2 SI
J. S. Paul, M. Steinrücken, and Y. S. Song
File S1
Supporting Information
PROOF OF EQUIVALENCE OF π̂SMC AND π̂PS,1
We now want to give a more detailed proof of Proposition 1 from the main text. With the notation
as in the paper we have:
Proposition 1. For an arbitrary single haplotype α ∈ H and haplotype configuration n, π̂SMC (α|n) =
π̂PS,1 (α|n).
Recall that the initial (stationary) density was given by
ζ (n) (S` = (t` , h` )) =
nh` − n t`
e 2 ,
2
(S.1)
the transtition density by
ρb
− 2 t`−1
δs`−1 ,s` +
φ(n)
ρb (s` | s`−1 ) = e
nα`
n
Z
t`−1 ∧t`
tb =0
ρb − ρb tb n − n (t` −tb )
,
e 2
e 2
2
2
(S.2)
and the emission probability by
h θ`
i
(`)
(n)
ξθ` (α[`] | s` ) = e 2 t` ·(P −I)
h` [`],α[`]
.
(S.3)
Since the configuration n is fixed, we will drop the superscript (n) in the sequel. As in the main text
we will also omit the recombination and mutation rate when unambiguous. Further, we will omit the
d· whenever we write down integrals. If not specified differently, equation-references refer to equations
from the main paper.
Proof of Proposition 1. We
start
by
showing
inductively
that
the
joint
density
fSMC (α[`0 : `], (t` , h` )) of observing the partial haplotype α[`0 : `] and being in the hidden state (t` , h` )
(basically) introduced in equations (4)-(6) satisfies a genealogical recursion f , defined as follows [c.f.,
Griffiths and Tavaré (1994)]:
Z t`
P
P
n+ u θu + u ρu
nh` δα[`0 :`],h` [`0 :`]
t
0
−
p
2
δtp ,t`
f (α[` : `], (t` , h` )) =
e
2
tp =0
X θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : `], (t` − tp , h` ))
2
a∈Eu
u∈L(`0 :`)
X ρu Z
0
+
f (α[` : u1 ], su1 ) f (α[u2 , `], (t` − tp , h` )) ,
(S.4)
2
s u1
0
u∈B(` :`)
P
P
where the sum u is over L(`0 : `), all loci between (and including) `0 and `, and u is over B(`0 : `),
the breakpoints between `0 and `. Here Sua (α) denotes the haplotype obtained by substituting the
allele a at locus u of α, and u = (u1 , u2 ). For `0 = ` (so that ` − `0 = 0),
Z t`
n+θ
θ X (`)
− 2 ` tp nh` δα[`],h` [`]
f (α[`], s` ) =
e
δtp ,t` +
Pa,α[`] f (a, (t` − tp , h` )) .
2
2
tp =0
a∈E`
J. S. Paul, M. Steinrücken, and Y. S. Song
3 SI
Substituting f = fSMC on the right-hand side,
Z t`
n+θ
θ` X (`)
− 2 ` tp nh` δα[`],h` [`]
δtp ,t` +
Pa,α[`] fSMC (a, (t` − tp , h` ))
e
2
2
tp =0
a∈E`
Z t`
n+θ`
n
δ
n+θ
θ` X (`)
h α[`],h` [`]
= e− 2 t` `
+
e− 2 tp
Pa,α[`] p(a | (t` − tp , h` )) · q1 (tu − tp , hu )
2
2
tp =0
a∈E`
m Z t`
∞ X
θ
X
nh` − n+θ` t`
θ` 2` (t` − tp )
(`) (`)
m
e 2
=
δα[`],h` [`] +
Pa,h[`] (P ) h [`],a
`
2
m!
tp =0 2
m=0 a∈E`
m+1 ∞
θ`
X
(`) m+1 nh` − n+θ` t`
2 (t` )
2
=
e
δα[`],h` [`] +
(P )
h` [`],α[`] (m + 1)!
2
m=0
h θ`
i
n
nh
(`)
= ` e− 2 t` · e 2 t` ·(P −I)
,
2
h` [`],α[`]
with the final result equal to ξ(α[`] | s` )ζ(s` ) = fSMC (α[`], s` ). Now, inductively assuming that
fSMC (α[`0 : `], s` ) = f (α[`0 : `], s` ) for 0 ≤ ` − `0 < m and all values of s` , let `0 < ` such that ` − `0 = m.
Substituting f = fSMC on the right-hand side of (S.4), we obtain
Z t`
P
P
n+ u θu + u ρu
0
0
−
tp nh` δα[` :`],h` [` :`]
2
e
δtp ,t`
2
tp =0
X θu X (u)
+
Pa,α[u] fSMC (Sua (α)[`0 : `], (t` − tp , h` ))
2
a∈Eu
u∈L(`0 :`)
X ρu Z
0
fSMC (α[` : u1 ], su1 ) fSMC (α[u2 , `], (t` − tp , h` )) .
(S.5)
+
2
s u1
0
u∈B(` :`)
We consider this expression one term at a time. Beginning with the first term:
Z t`
P
P
n+ u θu + u ρu
0
0
tp nh` δα[` :`],h` [` :`]
2
e−
δtp ,t`
2
tp =0
Z
Z t`−1
P
P
i
h θ` +ρb
n+ 0u θu + 0u ρu
0
n
δ 0
tp h`−1 α[` :`−1],h`−1 [` :`−1]
−
2
=
e
δtp ,t`−1 e− 2 tp δα[`],h` [`] δs`−1 ,s` , (S.6)
2
s`−1 tp =0
P0
P
where u is over L(`0 : ` − 1) and 0u is over B(`0 : ` − 1). Moving on to the second term, expand
using the definition (5) of fSMC , and then use the inductive hypothesis to replace the resulting fSMC
terms with the corresponding f terms:
Z t`
P
P
X θu X (u)
n+ u θu + u ρu
tp
2
e−
Pa,α[u] fSMC (Sua (α)[`0 : `], (t` − tp , h` ))
2
tp =0
a∈Eu
u∈L(`0 :`)
Z t`
P
P
X
n+ u θu + u ρu
θu X (u)
tp
2
e−
=
Pa,α[u]
2
tp =0
a∈Eu
u∈L(`0 :`−1)
Z
× ξ(α[`] | (t` − tp , h` ))
φ((t` − tp , h` ) | s`−1 )f (Sua (α)[`0 : ` − 1], s`−1 )
s`−1
Z
t`
+
e−
n+
P
u θu +
2
P
u ρu t
p
tp =0
θ` X
Pa,α[`]
2
a∈Eu
Z
× ξ(a | (t` − tp , h` ))
s`−1
φ((t` − tp , h` ) | s`−1 )f (α[`0 : ` − 1], s`−1 ).
4 SI
J. S. Paul, M. Steinrücken, and Y. S. Song
Concentrating on the first sub-term, making the substitution t`−1 → t`−1 + tp , and changing the order
of integration, we obtain
Z
Z
t` ∧t`−1
P0
P0
u θ u + u ρu t
p
2
θu X (u)
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tp , h`−1 ))
2
tp =0
a∈Eu
u∈L(`0 :`−1)
θ`
ρb
− 2 tp
− 2 tp
× e
p(α[`] | (t` − tp , h` )) · e
q((t` − tp , h` ) | (t`−1 − tp , h`−1 )) .
(S.7)
s`−1
e−
n+
X
Now concentrating on the second sub-term and expanding using definition (S.4) of f :
Z
t`
Z
θ` X
e
Pa,α[`] ξ(a | (t` − tp , h` ))
φ((t` − tp , h` ) | s`−1 )
2
tp =0
s`−1
a∈Eu
Z t`−1
P
P
n+ 0u θu + 0u ρu
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
tq
−
2
×
e
δtq ,t`−1
2
tq =0
X
θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tq , h`−1 ))
2
a∈Eu
u∈L(`0 :`−1)
Z
X
ρu
0
f (α[` : u1 ], su1 ) f (α[u2 , ` − 1], (t`−1 − tq , h`−1 ))
+
2
s u1
u∈B(`0 :`−1)
Z
Z t`−1
P
P
n+ 0u θu + 0u ρu
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
t
−
q
2
=
e
δtq ,t`−1
2
s`−1 tq =0
X
θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tq , h`−1 ))
2
a∈Eu
u∈L(`0 :`−1)
Z
X
ρu
0
f (α[` : u1 ], su1 ) f (α[u2 , ` − 1], (t`−1 − tq , h`−1 ))
+
2
s u1
u∈B(`0 :`−1)
Z tq ∧t`
X
ρ
θ
− 2b tp
− 2` tp θ`
Pa,α[`] ξ(a | (t` − tp , h` )) · e
×
e
φ((t` − tp , h` ) | (t`−1 − tp , h`−1 )) ,
2
tp =0
−
n+
P
u θu +
2
P
u ρu t
p
a∈Eu
(S.8)
with the equality obtained by making the substitutions t`−1 → t`−1 + tp and tq → tq + tp and then
changing the order of integration. Finally, moving onto the third term, expand using the definition (5) of fSMC , and then use the inductive hypothesis to replace the resulting fSMC terms with the
corresponding f terms:
Z
t`
n+
−
P
e
u θu +
2
P
u ρu t
p
tp =0
ρu
2
X
u∈B(`0 :`)
Z
t`
=
e−
n+
P
u θu +
2
P
u ρu t
p
tp =0
Z
fSMC (α[` : ul ], sul ) fSMC (α[ur , `], (t` − tp , h` ))
0
s ul
X
u∈B(`0 :`−1)
ρu
2
Z
f (α[`0 : ul ], sul )
s ul
Z
× ξ(α[`] | (t` − tp , h` ))
φ((t` − tp , h` ) | s`−1 )f (α[ur : ` − 1], s`−1 )
s`−1
Z
t`
+
tp =0
e−
n+
P
u θu +
2
P
u ρu t
p
ρb
2
Z
s`−1
f (α[`0 : ` − 1], s`−1 ) · f (α[`], (t` − tp , h` )).
J. S. Paul, M. Steinrücken, and Y. S. Song
5 SI
Concentrating on the first sub-term, making the substitution t`−1 → t`−1 + tp , and changing the order
of integration, we obtain:
Z
Z
t` ∧t`−1
−
e
s`−1
n+
P0
P0
u θ u + u ρu t
p
2
tp =0
X
u∈B(`0 :`−1)
ρu
2
Z
f (α[` : u1 ], su1 ) f (α[u2 : ` − 1], (t`−1 − tp , h`−1 ))
0
s u1
ρ
θ
− 2b tp
− 2` tp
ξ(α[`] | (t` − tp , h` )) · e
φ((t` − tp , h` ) | (t`−1 − tp , h`−1 )) .
× e
(S.9)
Now concentrating on the second sub-term and expanding using definition (S.4) of f :
Z
t`
ρb
f (α[`], (t` − tp , h` ))
2
tp =0
Z
Z t`−1
P
P
n+ 0u θu + 0u ρu
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
tq
−
2
δtq ,t`−1
×
e
2
s`−1 tq =0
X
θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tq , h`−1 ))
2
a∈Eu
u∈L(`0 :`−1)
Z
X
ρu
0
f (α[` : u1 ], su1 ) f (α[u2 , ` − 1], (t`−1 − tq , h`−1 ))
+
2
s u1
u∈B(`0 :`−1)
Z
Z t`−1
P
P
n+ 0u θu + 0u ρu
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
t
−
q
2
δtq ,t`−1
=
e
2
s`−1 tq =0
X
θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tq , h`−1 ))
2
a∈Eu
u∈L(`0 :`−1)
Z
X
ρu
0
f (α[` : u1 ], su1 ) f (α[u2 , ` − 1], (t`−1 − tq , h`−1 ))
+
2
s u1
u∈B(`0 :`−1)
Z tq ∧t`
θ`
ρb
ρb nh` − n (t` −tp )
×
e− 2 tp ξ(α[`] | (t` − tp , h` )) · e− 2 tp
e 2
,
2 2
tp =0
−
e
n+
P
u θu +
2
P
u ρu t
p
(S.10)
with the equality obtained by using the (one-locus) definition (6) for fSMC (α[`], (t` − tp , h` )), making
the substitutions t`−1 → t`−1 + tp and tq → tq + tp , and changing the order of integration.
Having appropriately expanded each term of our key expression (S.5), we aggregate common terms
across the resulting sub-expressions. Collecting the nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1] terms from (S.6),(S.8),
6 SI
J. S. Paul, M. Steinrücken, and Y. S. Song
and (S.10),
Z
s`−1
Z
t`−1
tp =0
e−
n+
P0
P0
u θ u + u ρu t
p
2
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
δtp ,t`−1
2
θ` +ρb
× e− 2 tp δα[`],h` [`] δs`−1 ,s`
Z tp ∧t`
θ`
ρb
θ` X
+
e− 2 tq
Pa,α[`] ξ(a | (t` − tq , h` )) · e− 2 tq φ((t` − tq , h` ) | (t`−1 − tq , h`−1 ))
2
tq =0
a∈Eu
Z tp ∧t`
ρ
θ`
− 2b tq ρb nh` − n
(t` −tq )
− 2 tq
2
ξ(α[`] | (t` − tq , h` )) · e
+
e
e
2 2
tq =0
Z t`−1
Z
P
P
n+ 0u θu + 0u ρu
0
n
δ 0
−
tp h`−1 α[` :`−1],h`−1 [` :`−1]
2
e
=
δtp ,t`−1
2
s`−1 tp =0
ρ
θ
− 2b t`−1
− 2` t`
× e
δs`−1 ,s` · e
δα[`],h` [`]
Z t`
X
θ
ρ
− 2` tz θ`
− 2b t`−1
e
δs`−1 ,s`
+e
Pa,α[`] ξ(a | (t` − tz , h` ))
2
tz =0
a∈Eu
Z tq
Z t`−1 ∧t`
X
θ
ρb − ρb tq nh` − n (t` −tq )
− 2` tz θ`
2
2
e
e
e
Pa,α[`] ξ(allele | (t` − tz , h` ))
+
2
2
2
tz =0
tq =0
a∈Eu
Z t`−1 ∧t`
ρb − ρb tq nh` − n (t` −tq ) − θ` tq
2
2
2
+
e
ξ(α[`] | (t` − tq , h` ))
e
e
2
2
tq =0
Z
Z t`−1
P
P
n+ 0u θu + 0u ρu
0
n
δ 0
tp h`−1 α[` :`−1],h`−1 [` :`−1]
−
2
=
e
δtp ,t`−1
2
s`−1 tp =0
Z t`−1 ∧t`
ρ
ρb − ρb tq nh` − n (t` −tq )
− 2b t`−1
× ξ(α[`] | s` ) e
δs`−1 ,s` +
e 2
e 2
2
2
tq =0
Z
Z t`−1
P
P
n+ 0u θu + 0u ρu
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
−
t
p
2
=
e
δtp ,t`−1 × ξ(α[`] | s` )φ(s` | s`−1 ) , (S.11)
2
s`−1 tp =0
where the first equality is obtained by making use of the δtp ,t`−1 and δs`−1 ,s` expressions and expanding
the q term using equation (S.2) and exchanging integrals, the second equality is obtained by combining
the first/second and third/fourth term along with the definition (S.3) of p, and final equality by again
making use of the equation (S.2).
Similarly, collecting the f (Sua (α)[`0 : `−1], (t`−1 −tq , h`−1 )) terms from the resulting sub-expressions
J. S. Paul, M. Steinrücken, and Y. S. Song
7 SI
(S.7),(S.8), and (S.10),
Z
s`−1
Z
t`−1
tp =0
e−
n+
P0
P0
u θ u + u ρu t
p
2
X
u∈L(`0 :`−1)
θu X (u)
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tp , h`−1 ))
2
a∈Eu
θ`
ρb
× I(tp ≤t` ) e− 2 tp ξ(α[`] | (t` − tp , h` )) · e− 2 tp φ((t` − tp , h` ) | (t`−1 − tp , h`−1 ))
Z tp ∧t`
ρb
θ`
θ` X
+
e− 2 tq
Pa,α[`] ξ(a | (t` − tq , h` )) · e− 2 tq φ((t` − tq , h` ) | (t`−1 − tq , h`−1 ))
2
tq =0
a∈Eu
Z tp ∧t`
θ`
ρ
− 2 tq
(t
−t
)
− 2b tq ρb nh` − n
q
`
+
e
e 2
ξ(α[`] | (t` − tq , h` )) · e
2 2
tq =0
Z
Z t`−1
P
P
X
n+ 0u θu + 0u ρu
θu X (u)
tp
2
=
e−
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tp , h`−1 ))
2
s`−1 tp =0
a∈Eu
u∈L(`0 :`−1)
ρb
θ
− 2 tp
− 2` tp
× I(tp ≤t` ) e
φ((t` − tp , h` ) | (t`−1 − tp , h`−1 )) e
ξ(α[`] | (t` − tp , h` ))
Z tp
X
ρ
θ
− 2b tp
− 2` tz θ`
Pa,α[`] ξ(a | (t` − tz , h` ))
+ I(tp ≤t` ) e
φ((t` − tp , h` ) | (t`−1 − tp , h`−1 ))
e
2
tz =0
a∈Eu
Z tq
Z tp ∧t`
X
θ`
ρb − ρb tq nh` − n (t` −tq )
θ
`
−
t
z
+
e 2
e 2
e 2
Pa,α[`] ξ(a | (t` − tz , h` ))
2
2
2
tq =0
tz =0
a∈Eu
Z tp ∧t`
ρb − ρb tq nh` − n (t` −tq ) − θ` tq
e 2 ξ(α[`] | (t` − tq , h` ))
e 2
e 2
+
2
2
tq =0
Z t`−1
Z
P
P
X
n+ 0u θu + 0u ρu
θu X (u)
tp
2
e−
=
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tp , h`−1 ))
2
s`−1 tp =0
a∈Eu
u∈L(`0 :`−1)
Z tp ∧t`
ρ
ρb − ρb tq nh` − n (t` −tq )
− 2b tp
2
2
× ξ(α[`] | s` ) I(tp ≤t` ) e
φ((t` − tp , h` ) | (t`−1 − tp , h`−1 )) +
e
e
2
2
tq =0
Z t`−1
Z
P
P
X
n+ 0u θu + 0u ρu
θu X (u)
tp
2
e−
=
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tp , h`−1 ))
2
s`−1 tp =0
a∈Eu
u∈L(`0 :`−1)
× ξ(α[`] | s` )φ(s` | s`−1 ) ,
(S.12)
where the first equality is obtained by expanding the φ term1 in the second term using equation (S.2),
the second equality is obtained by combining the first/second and third/fourth term along with the
definition (S.3) of ξ, and final equality by again making use of the equation (S.2) and considering
separately the case when tp ≤ t` and tp > t` .
The situation is identical when collecting terms with f (α[u2 , ` − 1], (t`−1 − tq , h`−1 )) from (S.9),
1
We use the following expansion for φ, which can be verified in the present context, namely that tq ≤ tp ≤ t`−1 and
tq ≤ t` :
ρb
φ((t` − tq , h` ) | (t`−1 − tq , h`−1 )) = I(tp ≤t` ) e− 2 (tp −tq ) · φ((t` − tp , h` ) | (t`−1 − tp , h`−1 ))
Z (tp ∧t` )−tq
ρb − ρ2b tz nh` − n2 (t` −tq −tz )
+
e
e
2
2
tz =0
8 SI
J. S. Paul, M. Steinrücken, and Y. S. Song
(S.8), and (S.10):
Z
s`−1
Z
t`−1
e−
n+
P0
P0
u θ u + u ρu t
p
2
tp =0
X
u∈B(`0 :`−1)
ρu
2
Z
f (α[`0 : u1 ], su1 ) f (α[u2 : ` − 1], (t`−1 − tp , h`−1 ))
s u1
ρb
θ`
× I(tp ≤t` ) e− 2 tp ξ(α[`] | (t` − tp , h` )) · e− 2 tp φ((t` − tp , h` ) | (t`−1 − tp , h`−1 ))
Z tp ∧t`
θ`
ρb
θ` X
+
e− 2 tq
Pa,α[`] ξ(a | (t` − tq , h` )) · e− 2 tq φ((t` − tq , h` ) | (t`−1 − tq , h`−1 ))
2
tq =0
a∈Eu
Z tp ∧t`
ρb
θ`
ρb nh` − n (t` −tq )
e 2
+
e− 2 tq ξ(α[`] | (t` − tq , h` )) · e− 2 tq
2 2
tq =0
Z
Z t`−1
Z
P0
P0
X
n+ u θu + u ρu
ρu
−
tp
0
2
=
e
f (α[` : u1 ], su1 ) f (α[u2 : ` − 1], (t`−1 − tp , h`−1 ))
2
s`−1 tp =0
s u1
u∈B(`0 :`−1)
× ξ(α[`] | s` )φ(s` | s`−1 ) .
(S.13)
Thus, combining equations (S.11),(S.12), and (S.13), we may re-write (S.5):
Z
Z
ξ(α[`] | s` )
t`−1
φ(s` | s`−1 ) ·
s`−1
−
e
tp =0
n+
P0
P0
u θ u + u ρu t
p
2
nh`−1 δα[`0 :`−1],h`−1 [`0 :`−1]
δtp ,t`−1
2
θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : ` − 1], (t`−1 − tp , h`−1 ))
2
a∈Eu
u∈L(`0 :`−1)
Z
X
ρu
0
+
f (α[` : u1 ], su1 ) f (α[u2 : ` − 1], (t`−1 − tp , h`−1 ))
2
s
u
0
1
u∈B(` :`−1)
Z
= ξ(α[`] | s` )
φ(s` | s`−1 )f (α[`0 : ` − 1], s`−1 )
X
s`−1
= fSMC (α[`0 : `], s` ),
where the first equality is obtained by definition (S.4) for f , and the second equality by using the
inductive hypothesis and the definition (5). Therefore, fSMC satisfies the recursion for f , and we
J. S. Paul, M. Steinrücken, and Y. S. Song
9 SI
conclude that fSMC = f . Moreover,
Z
Z Z t`
P
P
n+ u θu + u ρu
0
0
0
−
tp nh` δα[` :`],h` [` :`]
2
δtp ,t`
f (α[` : `], s` ) =
e
2
s`
s` tp =0
X θu X (u)
+
Pa,α[u] f (Sua (α)[`0 : `], (t` − tp , h` ))
2
a∈Eu
u∈L(`0 :`)
X ρu Z
0
+
f (α[` : u1 ], su1 ) f (α[u2 , `], (t` − tp , h` ))
2
s u1
u∈B(`0 :`)
X
1
P
P
=
nα 0
n + u∈L(α[`0 :`]) θu + u∈B(α[`0 :`]) ρu
α0 ∈H:
α0 [`0 :`]=α[`0 :`]
X
+
θu
u∈L(α[`0 :`])
X
+
u∈B(α[`0 :`])
X
Z
(u)
Pa,α[u]
a∈Eu
Z
f (Sua (α)[`0 : `], s` )
s`
0
Z
f (α[` : u1 ], su1 )
ρu
s u1
f (α[u2 , `], s` ) ,
s`
where the first equality is by definition (S.4), and the second
equality obtained by exchanging the
R
integrals and making the substitution t` → t` − tp . Thus, s` f (α[`0 : `], s` ) satisfies the recursion for
R
π̂PS,1 (Paul and Song, 2010, Equation (12)) and we conclude that s` f (α[`0 : `], s` ) = π̂PS,1 (α[`0 : `]).
Thus,
Z
Z
π̂SMC (α[`0 : `]) =
fSMC (α[`0 : `], s` ) =
f (α[`0 : `], s` ) = π̂PS,1 (α[`0 : `]),
s`
s`
thereby establishing the desired identity.
LITERATURE CITED
Griffiths, R. C. and Tavaré, S. 1994. Sampling theory for neutral alleles in a varying environment.
Philos. Trans. R. Soc. Lond. B Biol. Sci., 344, 403–410.
Paul, J. S. and Song, Y. S. 2010. A principled approach to deriving approximate conditional sampling
distributions in population genetics models with recombination. Genetics, 186, 321–338.