nchopin
commented
1 month ago Hi, That might be a bug. By `sampled covariance', you mean the empirical variance of the particle sample, which is used inside the (random walk) Metropolis step? This should (in principle) not be a problem: you may propose values outside the support of the distribution (i.e. outside the interval of a Uniform dist), but then these proposed values will be rejected because their prior density should be zero. So again maybe there is a bug in either Uniform of Cond. If you want, you can submit a MWR (=minimum working example), so that I can try to debug it.
glandfried
Thank you very much, Nicolas Chopin, for the package. I have carefully read your book. SMC is exactly what I needed to approximate the model evidence.
Before starting to use the package, I compared the results of the
particlespackage with my own implementation of the Bayesian linear (polynomial) regression model, which has analytical solutions for both the posterior and the evidence. They produced identical results. Amazing.Since then, I have implemented several models. However, I encountered difficulties every time I tried to implement a conditional prior.
Example.
We want to compare two traditional models used for evaluating diagnostic tests without a gold standard: the Hui-Walter Model, which assumes independence between tests, and the Dendukuri Model, which includes the covariance between pairs of tests as an additional hypothesis.
Evaluating the performance of a diagnostic test is straightforward when we have gold standard data. In such cases, the probability of a diagnosis given the true state, $P(Diagnosis|State)$, equals their contingency table (in the long run).
However, when the true state of the patient is unknown, we must evaluate the alternative tests based solely on their imperfect diagnoses. The observable outcome of two tests can be represented by vector $r = (n_0,n_1,n_2,n_3)$, where $n_0$, $n_1$, $n_2$, and $n_3$ are the number of joint diagnoses $00$, $01$, $10$, and $11$ respectively.
The Hui-Walter model assumes that the sensitivity ($s$) and specificity ($x$) of the tests are independent of each other. Then, the probability of a joint diagnosis for an individual in a population with prevalence $p$ is a vector $q = (q_0, q_1, q_2, q_3)$, where the probability of two negative diagnoses is
$$q_0 = p\cdot(1-s_a)\cdot(1-s_b) + (1-p)\cdot x_a \cdot x_b$$
Probabilities $q_1$ to $q_3$ can be seen in the next figure, where we specify the model using the factor graph notation.
Implementing this models was straightforward. Because I decide to feed the model with one data point at a time, I replace the multinomial distribution by a categorical one.
In contrast, the Dendukuri model was proposed to evaluate sensitivity and specificity in the presence of conditional dependencies by incorporating the covariance between tests as an additional hypothesis. The following figure specify this model using the factor graph notation.
The implementation of the Dendukuiri model is similar to the Hui-Walter model, but we must add a conditional distribution over the covariance variables.
This implementation runs, but encounters a problema. If we add the next code into the logpyt method of the class Dendukuiri,
we will see that there are negative probabilities.
The problem is that the sampled covariance for the sensitivity
csand specificitycxlies outside their lower and upper bound. I had a similar problem with another model. I have not found sufficient examples in the documentation to understand how to resolve this error.