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IPS254 Thaddeus Tarpey et al.
Hutson and Vexler (2018) present interesting results showing how the 4-
parameter beta-normal distribution can become “aliased” with a normal
distribution meaning that under particular parameter settings, the beta-
normal becomes almost indistinguishable from a normal distribution. It is
curious to note that strong aliasing can also occur using infinite mixtures (or
convolutions) of beta and normal distributions. In our work with modeling
placebo response (Tarpey and Petkova, 2010), if the population consists of
placebo responders and non-responders, then an outcome variable y among
drug-treated patients can be represented as a 2-component finite mixture
model by
= + + , (1)
1
0

where is the average drug effect, β1 is the average placebo effect, and is
0
a Bernoulli indicator of whether the patient is a placebo responder or not,
2
which is independent of ∼ (0, ). A realistic alternative model has the
placebo response varying continuously, in which case the 0-1 Bernoulli can be
replaced by a continuous latent beta variable x in (1) leading to a “latent”
regression model. Various parameter configurations in this latent regression
model produce distributions for y which are aliased with normal distributions.
One simple illustration is to set the beta parameters to = = 1 =
0
0, = 1 and let € ∼ (0, 1) (1). Then the pdf of y is () = Φ() −
1
1 13
Φ( − 1) which is essentially indistinguishable from the ( , ) distribution.
2 12
Similar to the estimation problems that occur with the beta-normal
distribution noted in Hutson and Vexler (2018), fitting the latent regression
model with x ∼ beta can lead to severe identifiability issues.

2. Methodology
The methodology we will investigate for the problem of psychiatric
nosology will be called Projection Pursuit Nosology. In practice, statistical
models are multidimensional as opposed to uni-dimensional. In a setting with
p-dimensional measures x, the impact of a psychiatric disease may exert itself
along a lower-dimensional subspace. The simplest and perhaps most useful
setting is when the disruption to health generates variation in a 1-dimensional
direction in the feature space.
For many variables, it may be reasonable to assume the measures vary
according to a normal law for healthy individuals but if a disease is present,
then that may have a skewing effect in one (or more) particular directions of
the feature space. Projection pursuit (e.g., Diaconis and Freedman, 1984) is a
statistical approach developed to handle such a statistical challenge.
Linear Pre-Conditioning in Clustering. Our approach will incorporate
information obtained by clinician-based diagnoses and information obtained
by measured features x. Specifically, we will use a clustering approach where
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