Parallax for dummies
Luis A. Aguilar (the main dummy)
◼ Dec/06/16
The problem
Imagine that somebody gave you the measured parallax of an object ϖ and its
associated error estimate σϖ .
The question is, Can I compute the distance as just 1/ϖ?
The answer
Well, the answer depends on how large the fractional error (σϖ /ϖ) is.
The PDF' s
Let' s assume that the TRUE distance is ro and let’s call its reciprocal ϖo , in other words: ϖo =1/ro .
Now the measured parallax ϖ represents a random draw of the following PDF (Probability Distribution
Function):
1
PDFϖ (ϖ) =
2 π σϖ
exp-
(ϖ - ϖo )2
,
2 σϖ2
(1)
which is a PDF in parallax space and it is a gaussian because parallax is the observable.
This gaussian has its mean equal to ϖo , which is given by the object’s true distance and σϖ which is
given by the measurement precision.
Since what we want are distances, rather than parallaxes, we should map this PDF to its corresponding
one in distance. For this we must remember that:
PDFϖ (ϖ) d ϖ
=
PDFr (r ) dr
where the absolute value is introduced on both sides since we are dealing with probabilities.
Since
dϖ
dr
we get,
=
d 1
dr r
=
1
r2
2
Parallax-2.nb
PDFr (r ) =
PDFϖ (1 / r )
r2
then,
PDFr (r ) =
1
2 π σϖ

1
((1 / r) - ϖo )2

exp
r2
2 σϖ2
(2)
which is the function plotted on the right panel of figure 1. Notice that the 1/r 2 divergence at the origin is
canceled by the exponential function.
Figure 1. A gaussian parallax PDF with mean of 1 and standard deviation of 0.3 (left) and the corresponding PDF for the
distance (right). The straightforward value of the reciprocal of the mean parallax (1 in this case) is not the mean, mode, or
median of the distance PDF. Furthermore, the distance PDF is highly skewed, which results in the reciprocal of the
measured ϖ being a biased estimate of the true distance.
The bias
So, our measured ϖ is a random draw from the PDF on the left of figure 1 and the question is what is its
relation to the true ϖo ?
Let' s find out.
Figure 2 shows the mean of (1/ϖ) for a random sample of n ϖ-values taken from the PDF of the left
panel of figure 1, divided by the reciprocal of the mean value of this PDF (which is the true value of the
distance r). The size of the sample increases logarithmically from n = 10, up to 106 . If 1/ϖ where an
unbiased estimator of the true distance, this ratio would hover around 1. As we can see, this is not the
case, regardless of sample size and furthermore, for the cases shown in this figure, the ratio is always
larger than 1, which means that 1/ϖ is an overestimation of the true distance.
Figure 2. The mean of the reciprocal of n sampled values of ϖ obtained from the PDF for the parallax (left panel of figure
1), divided by the reciprocal of the mean of this PDF, which is the true value of the distance, is shown as a function of
sample size n, from n = 10 to 1,000,000.
Parallax-2.nb
3
Figure 2. The mean of the reciprocal of n sampled values of ϖ obtained from the PDF for the parallax (left panel of figure
1), divided by the reciprocal of the mean of this PDF, which is the true value of the distance, is shown as a function of
sample size n, from n = 10 to 1,000,000.
Figure 3 shows the resulting histogram for the particular case with ϖtrue = 1 and σϖ = 0.2, using a
sample of 100,000 values:
Figure 3. A histogram of 100,000 values of 1/ϖ, where the ϖ have been sampled from a gaussian distribution with mean
of 1 and standard deviation of 0.3. Compare with the right panel of figure 1.
The mean of the distribution in figure 3 is 1.151. We should remember that the reciprocal of the true
distance is 1.
Up to now, we have been playing with a fixed error which translates into a fixed fractional error of 0.3.
What happens if we vary this?
The following plots show the effect of increasing the observational error while keeping the true distance
fixed.
4
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Parallax-2.nb
5
Figure 5. PDF’s in parallax (left) and distance (right) for increasing values of the fractional error in parallax. In all cases
the true distance is equal to 1.
A Bayesian Approach
Our problem is that we want an estimate of the true distance, or rather, the conditional probability
distribution function for r, given the measured ϖ and its associated error estimate σϖ : P (r ϖ, σϖ ).
However, since our measurement is in parallax space, what we have is: P (ϖ r , σϖ )which we will
take as our proposed measurement model, or likelihood.
Bayes Theorem tells us that:
P (r
ϖ, σϖ ) = P (r ) × P (ϖ
r , σϖ )
where P(r) is called the Prior and represents our a priori assumptions about the distribution of stars in
distance.
Bayes Theorem thus provides us with a tool to get a PDF distribution of the variable we want to get
(distance), based on a model for its PDF in the observable (parallax) which we know what form has.
For the prior we can try several possibilities like the Uniform Distance (UD), Uniform Space Density
(USD) and Exponentially Decreasing Space Density (EDSD) priors, which are given by:
PUD (r ) = 
1 / rlim , 0 < r < rlim
0,
otherwise
PUSD (r ) = 
3
3 r 2  rlim
, 0 < r < rlim
PEDSD (r ) = 
0,
otherwise
r 2  2 L 3  e-(r /L ) , 0 < r
0,
otherwise
Figure 7 below illustrates the use of these priors and compares it with the use of the reciprocal of the
measured parallax.
(3)
(4)
(5)
6
Parallax-2.nb
Figure 5. A comparison of the true vs. inferred distances for a mock catalog of 9,600 Red Clump stars. The four panels
show the results obtained using the Uniform distance (upper left), Uniform space density (upper right) and exponentially
decreasing space density (lower left) priors. The lower right panel shows the result of just taking the reciprocal of the
measured parallax. Note the change in scale in the axes. This is unpublished work of Ariadna Ribes Metidieri and Xavi
Luri from the University of Barcelona (October/2016).
Recommended bibliography
1. “Estimating distances from parallaxes”.
Bailer-Jones, C., (2015). PASP, 127, 994.
2. “Estimating distances from parallaxes II. Performance of Bayesian distance estimators on a Gaia-like
catalogue”
Astraatmadja, T.L., and Bailer-Jones, C., (2016). ApJ, 832, 137.
3. "Estimating distances from parallaxes III. Distances of two million stars in the Gaia DR1 catalogue”
Astraatmadja, T.L., and Bailer-Jones, C., (2016). Accepted for publication in PASP. Sept/29
(ArXiv160907369A).