About My Research

Here's an interesting problem in statistical inference:

 

[The Lost Submarine Problem] A submarine of length 2K has lost contact with its support vessel. It is assumed to be stationary. A hatch is located exactly in the center. The success of a rescue attempt depends on an accurate estimation of the hatch position. 

 

The crew of the support vessel notice two distinct bubbles which have emerged from the submarine. This allows the crew to confine its rescue attempt to the line intersecting the position of the bubbles, which can therefore be represented by coordinates x[1], x[2] in the interval [C - K, C + K] on that line, where C is the position of the hatch. Furthermore, since a bubble is equally likely to emerge from any place along the length of the submarine, x[1], x[2] form an independent sample from a uniform distribution on [C - K, C + K]. 

 

How can the crew best make use of this statistical model in any rescue attempt?

 

In one sense the answer to this question is trivial. We are able to confine any rescue attempt to the line intersecting the surface positions of the bubbles. Furthermore, we may restrict the search for C to the interval [max x[i] - K , min x[i] + K]. Then suppose the rescuers have one attempt (a constraint often imposed in this problem). The strategy is to estimate C using some  estimator c(x[1],x[2]), assume this is the true location, then proceed accordingly. The closer c(x[1],x[2]) is to C, the higher the probability of success. Unless there is a specific reason to assign a nonuniform prior on C, the best choice of c(x[1],x[2]) is clear. It is the unique location equivariant estimator which minimizes risk based on any symmetric loss function, which would be c(x[1],x[2]) = (x[1]+x[2])/2, that is, the midpoint between the two bubbles.  We then define V = x[2] - x[1], which is an ancillary statistic (that is, its distribution does not depend on C). 

 

At this point, as the problem is presented, the crew requests of a statistician an estimate of the precision of the estimate C, which is conventionally presented either as a confidence interval or a Bayesian credible interval [L, U]. Information about precision is contained in the ancillary statistic  V. If V = 2K, the maximum possible value, then the hatch position C can be located with certainty (the bubbles emerging from the two ends of the submarine). If V = 0, the minimum possible value, we can only restrict hatch position C to an interval of length 2K (the bubbles emerging next to each other). So V can be used to construct [L,U]. Whatever form the interval takes, it is constrained to have confidence level P(L < C < U) = 1 - a. In Morey et al (2016) four such intervals are presented, based on four distinct principles.  To make them comparable they are constrained to have confidence level 1 - a = 0.5. Needless to say, they have very different properties, but are derived using a variety of statistical principles commonly in use (Bayesian credible intervals; Neyman-Pearson lemma; standard deviation bounds; nonparametric procedure).  

 

Almudevar (2016) (https://arxiv.org/abs/2601.23171) includes a discussion of the "lost submarine problem", following Morey et al (2016). As the title of the latter paper suggests ("The fallacy of placing confidence in confidence intervals"), the example is intended to illustrate the futility of relying on the confidence interval as a formal inference statement. In my view, however, the misgivings expressed in Morey et al (2016) can be resolved using a decision theoretic approach. While it is true that a variety of statistical methods lead to a variety of confidence intervals, once we precisely define their purpose, a single optimal choice emerges. Furthermore, distinct purposes lead to distinct optimal choices. Therefore, that a variety of procedures exist is an advantage rather than a liability.

 

To make a long story short (although it is an interesting story) first note that the rescue crew is not making any use of the confidence interval, in the sense that it does not affect the rescue strategy in any way. What if it did? Suppose, first, that the crew decides to use that interval as a search range.  The probability of success is now controlled to be 1-a. The decision framework might now be an allocation of finite resources problem. So we choose the confidence interval which minimizes the expected length (we'll call it the minimum expected length (MEL) interval). This seems reasonable, since the probability of a successful rescue remains constant. In the preprint, the MEL interval is evaluated. 

 

The form of this MEL interval is somewhat surprising, until the decision theoretic aspect is considered (it is not one of the four considered above). The construction of the MEL interval includes some threshold t. Recall that larger V implies greater certainty with respect to the location of hatch position C. So if V > t, the confidence interval is made just wide enough to contain C with probability 1 (which can always be done). On the other hand, if V < t, the confidence interval is of width zero. The value of t depends on confidence interval 1-a. In other words, the MEL criterion simply allocates all weight to the shortest confidence intervals. 

 

It might seem unsatisfactory that based on the value of V the rescue crew might decide to do no search at all. So we'll rethink the decision criterion. The MEL criterion 

minimizes rescue crew effort. Suppose we alter the criterion so that the object is to carry out the rescue in as little time as possible (anticipating a limited oxygen supply within the submarine, for example). Again, as for the MEL criterion, we want the confidence intervals to be small as possible. However, there is one important difference. Now the length of the confidence interval only matters if it is correct (ie contains hatch position C). We'll call this the minimum expected length conditional on correctness (MELCC) criterion. If the confidence interval is correct, our loss is U - L. If it is not correct, our loss is some large number M (the size of the confidence interval now being irrelevant to this loss). The contribution of a rescue failure to risk (expected loss) is then aM for any choice of (L,U) constrained by the 1-a confidence level. Therefore, minimizing the MELCC criterion does not require knowledge of M, and turns out to be a well-defined problem.

 

The form of the MELCC interval is still somewhat surprising, but maybe less so than that of the MEL interval. It is, in fact, one of the four confidence intervals considered above, in particular, the one constructed from the standard error bounds. As it happens, it is the one which (uniquely) makes no use of ancillary statistic V. 

 

References

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Almudevar (2026) Revisiting the Lost Submarine Problem: A Decision Theoretic Approach, arXiv 2601.23171, https://arxiv.org/abs/2601.23171 

 

R. D. Morey, R. Hoekstra, J. N. Rouder, M. D. Lee and E-J. Wagenmakers (2016) The fallacy of placing confidence in confidence intervals, Psychonomic Bulletin & Review, vol. 23, pp 103-123.