a little data structure to store all the parameters. We use NamedTuple rather than dataclass, since we assume these are immutable. (Also, standard python dataclass does not work well with JAX, which requires parameters to be pytrees, as discussed in https://github.com/google/jax/issues/2371).
class HMM(NamedTuple):
trans_mat: jnp.array # A : (n_states, n_states)
obs_mat: jnp.array # B : (n_states, n_obs)
init_dist: jnp.array # pi : (n_states)
params = HMM(A, B, pi)
print(params)
print(type(params.trans_mat))
HMM(trans_mat=array([[0.95, 0.05],
[0.1 , 0.9 ]]), obs_mat=array([[0.16666667, 0.16666667, 0.16666667, 0.16666667, 0.16666667,
0.16666667],
[0.1 , 0.1 , 0.1 , 0.1 , 0.1 ,
0.5 ]]), init_dist=array([0.5, 0.5], dtype=float32))
<class 'numpy.ndarray'>
Sampling from the joint¶
Let’s write code to sample from this model. First we code it in numpy using a for loop. Then we rewrite it to use jax.lax.scan, which is faster.
def hmm_sample_numpy(params, seq_len, random_state=0):
def sample_one_step_(hist, a, p):
x_t = np.random.choice(a=a, p=p)
return np.append(hist, [x_t]), x_t
np.random.seed(random_state)
trans_mat, obs_mat, init_dist = params.trans_mat, params.obs_mat, params.init_dist
n_states, n_obs = obs_mat.shape
state_seq = np.array([], dtype=int)
obs_seq = np.array([], dtype=int)
latent_states = np.arange(n_states)
obs_states = np.arange(n_obs)
state_seq, zt = sample_one_step_(state_seq, latent_states, init_dist)
obs_seq, xt = sample_one_step_(obs_seq, obs_states, obs_mat[zt])
for _ in range(1, seq_len):
state_seq, zt = sample_one_step_(state_seq, latent_states, trans_mat[zt])
obs_seq, xt = sample_one_step_(obs_seq, obs_states, obs_mat[zt])
return state_seq, obs_seq
seq_len = 20
state_seq, obs_seq = hmm_sample_numpy(params, seq_len, random_state=0)
print(state_seq)
print(obs_seq)
[1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0]
[5 5 5 5 3 5 5 0 4 5 5 5 5 5 4 5 5 3 3 4]
Now let’s write a JAX version.
#@partial(jit, static_argnums=(1,))
def hmm_sample(params, seq_len, rng_key):
trans_mat, obs_mat, init_dist = params.trans_mat, params.obs_mat, params.init_dist
n_states, n_obs = obs_mat.shape
initial_state = jax.random.categorical(rng_key, logits=logit(init_dist), shape=(1,))
obs_states = jnp.arange(n_obs)
def draw_state(prev_state, key):
logits = logit(trans_mat[:, prev_state])
state = jax.random.categorical(key, logits=logits.flatten(), shape=(1,))
return state, state
rng_key, rng_state, rng_obs = jax.random.split(rng_key, 3)
keys = jax.random.split(rng_state, seq_len - 1)
final_state, states = jax.lax.scan(draw_state, initial_state, keys)
state_seq = jnp.append(jnp.array([initial_state]), states)
def draw_obs(z, key):
obs = jax.random.choice(key, a=obs_states, p=obs_mat[z])
return obs
keys = jax.random.split(rng_obs, seq_len)
obs_seq = jax.vmap(draw_obs, in_axes=(0, 0))(state_seq, keys)
return state_seq, obs_seq
seq_len = 20
state_seq, obs_seq = hmm_sample(params, seq_len, PRNGKey(1))
print(state_seq)
print(obs_seq)
---------------------------------------------------------------------------
TracerArrayConversionError Traceback (most recent call last)
<ipython-input-9-0d29578f3271> in <module>
1 seq_len = 20
----> 2 state_seq, obs_seq = hmm_sample(params, seq_len, PRNGKey(1))
3 print(state_seq)
4 print(obs_seq)
<ipython-input-8-373c99e1e67e> in hmm_sample(params, seq_len, rng_key)
16 keys = jax.random.split(rng_state, seq_len - 1)
17
---> 18 final_state, states = jax.lax.scan(draw_state, initial_state, keys)
19 state_seq = jnp.append(jnp.array([initial_state]), states)
20
[... skipping hidden 13 frame]
<ipython-input-8-373c99e1e67e> in draw_state(prev_state, key)
9
10 def draw_state(prev_state, key):
---> 11 logits = logit(trans_mat[:, prev_state])
12 state = jax.random.categorical(key, logits=logits.flatten(), shape=(1,))
13 return state, state
/opt/anaconda3/lib/python3.8/site-packages/jax/core.py in __array__(self, *args, **kw)
468
469 def __array__(self, *args, **kw):
--> 470 raise TracerArrayConversionError(self)
471
472 def __index__(self):
TracerArrayConversionError: The numpy.ndarray conversion method __array__() was called on the JAX Tracer object Traced<ShapedArray(int32[1])>with<DynamicJaxprTrace(level=1/0)>
While tracing the function draw_state at <ipython-input-8-373c99e1e67e>:10 for scan, this concrete value was not available in Python because it depends on the value of the argument 'prev_state'.
See https://jax.readthedocs.io/en/latest/errors.html#jax.errors.TracerArrayConversionError