You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
Copy file name to clipboardExpand all lines: tutorials/hidden-markov-models/index.qmd
+93-41Lines changed: 93 additions & 41 deletions
Original file line number
Diff line number
Diff line change
@@ -12,17 +12,19 @@ using Pkg;
12
12
Pkg.instantiate();
13
13
```
14
14
15
-
This tutorial illustrates training Bayesian [Hidden Markov Models](https://en.wikipedia.org/wiki/Hidden_Markov_model) (HMM) using Turing. The main goals are learning the transition matrix, emission parameter, and hidden states. For a more rigorous academic overview on Hidden Markov Models, see [An introduction to Hidden Markov Models and Bayesian Networks](http://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.pdf) (Ghahramani, 2001).
15
+
This tutorial illustrates training Bayesian [hidden Markov models](https://en.wikipedia.org/wiki/Hidden_Markov_model) (HMMs) using Turing.
16
+
The main goals are learning the transition matrix, emission parameter, and hidden states.
17
+
For a more rigorous academic overview of hidden Markov models, see [An Introduction to Hidden Markov Models and Bayesian Networks](https://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.pdf) (Ghahramani, 2001).
16
18
17
19
In this tutorial, we assume there are $k$ discrete hidden states; the observations are continuous and normally distributed - centered around the hidden states. This assumption reduces the number of parameters to be estimated in the emission matrix.
18
20
19
-
Let's load the libraries we'll need. We also set a random seed (for reproducibility) and the automatic differentiation backend to forward mode (more [here]({{<metausing-turing-autodiff>}}) on why this is useful).
21
+
Let's load the libraries we'll need, and set a random seed for reproducibility.
20
22
21
23
```{julia}
22
24
# Load libraries.
23
-
using Turing, StatsPlots, Random
25
+
using Turing, StatsPlots, Random, Bijectors
24
26
25
-
# Set a random seed and use the forward_diff AD mode.
27
+
# Set a random seed
26
28
Random.seed!(12345678);
27
29
```
28
30
@@ -32,52 +34,23 @@ In this example, we'll use something where the states and emission parameters ar
32
34
33
35
```{julia}
34
36
# Define the emission parameter.
35
-
y = [
36
-
1.0,
37
-
1.0,
38
-
1.0,
39
-
1.0,
40
-
1.0,
41
-
1.0,
42
-
2.0,
43
-
2.0,
44
-
2.0,
45
-
2.0,
46
-
2.0,
47
-
2.0,
48
-
3.0,
49
-
3.0,
50
-
3.0,
51
-
3.0,
52
-
3.0,
53
-
3.0,
54
-
3.0,
55
-
2.0,
56
-
2.0,
57
-
2.0,
58
-
2.0,
59
-
1.0,
60
-
1.0,
61
-
1.0,
62
-
1.0,
63
-
1.0,
64
-
1.0,
65
-
1.0,
66
-
];
37
+
y = [fill(1.0, 6)..., fill(2.0, 6)..., fill(3.0, 7)...,
We can see that we have three states, one for each height of the plot (1, 2, 3). This height is also our emission parameter, so state one produces a value of one, state two produces a value of two, and so on.
75
48
76
49
Ultimately, we would like to understand three major parameters:
77
50
78
51
1. The transition matrix. This is a matrix that assigns a probability of switching from one state to any other state, including the state that we are already in.
79
-
2. The emission matrix, which describes a typical value emitted by some state. In the plot above, the emission parameter for state one is simply one.
80
-
3. The state sequence is our understanding of what state we were actually in when we observed some data. This is very important in more sophisticated HMM models, where the emission value does not equal our state.
52
+
2. The emission parameters, which describes a typical value emitted by some state. In the plot above, the emission parameter for state one is simply one.
53
+
3. The state sequence is our understanding of what state we were actually in when we observed some data. This is very important in more sophisticated HMMs, where the emission value does not equal our state.
81
54
82
55
With this in mind, let's set up our model. We are going to use some of our knowledge as modelers to provide additional information about our system. This takes the form of the prior on our emission parameter.
83
56
@@ -127,18 +100,22 @@ We will use a combination of two samplers (HMC and Particle Gibbs) by passing th
127
100
128
101
In this case, we use HMC for `m` and `T`, representing the emission and transition matrices respectively. We use the Particle Gibbs sampler for `s`, the state sequence. You may wonder why it is that we are not assigning `s` to the HMC sampler, and why it is that we need compositional Gibbs sampling at all.
129
102
130
-
The parameter `s` is not a continuous variable. It is a vector of **integers**, and thus Hamiltonian methods like HMC and NUTS won't work correctly. Gibbs allows us to apply the right tools to the best effect. If you are a particularly advanced user interested in higher performance, you may benefit from setting up your Gibbs sampler to use [different automatic differentiation]({{<metausing-turing-autodiff>}}#compositional-sampling-with-differing-ad-modes) backends for each parameter space.
103
+
The parameter `s` is not a continuous variable.
104
+
It is a vector of **integers**, and thus Hamiltonian methods like HMC and NUTS won't work correctly.
105
+
Gibbs allows us to apply the right tools to the best effect.
106
+
If you are a particularly advanced user interested in higher performance, you may benefit from setting up your Gibbs sampler to use [different automatic differentiation]({{<metausing-turing-autodiff>}}#compositional-sampling-with-differing-ad-modes) backends for each parameter space.
131
107
132
108
Time to run our sampler.
133
109
134
110
```{julia}
135
111
#| output: false
112
+
#| echo: false
136
113
setprogress!(false)
137
114
```
138
115
139
116
```{julia}
140
117
g = Gibbs((:m, :T) => HMC(0.01, 50), :s => PG(120))
The p-values on the test suggest that we cannot reject the hypothesis that the observed sequence comes from a stationary distribution, so we can be reasonably confident that our transition matrix has converged to something reasonable.
173
+
174
+
## Efficient Inference With The Forward Algorithm
175
+
176
+
While the above method works well for the simple example in this tutorial, some users may desire a more efficient method, especially when their model is more complicated.
177
+
One simple way to improve inference is to marginalize out the hidden states of the model with an appropriate algorithm, calculating only the posterior over the continuous random variables.
178
+
Not only does this allow more efficient inference via Rao-Blackwellization, but now we can sample our model with `NUTS()` alone, which is usually a much more performant MCMC kernel.
179
+
180
+
Thankfully, [HiddenMarkovModels.jl](https://github.com/gdalle/HiddenMarkovModels.jl) provides an extremely efficient implementation of many algorithms related to hidden Markov models. This allows us to rewrite our model as:
181
+
182
+
```{julia}
183
+
using HiddenMarkovModels
184
+
using FillArrays
185
+
using LinearAlgebra
186
+
using LogExpFunctions
187
+
188
+
189
+
@model function BayesHmm2(y, K)
190
+
m ~ Bijectors.ordered(MvNormal([1.0, 2.0, 3.0], 0.5I))
191
+
T ~ filldist(Dirichlet(fill(1/K, K)), K)
192
+
193
+
hmm = HMM(softmax(ones(K)), copy(T'), [Normal(m[i], 0.1) for i in 1:K])
194
+
Turing.@addlogprob! logdensityof(hmm, y)
195
+
end
196
+
197
+
chn2 = sample(BayesHmm2(y, 3), NUTS(), 1000)
198
+
```
199
+
200
+
201
+
We can compare the chains of these two models, confirming the posterior estimate is similar (modulo label switching concerns with the Gibbs model):
202
+
```{julia}
203
+
#| code-fold: true
204
+
#| code-summary: "Plotting Chains"
205
+
206
+
plot(chn["m[1]"], label = "m[1], Model 1, Gibbs", color = :lightblue)
207
+
plot!(chn2["m[1]"], label = "m[1], Model 2, NUTS", color = :blue)
208
+
plot!(chn["m[2]"], label = "m[2], Model 1, Gibbs", color = :pink)
209
+
plot!(chn2["m[2]"], label = "m[2], Model 2, NUTS", color = :red)
210
+
plot!(chn["m[3]"], label = "m[3], Model 1, Gibbs", color = :yellow)
211
+
plot!(chn2["m[3]"], label = "m[3], Model 2, NUTS", color = :orange)
212
+
```
213
+
214
+
215
+
### Recovering Marginalized Trajectories
216
+
217
+
We can use the `viterbi()` algorithm, also from the `HiddenMarkovModels` package, to recover the most probable state for each parameter set in our posterior sample:
218
+
```{julia}
219
+
@model function BayesHmmRecover(y, K, IncludeGenerated = false)
220
+
m ~ Bijectors.ordered(MvNormal([1.0, 2.0, 3.0], 0.5I))
221
+
T ~ filldist(Dirichlet(fill(1/K, K)), K)
222
+
223
+
hmm = HMM(softmax(ones(K)), copy(T'), [Normal(m[i], 0.1) for i in 1:K])
0 commit comments