. In other words, equations 3.7, and 8.9 apply.
However, if you are interested in knowing about
both
, you must include their covariance. In our
example, the large negative covariance follows logically just from looking at a graph: if you fit some
points, all of which lie at positive
t
, then a more negative derived slope will raise the
y
-intercept.
Specifically, the large negative covariance means that positive departures of
A
0
are associated
with negative departures of
A
1
. So even though the
individual
values
δA
0
= +34 and
δA
1
= +9
are acceptable, you
cannot
conclude that the
pair
of values (
δA
0
, δA
1
) = (+34
,
+9) is acceptable,
because this pair has both positive. In contrast, what
is
acceptable here would be something like
(
δA
0
, δA
1
) = (+34
,
−
9).
We stress that the acceptable ranges of values depend on what you are interested in. This is
sort of like the observer’s influence in quantum mechanics. If you are interested in
A
1
alone, then
you can say
A
1
= 4
±
9 and, in making this statement, you have to realize that, as
A
1
varies over
this range,
A
0
can vary over (formally, at least) the range (
∞→−∞
): you just don’t give a damn
what
happens to
A
0
because you’re not interested.
But the moment you become interested and
restrict its possible range, that influences the possible range for
A
1
, too.
There is no simple relationship between the covariance matrix elements and the acceptable
ranges.
For two variables, the best way to express this is to construct the ellipses that define
the loci of constant Δ
χ
2
and present them on a graph with axes (
δa
0
, δa
1
) as in BR Figure 11.2
or NR Figure 14.5.4. For three variables, these ellipses become ellipsoids; for four, they become
four-dimensional volumes, etc.
We illustrate these concepts for the (
a
1
, a
2
) parameters in our numerical example. We sub-
tracted 7.75 from all times so that the covariance would be small enough to illustrate the difference
between the tangents to the ellipses and the end points of the ellipses.
Contours are calculated
as described in
§
9.5 and are at Δ
χ
2
= 1 and 2.3. The dashed horizontal and vertical lines are at
δ
a
=
±
σ
a
.
First consider the pair of vertical lines, which are drawn at
δa
1
=
±
σ
a
1
, where
σ
is the square
root of the variance of the parameters as described in equations 3.7, 4.11, 8.9, and 8.10.
If the
datapoints were projected downward, i.e. look at the marginal pdf of
δa
1
by taking small strips of
δa
1
and integrating over
δa
2
, the marginal pdf of
δa
1
is Gaussian; ditto for the other coordinate.
Thus, 68% of the points lie between these dashed lines. This is what we mean by the phrase “being
interested in knowing about
a
1
without regard to
a
2
”. If we allow
a
2
to vary so as to minimize
χ
2