Bayesian – Dealing with Multiple Definitions of a Node in WinBUGS

bayesianbugsmarkov chainwinbugs

So this question is about the BUGS modeling language. So you either know it or have no clue. I'm a newbie to this so it's been driving me mad. I want to define a simple two-state hidden Markov model (HMM) where the emission of each state follows a Normal distribution. I have an array data of Nxl dimension where each row is a subject and column is a continuous variable Time associated with the subject. So in WinBUGS, I would define the model as:

 for(i in 1:N) { # for each subject
    # Sample the initial observation
    Time[i, 1] ~ dnorm(mu[State[1]], tau[State[1]])

    for(j in 2:l[i]) { # for each observation
        # Sample the observed variables
        Time[i, j] ~ dnorm(mu[State[j]], tau[State[j]])
        # Sample the hidden states
        State[j] ~ dcat(P[State[j-1], ])
    }
}

# P is the transition matrix of the hidden Markov chain
P[1, 1:2] ~ ddirch(Pinit)
P[2, 1:2] ~ ddirch(Pinit)

# Set the initial states
State[1] ~ dcat(Pinit)

# Sample the initial params
mu[1] ~ dnorm(200, 1.0E-6)
mu[2] ~ dnorm(400, 1.0E-6)

# The precision params
tau[1] ~ dgamma(0.001, 0.001)
tau[2] ~ dgamma(0.001, 0.001)

Initial values have also been provided via R2WinBUGS. The model is syntactically correct. But when I run, I got this error: "multiple definitions of node State[2]"

Can you please tell me why and how to solve this?
I've searched around on the error but each has their own specific case and there's not a generic solution or explanation as to why this arises.

Best Answer

I don't know about HMM, but I can see that in every loop of i you are defining State[j]. This is causing the error in WinBUGS as it can not sample from each node (such as State[2]) more than once in each iteration in the MCMC.

You need to either somehow define State[j] out of the i loop (as you did for State[1]) in its own loop or switch to a [i,j] index for State. (Without knowing HMM, I am not sure which of these is the correct solution).

Related Solutions

Solved – Validate a Markov Chain with few states (model switching)

What you have to verify is your model, not your time series (though sample time series from your model can be used for this).

For example, you are assuming that your data is described by a first-order Markov chain, i.e., the probability of observing a given state only depends on the past state or, with other words, the process has a memory of 1. You can substantiate this assumption by showing that the process does not have a longer memory.

As an example, consider a Markov chain with only two states, called A and B. Let’s say that you have counted the subchains of length 3 with A in the middle and got:

A→A→A: 723 times
A→A→B: 387 times
B→A→A: 233 times
B→A→B: 130 times

If the Markov chain is of order 1, the proportions of subchains ending with A should not depend on whether the chain started with A or B. This is a general statistical task, which is often represented in form of a contingency table. The corresponding contigency table for this case would be:

               prior state
                 A    B   total
––––––––––––––––––––––––––––––––
next state A |  723  233   956
next state B |  387  130   517
total        | 1110  363  1473

This can, e.g., be tested with Fisher’s exact test, which for this data should yield that the next state and the prior state are independent and thus the state A in our example Markov chain does not preserve any memory of the preceding state (state B hower might do this, so we have to test it as well).

As you have six states, you would have to perform six such tests with 6×6 contingency tables. Be aware that this is multiple testing. Also, in theory your data can pass this test but have an even longer memory, but given your data, I consider this unlikely.

Another way would be comparing some statistical properties of your original data with your simulated ones (which should be the same), for example the distribution of state durations or waiting times between two instances of a state. Should these differ, there is something wrong about your model. (However, your model can still be wrong if the distributions comply.)

Of course you could go one step further and use your model to get a simulated time series of vehicle speeds and compare appropriate properties such as the autocorrelation or the frequency content (FFT) with your original data. Being stochastic makes the results somewhat different from those for regular time series, but they are still meaningful and should be comparable if your model is good.

Either way, the more independent properties you can reproduce, the better – but you will there may always be aspects your model fails to reproduce despite complying to the original data in many aspects. Be also careful that you do not use aspects for verification that your model fulfills by construction. For example the histogram for the individual Markov-chain states of your model will always comply with the one for the original data, as it’s determined by the transition rates.

Finally keep in mind that no model is perfect and that it may still be good for what you are doing despite failing to comply with the original data. For example if there is some low frequency content in your original data but your model fails to reproduce this, this may be alright, if what is important for your application happens on much smaller time scales.

Bayesian – Compute Likelihood of State Given Multiple Observations

This is correct: \begin{align}\mathbb P(S=S_i|\theta_1,\ldots,\theta_k) \propto \frac{\mathbb P(S=S_i|\theta_1)p(\theta_1)}{\pi_i}\cdots \frac{\mathbb P(S=S_i|\theta_k)p(\theta_k)}{\pi_i} \pi_i\\ \propto \mathbb P(S=S_i|\theta_1)\cdots \mathbb P(S=S_i|\theta_k) \pi_i^{-k+1} \end{align} and the $p(\theta_j)$'s play no role there.

Best Answer

Related Solutions

Solved – Validate a Markov Chain with few states (model switching)

Bayesian – Compute Likelihood of State Given Multiple Observations

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