Fiducial distribution and sequential monte-carlo algorithm

Question

I have spent a little time reading "Generalized fiducial inference for normal mixed models" by Cisewski and Hannig. First of all I am interested in understanding how to simulate the fiducial distribution of the parameters. But I don't understand this point.

Consider for instance the one-way random effects model $y_{ij}=\mu + \alpha_i + \epsilon_{ij}$ with $\alpha_i \sim {\cal N}(0,\sigma_a)$ independent of $\epsilon_{ij} \sim {\cal N}(0,\sigma)$.

• First of all the authors write $\alpha_i= \sigma_a u_{i}$ and $\epsilon_{ij}=\sigma v_{ij}$ with $u_{i}$ and $v_{ij}$ independent standard normal random variables.

• In fact they consider the data are rounded. For instance when $y_{ij}=1.32$ then actually the $(i,j)$-datum is the interval $[a_{ij}, b_{ij}[=[1.315, 1.325[$ because the "real" observation is rounded with two decimals. Thus the model is $$a_{ij} \leq \mu + \sigma_a u_i+\sigma v_{ij} < b_{ij}.$$.

• Denote by $Z$ the random vector made of all the random components of the model: all the $u_i$'s and all the $v_{ij}'s$.

• Define the set $Q\bigl(a,b, u, v\bigr) = \bigl\{ (\mu,\sigma_a,\sigma) \mid a \leq \mu + \sigma_a u + \sigma v < b\bigr\}$ and then define the random set $$Q(\text{data}, Z) = \bigcap_{i,j}Q\bigl(a_{ij},b_{ij},u_i,v_{ij}\bigr)$$ (with randomness contained in $Z$, and the data are fixed). When this set is nonempty we denote by $V\left(Z\right)$ an element of this set selected according to a specified rule (at least in a measurable way). The (generalized) fiducial distribution of $(\mu,\sigma_a,\sigma_b)$ is the conditional distribution of $V(Z)$ given $Q(\text{data}, Z) \neq \varnothing$.

• Then the authors propose a sequential Monte-Carlo (SMC) generator of the fiducial distribution, and this is the point that confuses me. It starts on page 9 of the paper.

• The authors claim that generating the fiducial distribution is equivalent to generating $Z$ conditional on $Q(\text{data}, Z) \neq \varnothing$. I agree with this part but they do not propose a measurable rule $V$, or I have missed this point while reading the paper (or is it an obvious point ?).

• For $t=1, \ldots, n$ where $n$ is the sample size, denote by $Z_{1:t}$ the random components corresponding to the first $t$ data values (my notation is not appropriate here). Then the authors propose a sequential algorithm whose first step at time $t$ requires them to simulate the distribution $\pi_t$ of $Z_{1:t}$.

• There are some undefined notations in the preprinted version whose link is given above: $Q_t$ and $Q_t^{(J)}$ and the definition of the set $C_t$ is not well-written. It is easy to guess the definition of $Q_t$: for $t=1,\ldots,n$ the random set $Q_t$ is constructed similarly to $Q(\text{data}, Z)$ but only with the first $t$ available data points and their associated random components in the model (my notation is not appropriate for defining $Q_t$), and $C_t$ are the possible values of the random components for which $Q_t$ is not empty.

• Then $\pi_t$ is the independent product of the standard normal distributions truncated to $C_t$. I am totally lost here. In particular, does the algorithm require one to derive $C_t$ explicitly ?

I would be glad if you someone could clear up the points that I find confusing, or give me a link to a similar algorithm which is more detailed.

Answer 1

The whole goal was to generate standard normal conditional on the $C$ (using all of the data). That proved to be a rather difficult problem and therefore we have chosen to do this sequentially. You have correctly understood the meaning of $C_t$ and $Q_t$.

By the way, $C_t$ is formally defined in the middle of the page 9. You are correct the definition of $Q_t$ has slipped through the cracks of several revisions but you did guess its meaning correctly. Thank you for pointing this out.

The main idea is that it is easy to move from $C_t$ to $C_{t+1}$—that is all the business with $m_t$ and $M_t$ that follows. The reason why we choose sequential MC is that we can easily generate from the conditional distribution $Z_{t+1} I_{C_{t+1}}$ given $Z_t$ [using $m_t$ and $M_t$]. However there is a need to re-weight due to the fact that marginal distribution of $Z_{t+1} I_{C_{t+1}}$ marginalized to the time $t$ is different than the distribution of $Z_t I_{C_t}$.

The details of how much to re-weight is on the page 10 with more details in the Appendix C.

Answer 2

Thank you for your questions. I just want to elaborate on a couple points you made above:

1. I just want to clarify your description of the interval, or "fat", data: each observation is considered an interval rather than an exact value, but the intervals are fixed and defined based on the data collection/data storing process. The clarification is for your example where you say if your observation is 1.32, then the interval would be [1.315, 1.325). In this illustration, you are making the observation the midpoint of the interval and allowing more significant digits than the original datum, which is not what we use. Let's suppose your observation 1.32 is really 1.32 meters, and your measuring devise is a ruler than has tick marks down to millimeters. Then the observation is recorded as 1.32 meters, but really, it could be 1.312 m or 1.318 m (this is supposing that the practice is to round to the next highest mm). Hence, the observation "1.32" would be placed in the interval [1.31, 1.32). The point is that the intervals are not arbitrarily defined around each point, but are set in advance as a fixed grid (in this case, the fixed grid is determined by the ruler).

2. I also want to note that there is no assumption about the distribution across an interval. This leads into your other question about V(). We maintain the intervals throughout the algorithm, and at t = n we have a sample of weighted particles that, geometrically, are polyhedrons on the parameter space. One can then select a value within each weighted polyhedron according to some rule, V(). In the simulation study in our paper, we randomly select either the upper or lower endpoint of the marginal intervals for each dimension of the parameter space. There are other options here, and Hannig (2009) "On Generalized Fiducial Inference" describes some alternatives.