A Uniformly Consistent Estimator of non-Gaussian Causal Effects Under the
k-Triangle-Faithfulness Assumption

Shuyan Wang Peter Spirtes

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Kalisch and Bühlmann (2007) showed that for linear Gaussian models, under the
Causal Markov Assumption, the Strong Causal Faithfulness Assumption, and the
assumption of causal sufficiency, the PC algorithm is a uniformly consistent
estimator of the Markov Equivalence Class of the true causal DAG for linear
Gaussian models; it follows from this that for the identifiable causal effects
in the Markov Equivalence Class, there are uniformly consistent estimators of
causal effects as well. The $k$-Triangle-Faithfulness Assumption is a strictly
weaker assumption that avoids some implausible implications of the Strong
Causal Faithfulness Assumption and also allows for uniformly consistent
estimates of Markov Equivalence Classes (in a weakened sense), and of
identifiable causal effects. However, both of these assumptions are restricted
to linear Gaussian models. We propose the Generalized $k$-Triangle
Faithfulness, which can be applied to any smooth distribution. In addition,
under the Generalized $k$-Triangle Faithfulness Assumption, we describe the
Edge Estimation Algorithm that provides uniformly consistent estimates of
causal effects in some cases (and otherwise outputs “can’t tell”), and the
Very Conservative $SGS$ Algorithm that (in a slightly weaker sense) is a
uniformly consistent estimator of the Markov equivalence class of the true
DAG.

##  1 Introduction

It has been proved that under the Causal Markov, Faithfulness assumptions and
Causal Sufficiency Assumption, there are no uniformly consistent estimators of
Markov equivalence classes of causal structures represented by directed
acyclic graphs (DAG)(Robins et al., 2003). Kalisch and Bühlmann (2007) showed
that for linear Gaussian models, under the Causal Markov Assumption, the
Strong Causal Faithfulness Assumption, and the assumption of causal
sufficiency, the PC algorithm is a uniformly consistent estimator of the
Markov Equivalence Class of the true causal DAG for linear Gaussian models; it
follows from this that for the identifiable causal effects in the Markov
Equivalence Class, there are uniformly consistent estimators of causal effects
as well. The $k$-Triangle-Faithfulness Assumption is a strictly weaker
assumption that avoids some implausible implications of the Strong Causal
Faithfulness Assumption and also allows for uniformly consistent estimates of
Markov Equivalence Classes (in a weakened sense), and of identifiable causal
effects.

However, both of these assumptions are restricted to linear Gaussian models.
We propose the Generalized k-Triangle Faithfulness, which can be applied to
any smooth distribution. In addition, under the Generalized k-Triangle
Faithfulness Assumption, we describe the Edge Estimation Algorithm that
provides uniformly consistent estimates of causal effects in some cases (and
otherwise outputs ”can’t tell”), and the Very Conservative SGS Algorithm that
(in a slightly weaker sense) is a uniformly consistent estimator of the Markov
equivalence class of the true DAG.

##  2 The Basic Assumptions for Causal Discovery

###  2.1 DAG and Causal Markov Condition

We use directed acyclic graphs to represent causal relations between
variables. A directed graph $G=\langle\mathbf{V,E}\rangle$ consists of a set
of nodes V and a set of edges $\mathbf{E\subset V\times V}$. If there is an
edge $\langle A,B\rangle\in\mathbf{E}$ , we write $A\rightarrow B$. $A$ is a
parent of $B$, and $B$ is a child of $A$, the edge is out of $A$ and into $B$,
and $A$ is the source and $B$ is the target. A directed path from $X$ to $Y$
is a sequence of ordered edges where the source of the first edge is $X$, the
target of the last edge is $Y$, and if there are n edges in the sequence, for
$1\leq i<n$, the target of the $i$th edge is the source of the $i+1$th edge;
$X$ is an ancestor of $Y$, and $X$ is a descendant of $Y$.

If a variable $Y$ is in a structure $X\rightarrow Y\leftarrow Z$, and there is
no edge between $X$ and $Z$, we call $\langle X,Y,Z\rangle$ an unshielded
collider; if there is also an edge between $X$ and $Z$,then $\langle
X,Y,Z\rangle$ is a triangle and we call $\langle X,Y,Z\rangle$ a shielded
collider. If $\langle X,Y,Z\rangle$ is a triangle but $Y$ is not a child of
both $X$ and $Z$, we call $\langle X,Y,Z\rangle$ a shielded non-collider; if
there is no edge between $X$ and $Z$, then $\langle X,Y,Z\rangle$ is a
unshielded non-collider.

A Bayesian network is an ordered pair $\langle P,G\rangle$ where $P$ is a
probability distribution over a set of variables V in $G$. A distribution $P$
over a set of variables $\mathbf{V}$ satisfies the  (local) Markov condition
for $G$ if and only if each variable in $\mathbf{V}$ is independent of its
non-parents and non-descendants, conditional on its parents. Given $M=\langle
P,G\rangle$, $P_{M}$ denotes $P$ and $G_{M}$ denotes $G$. Two acyclic directed
graphs (DAG) $G_{1}$ and $G_{2}$ are  Markov equivalent if conditional
independence relations entailed by Markov condition in $G_{1}$ are the same as
in $G_{2}$. It has been proven that two $DAG$s are Markov equivalent if and
only if they have the same adjacencies and same unshielded colliders (Verma
and Pearl, 1990). A pattern $O$ is an undirected graph that represents a set
$M$ of Markov equivalent DAGs: an edge $X\rightarrow Y$ is in $O$ if it is in
every DAG in $M$; if $X\rightarrow Y$ is in some DAG and $Y\rightarrow X$ in
some other DAG in $M$, then $X-Y$ in $O$ (Spirtes and Zhang, 2016)

We assume causal sufficiency, which means that V contains all direct common
causes of variables in V.

###  2.2 Faithfulness, linear Gaussian case and k-Triangle-Faithfulness

Given a $\langle P,G\rangle$ that satisfies Markov Condition, we say that $P$
is faithful to $G$ if any conditional independence relation that holds in $P$
is entailed by $G$ by the Markov Condition. We further make the Causal Markov
and Faithfulness assumption (Spirtes et al., 2001)::

Causal Markov Assumption: If the true causal model $M$ of a population is
causally sufficient, each variable in $V$ is independent of its non-parents
and non-descendants, conditional on its its parents in $G_{M}$ (Spirtes and
Zhang, 2016).

Causal Faithfulness Assumption:  all conditional independence relations that
hold in the population are consequences of the Markov condition from the
underlying true causal DAG.

In this paper we talk about cases where $P_{M}$ over $\mathbf{V}$ for
$G=\langle\mathbf{V,E}\rangle$ respects the Causal Markov Assumption. If
$P_{M}$ is faithful to $G_{M}$ and all variables in $M$ are Gaussian and all
causal relations are linear, that is, any $X_{i}\in\mathbf{V}$ can be written
as:

$X_{i}=\underset{X_{j}\in Pa_{M}(X_{i})}{\sum}a_{i,j}X_{j}+\epsilon_{j}$

where $Pa_{M}(X)$ denotes the set of parents of $X$ in $G_{M}$, the set of $X$
variables is jointly standard Gaussian, $a_{i,j}$ is a real valued
coefficient, and the set of $\epsilon_{j}$ are jointly Gaussian and jointly
independent, conditional correlation between any two variables
$\rho_{X,Y|\mathbf{A}}=0$ where $X,Y\in\mathbf{V}$ and $\mathbf{A\subset
V}\setminus{\\{X,Y}\\}$ implies that there is no edge between $X$ and $Y$.
Based on the equation above, we define in the linear Gaussian case the edge
strength $e_{M}(X_{j}\rightarrow X_{i})$ as the corresponding coefficient
$a_{i,j}$.

It has been proved that under the Causal Markov and Faithfulness assumptions,
there are no uniformly consistent estimators of Markov equivalence classes of
causal structures represented by DAG (Robins et al., 2003). Kalisch and
Bühlmann (2007) showed that such uniform consistency is achieved by the $PC$
algorithm if the underlying DAG is sparse relative to the sample size under a
strengthened version of Faithfulness Assumption. This Strong Causal
Faithfulness Assumption  in the linear Gaussian case bounds the absolute value
of any partial correlation not entailed to be zero by the true causal DAG away
from zero by some positive constants. It has the implausible consequence that
it puts a lower bound on the strength of edges, since a very weak edge entails
a very weak partial correlation. However, the Strong Causal Faithfulness
Assumption  can be weakened to the strictly weaker (for some values of $k$)
k-Faithfulness Assumption while still achieving uniform consistency.
Furthermore, at the cost of having a smaller subset of edges oriented, the
k-Faithfulness Assumption can be weakened to the k-Triangle-Faithfulness
Assumption, while still achieving uniform consistency and can be relaxed while
preserving the uniform consistency: the k-Triangle-Faithfulness Assumption
(Spirtes and Zhang, 2014) only bounds the conditional correlation between
variables in a triangle structure from below by some functions of the
corresponding edge strength:

k-Triangle-Faithfulness Assumption: Given a set of variables V, suppose the
true causal model over V is $M=\langle P,G\rangle$, where $P$ is a Gaussian
distribution over V, and G is a DAG with vertices V. For any variables X, Y, Z
that form a triangle in G:

  * •

if Y is a non-collider on the path $\langle X,Y,Z\rangle$, then
$|\rho_{M}(X,Z|\mathbf{W})|\geq K\times|e_{M}(X-Z)|$ for all $\mathbf{W\subset
V}$ that do not contain Y; and

  * •

if Y is a collider on the path $\langle X,Y,Z\rangle$, then
$|\rho_{M}(X,Z|\mathbf{W})|\geq K\times|e_{M}(X-Z)|$ for all $\mathbf{W\subset
V}$ that do contain Y.

where the $X-Z$ represents the edge between $X$ and $Z$ but the direction is
not determined.(Sprites and Zhang, 2014)

The k-Triangle-Faithfulness Assumption is strictly weaker than the Strong
Faithfulness Assumption in several respects: the Strong faithfulness
Assumption does not allow edges to be week anyywhere in a graph, while the
k-Triangle-Faithfulness Assumption only excludes conditional correlations
$\rho(X,Z|\mathbf{W})$ from being too small if $X$ and $Z$ are in some
triangle structures $\langle X,Y,Z\rangle$ and $X-Z$ is not a weak edge; for
every $\epsilon$ used in the Strong Faithfulness Assumption as the lower bound
for any partial correlation, there is a $k$ for the k-Triangle-Faithfulness
Assumption that gives a lower bound smaller than $\epsilon$.

###  2.3 VCSGS Algorithm

The algorithm we use to infer the structure of the underlying true causal
graph is Very Conservative SGS ($VCSGS$) algorithm, which takes uniformly
consistent tests of conditional independence as input:

VCSGS Algorithm

  1. 1.

Form the complete undirected graph $H$ on the given set of variables
$\mathbf{V}$.

  2. 2.

For each pair of variables $X$ and $Y$ in $\mathbf{V}$, search for a subset
$\mathbf{S}$ of $\mathbf{V}\setminus\\{X,Y\\}$ such that $X$ and $Y$ are
independent conditional on $\mathbf{S}$. Remove the edge between $X$ and $Y$
in $H$ if and only if such a set is found.

  3. 3.

Let $K$ be the graph resulting from Step 2. For each unshielded triple
$\langle X,Y,Z\rangle$ (the only two variables that are not adjacent are $X$
and $Z$ ),

    1. (a)

If $X$ and $Z$ are not independent conditional on any subset of
$\mathbf{V}\setminus{X,Z}$ that contains $Y$, then orient the triple as a
collider: $X\rightarrow Y\leftarrow Z$.

    2. (b)

If $X$ and $Z$ are not independent conditional on any subset of
$\mathbf{V}\setminus{X,Z}$ that does not contain Y, then mark the triple as a
non-collider.

    3. (c)

Otherwise, mark the triple as ambiguous.

  4. 4.

Execute the following orientation rules until none of them applies:

    1. (a)

If $X\rightarrow Y-Z$, and the triple $\langle X,Y,Z\rangle$ is marked as a
non-collider, then orient $Y-Z$ as $Y\rightarrow Z$.

    2. (b)

If $X\rightarrow Y\rightarrow Z$ and $X-Z$, then orient $X-Z$ as $X\rightarrow
Z$.

    3. (c)

If $X\rightarrow Y\rightarrow Z$, another triple $\langle X,W,Z\rangle$ is
marked as a non-collider, and $W-Y$, then orient $W-Y$ as $W\rightarrow Y$.

  5. 5.

Let $M$ be the graph resulting from step 4. For each consistent disambiguation
of the ambiguous triples in $M$ ($i.e.$, each disambiguation that leads to a
pattern), test whether the resulting pattern satisfies the Markov condition.
If every pattern does, then mark all the ‘apparently non-adjacent’ pairs as
‘definitely non-adjacent’.

It has been proved that under the k-Triangle-Faithfulness Assumption, $VCSGS$
algorithm is uniformly consistent in the inference of graph structure.
Furthermore, a follow-up algorithm that estimates edge strength given the
output of $VCSGS$ also reaches uniform consistency. We are going to prove that
the uniform consistency of the estimation of the causal influences under the
k-Triangle-Faithfulness Assumption can be extended to discrete and
nonparametric cases as long as there are uniformly consistent tests of
conditional independence (which in the general case requires a smoothness
assumption), by showing that missed edges in the inference of causal structure
are so weak that the estimations of the causal influences are still uniformly
consistent.

##  3 Nonparametric Case

For nonparametric case, we consider variables supported on $[0,1]$. We first
define the strength of the edge $X\rightarrow Y$ as the maximum change in $L1$
norm of the probability of $Y$ when we condition on different values of $X$
while holding everything else constant:

If $x\in Pa(Y):$ $e(X\rightarrow
Y):=max_{pa_{\setminus\\{x\\}}(Y)\in[Pa(Y)\setminus\\{X\\}]}max_{x_{1},x_{2}\in[X]}||p_{Y|x_{1},pa_{\setminus\\{x\\}}(Y)}-p_{Y|x_{2},pa_{\setminus\\{x\\}}(Y)}||_{1}$

where $[X]$ denotes the set of values that $X$ takes,
$[Pa(Y)]\subset[0,1]^{|Pa(Y)|}$ the set of values that parents of Y take,
$p_{Y|x_{1},pa_{\setminus\\{x\\}}(Y)}$ the probability distribution of
$Y|X=x_{1},Pa(Y)\setminus\\{X\\}=pa_{\setminus\\{x\\}}(Y)$ and
$p_{y|x_{1},pa_{\setminus\\{x\\}}(Y)}$ the density of
$p_{Y|x_{1},pa_{\setminus\\{x\\}}(Y)}$ for $Y=y$. Since we are conditioning on
the set of parents, the conditional probability is equal to the manipulated
probability.

Then we can make the k-Triangle-Faithfulness Assumption: given a set of
variables V, where the true causal model over V is $M=\langle P,G\rangle$, $P$
is a distribution over V, and G is a DAG with vertices V, for any variables X,
Y, Z that form a triangle in G:

  * •

if Z is a non-collider on the path $\langle X,Z,Y\rangle$, given any subset
$\mathbf{W\subset V}\setminus\\{X,Y,Z\\}$,
$min_{\mathbf{w\in[W]}}min_{x_{1},x_{2}\in[X]}||p_{Y|\mathbf{w},x_{1}}-p_{Y|\mathbf{w},x_{2}}||_{1}\geq
K_{Y}e(X\rightarrow Y)$ for some $K_{Y}>0$

  * •

if Z is a collider on the path $\langle X,Z,Y\rangle$, then for every
$y\in[Y]$, given any subset $\mathbf{W\subset
V}\setminus\\{X,Y,Z\\}$,$min_{\mathbf{w\in[W]}}min_{z\in[Z]}min_{x_{1},x_{2}\in[X]}||p_{Y|\mathbf{w},x_{1},z}-p_{Y=y|\mathbf{w},x_{2},z}||_{1}\geq
K_{Y}e(X\rightarrow Y)$ for some $K_{Y}>0$

In order to have uniformly consistent tests of conditional independence, we
make smoothness assumption for continuous variables with the support on
$[0,1]$:
TV Smoothness(L):  Let $\mathcal{P}_{[0,1],TV(L)}$ be the collection of
distributions $p_{Y,\mathbf{A}}$, such that for all
$\mathbf{a},\mathbf{a^{\prime}}\in[0,1]^{|\mathbf{A}|}$, we have:
$||p_{Y|\mathbf{A=a}}-p_{Y|\mathbf{A=a^{\prime}}}||_{1}\leq
L||\mathbf{a-a^{\prime}}||_{1}$

Given the TV smoothness(L), $p$ is continuous. Furthermore, since $[0,1]^{d}$
($d\in\mathbb{N}$) is compact, for any $\mathbf{W,U\subset V}$ (the set of all
variables in the true causal graph) , $p_{U|W}$ attains its max and min on its
support. Since $\mathbf{|V|}$ is finite, we can further assume conditional
densities are non-zero (NZ(T)):
for any $X\in\mathbf{V}$, $\mathbf{U\subset V}$, $p_{X|\mathbf{U}}\geq T$ for
some $1>T>0$.

Notice that by TV Smoothness(L) and that variables have support on $[0,1]$, we
can derive an upper bound for probability of any variable given its parents:

$p_{y|{Pa(Y)=pa(Y)}}\leq||p_{Y|Pa(Y)=pa(Y)}||_{1}\newline
\leq||p_{Y|Pa(Y)=pa^{\prime}(Y)}||_{1}+L||pa(Y)-pa^{\prime}(Y)||_{1}\newline
\leq(1+L|Pa(Y)|)$

Although the discrete probability case does not have support on $[0,1]$, and
its probability is not continuous, it still satisfies the TV smoothness(L)
assumption: for instance, if the discrete variables have support only on
integers, we can set $L=1$. By replacing the density $p_{X|\mathbf{U}}$ with
the probability $P(X|\mathbf{U})$ in NZ(T) assumption, we have a NZ(T)
assumption for the discrete case. Therefore the proof of uniform consistency
for the nonparametric case in the rest of the paper also works for the
discrete case.

###  3.1 Uniform Consistency in the inference of structure

We use $L1$ norm to characterize dependence :
$\epsilon_{X,Y|\mathcal{A}}=||p_{X,Y,\mathcal{A}}-p_{X|\mathcal{A}}p_{Y|\mathcal{A}}p_{\mathcal{A}}||_{1}$.We
want a test $\psi$ of $H_{0}:\epsilon=0$ versus $H_{1}:\epsilon>0$. $\psi$ is
a family of functions: $\psi_{0}...\psi_{n}...$ one for each sample size, that
takes an i.i.d sample $V_{n}$ from the joint distribution over $\mathbf{V}$.
Then the test is uniformly consistent w.r.t. a set of distributions $\Omega$
for :

  * •

$lim_{n\rightarrow\infty}sup_{P\in\mathcal{P}_{[0,1],TV(L)},\epsilon(P)=0}P^{n}(\psi_{n}(V_{n})=1)=0$

  * •

for every $\delta>0$,
$lim_{n\rightarrow\infty}sup_{\epsilon(P)\geq\delta}P^{n}(\psi_{n}(V_{n})=0)=0$

With the TV Smoothness(L) assumption, there are uniformly consistent tests of
conditional independence, such as a minimax optimal conditional independence
test proposed by Neykov et al.(2020).

Given any causal model $M=lrangleP,G\rangle$ over $\mathbf{V}$, let $C(n,M)$
denote the (random) output of the $VCSGS$ algorithm given an$i.i.d.$ sample of
size size $n$ from the distribution $P_{M}$, then there are three types of
errors that it can contain that will mislead the estimation of causal
influences:

  1. 1.

$C(n,M)$ errs in kind I if it has an adjacency not in $G_{M}$;

  2. 2.

$C(n,M)$ errs in kind II if every adjacency it has is in $G_{M}$, but it has a
marked non-collider not in $G_{m}$;

  3. 3.

$C(n,M)$ errs in kind III if every adjacency and marked non-collider it has is
in $G_{M}$, but it has an orientation not in $G_{M}$

If $C(n.M)$ errs in either of these three way, there will be variable $X$ and
$Y$ in $C(n,M)$ such that $X$ is treated as a parent of $Y$ but is not in the
true graph $G_{M}$; if there is no undirected edge connecting $Y$ in this
$C(n,M)$, the algorithm will estimate the causal influence of “ parents” of
$Y$ on $Y$, but such estimation does not bear useful information since
intervening $X$ does not really influence $Y$. Notice that missing an edge is
not listed as an mistake here, and we are going to prove later that the
estimation of causal influence can still be used to correctly predict the
effect of intervention even if the algorithm misses edges.

Let $\phi^{k,L,T}$ be the set of causal models over V under $k$-Triangle-
Faithfulness Assumption, TV smoothness(L) and the assumptions of NZ(T).

We will prove that under the causal sufficiency of the measured variables V,
causal Markov assumption, k-Triangle-Faithfulness, TV smoothness(L) assumption
and NZ(T) assumption,

$\underset{n\rightarrow\infty}{lim}\underset{M\in\phi^{k,L,C}}{sup}P^{n}_{M}(C(n,M)errs)=0$

We begin by proving a useful lemma that bounds $\epsilon_{X,Y|\mathbf{A}}$
with strengths of the edge $X\rightarrow Y$:

######  Lemma 3.1.

Given an ancestral set $\mathbf{A\subset V}$ that contains the parents of $Y$
but not $Y$:

If $X$ is a parent of $Y$:

$T^{|\mathbf{A}|}e(X\rightarrow
Y)\leq\epsilon_{X,Y|\mathbf{A}\setminus\\{X\\}}\leq e(X\rightarrow Y)$

###### Proof.

$\epsilon_{X,Y|\mathbf{A}\setminus\\{X\\}}$
$=\int_{\mathbf{A}\setminus\\{X\\}}\int_{X}\int_{Y}|p_{x,y,\mathbf{a}\setminus\\{x\\}}-\newline
p_{y|\mathbf{a}\setminus\\{x\\}}p_{x|\mathbf{a}\setminus\\{x\\}}p_{\mathbf{a}\setminus\\{x\\}}|dydxd\mathbf{a}\setminus\\{x\\}$

$=\int_{\mathbf{A}}p_{\mathbf{a}}||p_{y|x,\mathbf{A}\setminus\\{X\\}}-p_{y|\mathbf{A}\setminus\\{X\\}}||_{1}d\mathbf{a}$

$\leq\mathbb{E}_{{\mathbf{A}}\sim p_{\mathbf{A}}}[e(X\rightarrow Y)]$

$=e(X\rightarrow Y)$

$\epsilon_{X,Y|\mathbf{A}\setminus\\{X\\}}$
$=\int_{\mathbf{A}\setminus\\{X\\}}\int_{X}\int_{Y}|p_{x,y|\mathbf{a}\setminus\\{x\\}}-\newline
p_{y|\mathbf{a}\setminus\\{x\\}}p_{x|\mathbf{a}\setminus\\{x\\}}|dydxd\mathbf{a}\setminus\\{x\\}$

$=\int_{\mathbf{A}\setminus\\{X\\}}p_{\mathbf{a}\setminus\\{x\\}}\int_{x}p_{x|\mathbf{a}\setminus\\{x\\}}||p_{Y|x,\mathbf{a}\setminus\\{x\\}}-\newline
p_{Y|\mathbf{a}\setminus\\{x\\}}||_{1}dxd\mathbf{a}\setminus\\{x\\}$

$\geq T^{|\mathbf{A}|}e(X\rightarrow Y)$

The last step is derived using a direct conclusion from NZ(T):

for any $\mathbf{V\supset W}=\\{W_{1},W_{2}...W_{n}\\}$, by the Chain Rule:
$p_{\mathbf{W}}=\prod_{i=1}^{n}p_{W_{i}|W_{i+1}...W_{n}}\geq T^{n}$

∎

We are going to prove for each case that the probability for $C(n,M)$ to make
each of the three kinds of mistakes uniformly converges to zero. Since the
proofs for the kind I and kind III errors are almost the same as the proof for
the linear Gaussian case (Spirtes and Zhang, 2014), we are only going to prove
kind II here.

######  Lemma 3.2.

Given causal sufficiency of the measured variables $\mathbf{V}$, the Causal
Markov, k-Triangle-Faithfulness, TV smoothness(L) assumption and NZ(T)
assumption:

$\underset{n\rightarrow\infty}{lim}\underset{M\in\phi^{k,L,C}}{sup}P^{n}_{M}(C(n,M)\textit{
errs in kind II})=0$

###### Proof.

For any $M\in\phi^{k,L,C}$, if it errs in kind II then it contains a marked
non-collider $\langle X,Z,Y\rangle$ that is not in $G_{M}$. Since it’s been
proved (Spirtes and Zhang, 2014):

$\underset{n\rightarrow\infty}{lim}\underset{M\in\phi^{k,L,C}}{sup}P^{n}_{M}(C(n,M)\textit{
errs in kind I})=0$

the errors of kind II can be one of the two cases:

$(I)$ $\langle X,Z,Y\rangle$ is an unshielded collider in $G_{M}$;

$(II)$ $\langle X,Z,Y\rangle$ is a shielded collider in $G_{M}$;

the proof for case (I) is the same as the proof for the $C(n,M)$ errs in kind
I (Spirtes and Zhang, 2014), so we are going to prove here that the
probability of case (II) uniformly converges to zero as sample size increases.

We are going to prove by contradiction. Suppose that the probability that
$VCSGS$ making a mistake of kind II does not uniformly converge to zero. Then
there exists $\lambda>0$, such that for every sample size $n$, there is a
model $M(n)$ such that the probability of $C(n,M(n))$ containing an unshielded
non-collider that is a shielded collider in $M(n)$ is greater than $\lambda$.
Let that triangle be $\langle X^{M(n)},Z^{M(n)},Y^{M(n)}\rangle$ with
$X^{M(n)}$ being the parent of $Y^{M(n)}$ in $M(n)$.

The algorithm will identify the triple as an unshielded non-collider only if:

$(i)$ there is a set $\mathbf{U}^{M(n)}\subset\mathbf{V}^{M(n)}$ containing
$Z^{M(n)}$, such that the test of
$\epsilon_{X^{M(n)},Y^{M(n)}|\mathbf{U}^{M(n)}}=0$ returns 0, call this test
$\psi_{n0}$;

$(ii)$ there is an ancestral set $\mathbf{W}^{M(n)}$ that contains $X^{M(n)}$
and $Y^{M(n)}$ but not $Z^{M(n)}$, such that for set
$\mathbf{A}^{M(n)}=\mathbf{W}^{M(n)}\setminus\\{X^{M(n)},Y^{M(n)}\\}$, the
test $\epsilon_{X^{M(n)},Y^{M(n)}|\mathbf{A}^{M(n)}}=0$ returns 1, call this
test $\psi_{n1}$.

If what we want to proof does not hold for the algorithm, for all $n$ there is
a model $M(n)$:
(1)$P^{n}_{{M(n)}}(\psi_{n0}=0)>\lambda$
(2)$P^{n}_{{M(n)}}(\psi_{n1}=1)>\lambda$

(1) tells us that there is some $\delta_{n}$ such that
$|\epsilon_{X^{M(n)},Y^{M(n)}|\mathbf{U}^{M(n)}}|<\delta_{n}$ and
$\delta_{n}\rightarrow 0$ as $n\rightarrow\infty$ since the test is uniformly
consistent. So we have:
$\delta_{n}>\epsilon_{X^{M(n)},Y^{M(n)}|\mathbf{U}^{M(n)}}=\newline
\mathbb{E}_{\mathbf{U}^{M(n)}}\sim
p_{\mathbf{U}^{M(n)}}[\int_{X^{M(n)}}p_{x^{M(n)}|\mathbf{U}^{M(n)}}||p_{Y^{M(n)}|x^{M(n)},\mathbf{U}^{M(n)}}-\newline
p_{Y^{M(n)}|\mathbf{U}^{M(n)}}||_{1}dx^{M(n)}]\geq\newline
\min_{x_{1}^{M(n)},x_{2}^{M(n)}\in[X^{M(n)}]}||p_{Y^{M(n)}|x_{1}^{M(n)},\mathbf{U}^{M(n)}}-\newline
p_{Y^{M(n)}|x_{2}^{M(n)},\mathbf{U}^{M(n)}}||_{1}\newline \geq
K_{Y^{M(n)}}e_{M}(X^{M(n)}\rightarrow Y^{M(n)})$
The last step is by k-Triangle-Faithfulness

So $e_{M}(X^{M(n)}\rightarrow Y^{M(n)})<\dfrac{\delta_{n}}{K_{Y^{M(n)}}}$.

By Lemma 3.1,
$\epsilon_{X^{M(n)},Y^{M(n)}|\mathbf{A}^{M(n)}}<e_{M}(X^{M(n)}\rightarrow
Y^{M(n)})$.

Therefore,
$\epsilon_{X^{M(n)},Y^{M(n)}|\mathbf{A}^{M(n)}}<\dfrac{\delta_{n}}{K_{Y}^{M(n)}}\rightarrow
0$ as $n\rightarrow\infty$ ∎

######  Theorem 3.3.

Given causal sufficiency of the measured variables $\mathbf{V}$, the Causal
Markov, k-Triangle-Faithfulness, TV smoothness(L) and NZ(T) assumptions:

$\underset{n\rightarrow\infty}{lim}\underset{M\in\phi^{k,L,C}}{sup}P^{n}_{M}(C(n,M)\textit{
errs })=0$

###### Proof.

Since we have proved that the probability for $C(n,M)$ to make any of the
three kinds of mistakes uniformly converges to 0, the theorem directly
follows. ∎

###  3.2 Uniform consistency in the inference of causal effects

Edge Estimation Algorithm:

For each vertex $Y$ such that all of the edges containing $Y$ are oriented in
$C(n,M)$ (output $VCSGS$), if $Pa(Y)$ is the parent set of $Y$ in $C(n,M)$, we
use histogram to estimate $p(y_{i}|{Pa(Y)=pa(Y)})$111we denote the density of
$p_{Y|Pa(Y)=pa(Y)}$ at $Y=y$ as $p(y|{Pa(Y)=pa(Y)})$ in this section to match
with the result of estimation. for $y_{i}\in[Y]$ and ${pa(Y)\in[Pa(Y)]}$; for
any of the remaining edges, return ‘Unknown’; if any estimation of conditional
probability violates TV smoothness(L), abandon the output and return
‘Unknown’.

Defining the distance between $M_{1}$ and $M_{2}$

The method for estimation for $p(y|{Pa(Y)=pa(Y)})$ is: we first get
$\hat{p}(Y=y,{Pa(Y)=pa(Y)})$ and $\hat{p}({Pa(Y)=pa(Y)})$ by histogram, then
we get:

$\hat{p}(y|{Pa(Y)=pa(Y)})=\dfrac{\hat{p}(Y=y,{Pa(Y)=pa(Y)})}{\hat{p}({Pa(Y)=pa(Y)})}$

Let $M_{1}$ be the output of the Edge Estimation Algorithm, and $M_{2}$ be a
causal model, we define the conditional probability distance,
$d[M_{1},M_{2}]$, between $M_{1}$ and $M_{2}$ to be:

$d[M_{1},M_{2}]$
$=\underset{\begin{subarray}{c}{Y\in\mathbf{V}},\\\ {y_{i}\in[Y]},\\\
{pa_{M_{1}}(Y)}\\\ {\in[Pa_{M_{1}}(Y)]},\\\ {pa_{M_{2}}(Y)}\\\
{\in[Pa_{M_{2}}(Y)]},\\\ {pa_{M_{1}}\subset
pa_{M_{2}}}\end{subarray}}{max}|\hat{p}_{M_{1}}(y_{i}|pa_{M_{1}}(Y))-p_{M_{2}}(y_{i}|pa_{M_{2}}(Y))|$

where $Pa_{M}(Y)$ denotes the parent set of $Y$ in causal model $M$. By
convention
$|\hat{P}_{M_{1}}(y_{i}|pa_{M_{1}}(Y))-P_{M_{2}}(y_{i}|pa_{M_{2}}(Y))|=0$ if
$\hat{P}_{M_{1}}(y_{i}|pa_{M_{1}}(Y))$ is “Unknown”.

Now we want to show, the edge estimation algorithm is uniformly consistent in
the sense that for every $\delta>0$,

$\underset{n\rightarrow\infty}{lim}$ $\underset{M\in\phi^{k,L,C}}{sup}$
$P^{n}_{M}(d[\hat{O}(M),M]>\delta)=0$

Here $M$ is any causal model satisfying causal sufficiency of the measured
variables $\mathbf{V}$, the Causal Markov, k-Triangle-Faithfulness, TV
smoothness(L) and NZ(T) assumptions and $\hat{O}(M)$ is the output of the
algorithm given an iid sample from $P_{M}$.

###### Proof.

Let $\mathcal{O}$ be the set of possible graphs of $VCSGS$. Since given
$\mathbf{V}$, there are only finitely many outputs in $\mathcal{O}$, it
suffices to prove that for each output $O\in\mathcal{O}$,

$\underset{n\rightarrow\infty}{lim}\underset{M\in\phi^{k,L,C}}{sup}P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=O)P_{M}^{n}(C(n,M)=O)=0$

Now partition all the $M$ into three sets given $O$ :

  * •

$\Psi_{1}=\\{M|$ all adjacencies, non-adjacencies and orientations in O are
true in$M\\}$;

  * •

$\Psi_{2}=\\{M|$ only some adjacencies, or orientations in O are not true in
$M\\}$;

  * •

$\Psi_{3}=\\{M|$ only some non-adjacencies in O are not true in $M\\}$.

It suffices to show that for each $\Psi_{i}$,

$\underset{n\rightarrow\infty}{lim}$ $\underset{M\in\Psi_{i}}{sup}$
$P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=O)P_{M}^{n}(C(n,M)=O)=0$

$\Psi_{1}$:

For any $M\in\Psi_{1}$, if the conditional probabilities of a vertex $Y$ in
$O$ can be estimated (so not “Unknown”), it means that $Pa_{O}(Y)=Pa_{M}(Y)$.
Recall that the histogram estimator is close to the true density with high
probability:

for any $\lambda<1$,
$sup_{P\in\mathcal{P}_{TV(L)}}P^{n}(||\hat{p}_{h}(x)-p(x)||_{\infty}\leq
f(n,\lambda)\leq O((\dfrac{\log n}{n})^{\frac{1}{2+d}}))\geq 1-\lambda$

where $f(n,\lambda)$ is continuous and monotonically decreasing wrt $n$ and
$\epsilon$ and $h\propto n^{2/(2+d)}$ (the number of bins) where $d$ is the
dimentionality of the $x$. For instance, $d=|Pa(Y)|+1$ when estimating
$P(Y,Pa(Y))$.

Given a $\delta>0$, $f(n,\epsilon)=\delta$ entails that
$sup_{P\in\mathcal{P}_{[0,1],TV(L)}}P^{n}(||\hat{p}_{h}(x)-p(x)||_{\infty}>\delta)<\lambda$.
By monotonicity of $f(n,\lambda)$, when $n>n_{f}$ s.t.
$f(n_{f},\lambda)=\delta$,
$sup_{P\in\mathcal{P}_{[0,1],TV(L)}}P^{n}(||\hat{p}_{h}(x)-p(x)||_{\infty}>\delta)<\lambda$.

Therefore the histogram estimators of
$p_{M}(y,Pa_{M}(Y)=pa_{M}(Y))$ and $p_{M}(Pa_{M}(Y)=pa_{M}(Y))$ are uniformly
consistent. Next we are going to use the lemma below:

######  Lemma 3.4.

If $\hat{p}(Y=y,{Pa(Y)=pa(Y)})$and $\hat{p}({Pa(Y)=pa(Y)})$ are uniformly
consistent
estimators of $p(Y=y,{Pa(Y)=pa(Y)})$and
$p({Pa(Y)=pa(Y)})$, then

$\hat{p}(y|{Pa(Y)=pa(Y)})=\dfrac{\hat{p}(Y=y,{Pa(Y)=pa(Y)})}{\hat{p}({Pa(Y)=pa(Y)})}$

is a uniformly consistent estimator for
$p(y|{Pa(Y)=pa(Y)})$

###### Proof.

Recall that:

for any $\lambda<1$,
$sup_{P\in\mathcal{P}_{TV(L)}}P^{n}(||\hat{p}_{h}(x)-p(x)||_{\infty}\leq
f(n,\lambda)=O((\dfrac{\log n}{n})^{\frac{1}{2+d}}))\geq 1-\lambda$ $(*)$

where $f(n,\lambda)$ is continuous and monotonically decreasing wrt $n$ and
$\lambda$ and $h\propto n^{2/(2+d)}$ (the number of bins) where $d$ is the
dimentionality of the $x$. For instance, $d=|Pa(Y)|+1$ when estimating
$P(Y,Pa(Y))$.

For any $\delta>0$, $f(n,\lambda)=\delta$ entails that
$sup_{P\in\mathcal{P}_{[0,1],TV(L)}}P^{n}(||\hat{p}_{h}(x)-p(x)||_{\infty}>\delta)<\lambda$.
By monotonicity of $f(n,\lambda)$, when $n>n_{f}$ s.t.
$f(n_{f},\lambda)\leq\delta$,
$sup_{P\in\mathcal{P}_{[0,1],TV(L)}}P^{n}(||\hat{p}_{h}(x)-p(x)||_{\infty}>\delta)<\lambda$.

Let $d=|Pa(Y)|$, notice that for any $\lambda$, with probability at least
$1-\lambda$,

$|\dfrac{\hat{p}(Y=y,{Pa(Y)=pa(Y)})}{\hat{p}({Pa(Y)=pa(Y)})}-p(y|{Pa(Y)=pa(Y)})|$222here
we use $\hat{p}(x)$ instead of $\hat{p}_{h}(x)$ because $h$ is dependent on
the dimension of $x$

$=\dfrac{1}{\hat{p}({Pa(Y)=pa(Y)})}|\hat{p}(Y=y,{Pa(Y)=pa(Y)})-p(y|{Pa(Y)=pa(Y)})\hat{p}({Pa(Y)=pa(Y)})|$

$\leq\dfrac{1}{T^{d}}|p(Y=y,{Pa(Y)=pa(Y)})+\newline O((\dfrac{\log
n}{n})^{\frac{1}{3+d}})-p(y|{Pa(Y)=pa(Y)})(p({Pa(Y)=pa(Y)})-O((\dfrac{\log
n}{n})^{\frac{1}{2+d}}))|$ (By $(*)$ and the fact that the estimation result
can only be valid if it satisfies TV smoothness(L))333Recall that $\mathbf{V}$
denotes the set of variables in the true graph

$\leq\dfrac{1}{T^{d}}|O((\dfrac{\log n}{n})^{\frac{1}{3+d}})+O((\dfrac{\log
n}{n})^{\frac{1}{2+d}})|$

$=O((\dfrac{\log n}{n})^{\frac{1}{3+d}})$

The second to the last step is derived because $p(y|{Pa(Y)=pa(Y)})$is upper
bounded by $(1+L|Pa(Y)|)$ by TV smoothness.

We have:

$sup_{P\in\mathcal{P}_{TV(L)}}P^{n}(|\dfrac{\hat{p}(Y=y,{Pa(Y)=pa(Y)})}{\hat{p}({Pa(Y)=pa(Y)})}-$
$p(y|{Pa(Y)=pa(Y)})|\leq O((\dfrac{\log n}{n})^{\frac{1}{3+d}}))\geq
1-\lambda$

So $\hat{p}(y|{Pa(Y)=pa(Y)})=$
$\dfrac{\hat{p}(Y=y,{Pa(Y)=pa(Y)})}{\hat{p}({Pa(Y)=pa(Y)})}$ is a uniform
consistent estimator for $p(y|{Pa(Y)=pa(Y)})$

∎

By lemma 3.4, we conclude that the $\hat{p}_{M}(y|Pa_{M}(Y)=pa_{M}(Y))$ is a
uniformly consistent estimator for

$p_{M}(y|Pa_{M}(Y)=pa_{M}(Y))=\frac{p_{M}(y,Pa_{M}(Y)=pa_{M}(Y))}{p_{M}(Pa_{M}(Y)=pa_{M}(Y))}$,

So:

$\underset{n\rightarrow\infty}{lim}\underset{M\in\Psi_{1}}{sup}P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=$
$O)P_{M}^{n}(C(n,M)=O)\newline
\leq\underset{n\rightarrow\infty}{lim}\underset{M\in\Psi_{1}}{sup}P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=O)=0$

$\Psi_{2}:$ the proof is exactly the same as for the discrete case.
$\Psi_{3}:$

Let $O(M)$ be the population version of $\hat{O}(M)$. Since the histogram
estimator is uniformly consistent over $||\hat{p}_{h}-p||_{\infty}$ and there
are finitely many parent-child combinations, for every $\lambda>0$ there is a
sample size $N_{1}$, such that for $n>N_{1}$, and all $M\in\Psi_{3}$,

$P^{n}_{M}(d[\hat{O}(M),O(M)]>\delta/2|C(n,M)=O)<\lambda$

Since only some non-adjacencies in $O$ are not true in $M$, we know that for
any vertex $Y$ that have some estimated conditional probabilities given its
parents in $O$, $Pa_{O(M)}(Y)\subset Pa_{M}(Y)$ where $Pa_{O(M)}(Y)$ denotes
the parent set of $Y$ in the $O$ when the underlying probability is
$P_{M}$(i.e., $M$ is the true causal model). Since $Pa_{M}(Y)\not\subset
Pa_{O(M)}(Y)$, for any $y_{i}\in[Y]$ and $pa_{O(M)}(Y)\in[Pa_{O(M)}(Y)]$,
$P_{O}(y_{i}|Pa_{O(M)}(Y)=pa_{O(M)}(Y))$ is a marginalization of
$p_{M}(y_{i}|Pa_{M}(Y)=pa_{M}(Y))$. Therefore, the distance between $O(M)$ and
$M$ is:

$d[O(M),M]=\newline \underset{\begin{subarray}{c}{Y\in\mathbf{V}},\\\
{y_{i}\in[Y]},\\\ {pa_{O(M)}(Y)\in}\\\ {[Pa_{O(M)}(Y)]},\\\ {pa_{M}(Y)\in}\\\
{[Pa_{M}(Y)]},\\\ {pa_{O(M)}\subset
pa_{M}}\end{subarray}}{max}|p_{O(M)}(y_{i}|pa_{O(M)}(Y))-p_{M}(y_{i}|pa_{M}(Y))|$

Given the $Y$ corresponding to the equation above, let
$Pa_{M}(Y)=\\{A_{1}...A_{g}..A_{g+h}\\}$ and
$Pa_{O(M)}(Y)=\\{A_{1}...A_{g}\\}$. Since $P_{O(M)}(y_{i}|pa_{O(M)}(Y))$ is
the marginalization of all $P_{M}(y_{i}|pa_{M}(Y))$, we have:

$|p_{O(M)}(y_{i}|pa_{O(M)}(Y))-p_{M}(y_{i}|pa_{M}(Y))|\leq$

$\underset{\begin{subarray}{c}pa_{M}(Y)_{1},\\\ pa_{M}(Y)_{2}\\\
\in[Pa_{M}(Y)],\\\ s.t.{pa_{O(M)}\subset}\\\ {pa_{M}(Y)_{1}\cap
pa_{M}(Y)_{2}}\end{subarray}}{max}|p_{M}(y_{i}|pa_{M}(Y)_{1})-p_{M}(y_{i}|pa_{M}(Y)_{2})|$

$<\sum_{j=1}^{j=h}e_{M}(A_{g+j}\rightarrow Y)$

If $\sum_{j=1}^{j=h}e_{M}(A_{g+j}\rightarrow Y)<\delta/2$, then we have:

$d[O(M),M]<\delta/2$

For all such $M$, there is a $N_{1}$, such that for any $n>N_{1}$:

$P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=O)$
$\leq P^{n}_{M}(d[\hat{O}(M),O(M)]+d[O(M),M]>\delta|C(n,M)=O)$
$\leq P^{n}_{M}(d[\hat{O}(M),O(M)]>\delta/2|C(n,M)=O)<\epsilon$

If $\sum_{j=1}^{j=h}e_{M}(A_{g+j}\rightarrow Y)\geq\delta/2$, then there is at
least an $w\in\\{1,2...h\\}$, $s,t.$ $e_{M}(A_{g+w}\rightarrow
Y)>\dfrac{\delta}{2h}$. By Lemma 3.1:

$\epsilon_{A_{g+w},Y|\mathbf{U}}\geq
T^{|\mathbf{U}|+1}e_{M}(A_{g+w}\rightarrow
Y)>T^{|\mathbf{U}|+1}\dfrac{\delta}{2h}$.

where the $\mathbf{U}\cup\\{A_{g+w},Y\\}$ is some ancestral set not containing
any descendant of $Y$.

Since the density estimation does not turn “unknown”, we know that in step 5
of $VCSGS$ the test of $\epsilon_{A_{g+w},Y|U}=0$ returns 0 while
$\epsilon_{A_{g+w},Y|U}\geq T^{|\mathbf{U}|+1}\dfrac{\delta}{2h}$. Since the
test is uniformly consistent, it follows that there is a sample size $N_{2}$
such that

$P^{N_{2}}_{M}(\epsilon_{A_{g+w},Y|U}=0)<\lambda$

for any $n>N_{2}$ and therefore for all such M,

$P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=O)<\lambda$

Let $N=max(N_{1},N_{2})$, for $n>N$,

$\underset{n\rightarrow\infty}{lim}\underset{M\in\Psi_{3}}{sup}P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=$
$O)P_{M}^{n}(C(n,M)=O)\newline
\leq\underset{n\rightarrow\infty}{lim}\underset{M\in\Psi_{3}}{sup}P^{n}_{M}(d[\hat{O}(M),M]>\delta|C(n,M)=O)=0$

∎

##  4 Discussion

We have shown that there is a uniformly consistent estimator of causal
structure and some causal effects for nonparametric distributions under the
$k$-Triangle-Faithfulness assumption, which is sometimes stronger than the
Faithfulness Assumption and weaker than the Strong Faithfulness Assumption.
There are a number of open questions, such as whether the causal sufficiency
assumption can be relaxed, so we allow the existence of latent variables and
whether there are similar results under assumptions weaker than the Causal
Faithfulness Assumption, such as the Sparsest Markov Representation Assumption
(Solus et al.,2016).

## References

  * 1 M. Neykov, S. Balakrishnan, and L. Wasserman (2020).  Minimax Optimal Conditional Independence Testing.
  * 2 J. M. Robins, R. Scheines, P. Spirtes, and L. Wasserman(2003).  Uniform consistency in causal inference.  Biometrika, 90(3):491–515.
  * 3 P. Spirtes, C. Glymour and R. Scheines(2001).  Causation, Prediction, and Search,, 2nd edn. Cambridge, MA: MIT Press..
  * 4 P. Spirtes, and J. Zhang (2014).  A Uniformly Consistent Estimator of Causal Effects under the k-Triangle-Faithfulness Assumption.
  * 5 P. Spirtes, and K. Zhang (2016).  Causal discovery and inference: concepts and recent methodological advances.  In D. S. Touretzky (ed.), Applied Informatics, 3(3) 1-28.
  * 6 L. Solus, Y. Wang and C. UhlerP (2017).  Consistency Guarantees for Greedy Permutation-Based Causal Inference Algorithms. arXiv:1702.03530 .
  * 7 T. Verma, and P. Pearl (1990).  Equivalence and synthesis of causal models. Proceedings of the 6th Conference on Uncertainty in Artificial Intelligence, 220-227.


# FISSION FRAGMENT MASS AND KINETIC ENERGY YIELDS OF FERMIUM ISOTOPES
††thanks: Presented at the Mazurian Lakes Conference on Physics, 2023, Poland.

K. Pomorskia111Email<EMAIL_ADDRESS>ORCID ID:
0000-0002-5557-6037 A. Dobrowolskia B. Nerlo-Pomorskaa M. Wardaa A. Zdeba J.
Bartelb H. Moliqueb C. Schmittb Z. G. Xiaoc Y. J. Chend L. L. Liud aInstitute
of Physics, Maria Curie Skłodowska University, Lublin, Poland
b IPHC, University of Strasbourg, Strasbourg, France
c Department of Physics, Tsinghua University, Beijing, China
d China Institute of Atomic Energy, Beijing, China

###### Abstract

A rapidly converging 4-dimensional Fourier shape parametrization is used to
model the fission process of heavy nuclei. Potential energy landscapes are
computed within the macroscopic-microscopic approach, on top of which the
multi-dimensional Langevin equation is solved to describe the fission
dynamics. Charge equilibration at scission and de-excitation by neutron
evaporation of the primary fragments after scission is investigated. The model
describes various observables, including fission-fragment mass, charge, and
kinetic energy yields, as well as post-scission neutron multiplicities and,
most importantly, their correlations, which are crucial to unravel the
complexity of the fission process. The parameters of the dynamical model were
tuned to reproduce experimental data obtained from thermal neutron-induced
fission of 235U, which allows us to discuss the transition from asymmetric to
symmetric fission along the Fm isotopic chain.

24.75.+i, 25.85.-w,28.41.Ak

Since the discovery of the fission phenomenon in 1938, it has been commonly
accepted that the most probable mass of the heaviest fragment produced in
spontaneous or low-energy fission is located at $A\approx 140$. However, a
series of experiments by Hulet and co-workers [1] have shown that this is not
always the case. In the spontaneous fission of Fermium isotopes, in
particular, one notices a rapid change of the fragment mass-yield systematics
with growing neutron number. For light Fm isotopes, one observes a mass
asymmetry of the fission fragments typical for actinides. Yet, at $A=258$, the
fission fragment mass distribution changes rapidly and becomes symmetric and
narrow. Also, the fragments’ total kinetic energy (TKE) yield becomes
significantly larger than in the lighter Fm isotopes. This new trend in the
fragment distribution is also observed for No isotopes. This discovery of
Hulet and co-workers constitutes a challenge for nuclear theoreticians.
Several attempts have been made to explain this phenomenon, an excellent
review of which is presented in the work of Albertsson et al. [2]. In the
present paper, we will show that this rapid change of the fission yield
systematics in the Fermium isotopes can be well characterized when using a
new, very efficient Fourier-type parametrization [3] of the shapes of
fissioning nuclei, combined with the WKB method to describe the penetration of
the fission barrier and a Langevin type calculation [4] of the fragment
yields.

Our “Fourier-over-Spheroid” (FoS) parametrization, which describes the shape
of the nucleus relative to a spheroidal deformation, was recently introduced
in Refs. [5, 4] through the shape function:

$\rho_{s}^{2}(z)=\frac{R_{0}^{2}}{c}\,f\left(\frac{z-z_{\rm
sh}}{z_{0}}\right)$ (1)

which describes, in cylindrical coordinates, the location of a surface point
of an axially symmetric shape as function of the symmetry $z$ coordinate. In
this expression, $R_{0}$ corresponds to the radius of the spherical nucleus
having the same volume, and the parameter $c$ determines the elongation of the
shape with half-axis $z_{0}=cR_{0}$. The shift parameter $z_{\rm sh}$
guarantees that the center of mass of the shape is located at the origin of
the coordinate system ($z_{\rm sh}-z_{0}\leq z\leq z_{\rm sh}+z_{0}$). The
function $f(u)$ defines a shape having half-length $c=1$:

$\begin{array}[]{rl}f(u)&=1-u^{2}-\left[{a_{4}\over 3}-{a_{6}\over
5}+\dots\right]\\!\cos\left({\pi\over
2}u\right)-a_{3}\sin(\pi\,u)\\\\[2.15277pt] &-a_{4}\cos\left({3\pi\over
2}u\right)-a_{5}\sin(2\pi\,u)-a_{6}\cos\left({5\pi\over
2}u\right)-\dots~{},\end{array}$ (2)

where $-1\leq u\leq 1$. The first two terms in $f(u)$ describe a circle, and
the third ensures volume conservation for arbitrary deformation parameters
$\\{a_{3},\;a_{4},\;\dots\\}$. The parameters $a_{3}$ and $a_{4}$ allow for
reflection asymmetric and necked-in shapes, with higher-order parameters
$a_{n},\;n\geq 5$ responsible mostly for the fission fragments deformations.
Shapes breaking axial symmetry can easily be described through the shape
function

$\varrho_{s}^{2}(z,\varphi)=\rho_{s}^{2}(z)\,{1-\eta^{2}\over
1+\eta^{2}+2\eta\cos(2\varphi)}\;.$ (3)

with the parameter $\eta=(b-a)/(b+a)$, where $a$ and $b$ are the half-axis of
the ellipsoid obtained as the cross-section of the non-axial shape for
constant $z$ (see e.g. [3]). The parametrization (3) is similar but more
general than the $\gamma$-deformation of Åge Bohr. Equation (3) describes the
same class of shapes as the original Fourier deformation parameter set of Ref.
[3] but is better adapted for performing numerical calculations around the
scission point of fissioning nuclei.

Figure 1: Potential energy surface of 258Fm on the ($c,\,a_{4}$) plane. Each
point is minimized with respect to the non-axial ($\eta$) and the reflectional
($a_{3}$) deformations.

Using the above FoS parametrization, we have performed extensive macroscopic-
microscopic calculations of the PESs for even-even nuclei with $90\leq{\rm
Z}\leq 122$ [4] in the 4D $\\{\eta,c,a_{3},a_{4}\\}$ deformation space. The
Lublin-Strasbourg Drop (LSD) formula [6] was used to evaluate the macroscopic
part of the energy, while the microscopic one are obtained using the single-
particle spectra of a Yukawa-folded mean-field Hamiltonian [7] and the
Strutinsky shell and BCS pairing corrections. An example of the obtained
potential energy surface (PES) is presented in Fig. 1, where the energy of
258Fm (renormalized to the LSD energy of the spherical nucleus) is presented
as function of the elongation $c$ and the neck-parameter $a_{4}$. Each point
of the surface is minimized with respect to the non-axial ($\eta$) and
reflectional asymmetry ($a_{3}$) deformations. The ground-state of 258Fm is
found to be located at $c=1.13$ and $a_{4}=0.03$, with $a_{3}=\eta=0$.
Interestingly, two-second saddles are found: one at $c=1.41$ and $a_{4}=0.27$
leading to compact symmetric fission and a second one, around 1 MeV higher, at
$c=1.55$ and $a_{4}=0.12$, which opens a path leading both to asymmetric mass
fragments, but bifurcates also into another valley characterized by a
symmetric mass split with very elongated fragments. The shapes of the forming
fission fragments corresponding to these three valleys are characterized on
top of Fig. 1. The FoS parametrization scission line is found around
$a_{4}\approx 0.72$.

Figure 2: Second fission barrier heights of 254-262Fm corresponding to the
asymmetric (a) and symmetric (s) saddle points. Figure 3: Fission fragment
mass yields of Fermium isotopes. The experimental data (*) are taken from Refs
[1, 11, 12, 13].

The competition in energy between the different second saddles and subsequent
fission valleys in Fm isotopes decides which fission path will be more
populated. The second barrier height, defined as the energy difference between
the 2nd saddle and the ground state of the Fm isotopes, is shown in Fig. 2.

Using the sets of Langevin equations defined in the FoS deformation space, one
obtains the mass and total kinetic energy of the fission fragments [8, 9]. We
have chosen the corresponding symmetric or asymmetric exit point from the
fission barrier as a starting point of the Langevin trajectories.

The final mass and TKE yields ($Y_{\rm th}$) were obtained by weighting the
yields obtained for both starting points ($Y_{a}$ and $Y_{s}$) using the
penetration probability ($P_{a}$ and $P_{s}$) of the corresponding fission
barrier:

$Y_{\rm th}(A_{f})=P_{a}\cdot Y_{a}(A_{f})+P_{s}\cdot Y_{s}(A_{f})~{}.$ (4)

The relative probabilities $P_{i}$ are given through the corresponding action
integral $S_{i}$ determined in the WKB approximation:

$P_{i}=\exp(-S_{i})/[\exp(-S_{a})+\exp(-S_{s})]$ (5)

evaluated along the path $i$ (confer e.g. to Ref [10]).

Figure 4: Fission fragment yield (thick solid line) of 258Fm as function of
the fragment TKE. The experimental data (histogram) are taken from Ref. [1]. A
thin solid line shows the TKE distribution corresponding to the asymmetric
path, and the dotted line the one for the compact symmetric path.

Our estimates of the mass yields of 246-262Fm isotopes are compared with the
experimental distributions in Fig. 3. The corresponding fission fragment TKE
yield of 258Fm is shown in Fig. 4. The predicted yield (thick solid line)
reproduces the general trend observed in the measured TKE distribution [1].

Taking into account the charge equilibration effect, the mass and deformation,
and the excitation energy of the fragments, one can estimate the post-fission
neutron multiplicities as described in Ref. [4]. The multiplicities of
neutrons accompanying the spontaneous fission of 258Fm, the corresponding
isotope and TKE yields and the elongation of nucleus at scission are shown in
Fig. 5 as function of the fragment neutron $N_{f}$ and proton $Z_{f}$ numbers,
which illustrates the possibilities of our numerical code to account for the
different aspects of the nuclear fission process.

Figure 5: Maps of the post-fission neutron multiplicities (top l.h.s.),
fission fragment yield (top r.h.s), TKE (bottom l.h.s.), and elongation at
scission (bottom r.h.s.) for 258Fm.

## Conclusions

The following conclusions can be drawn from our investigation:

  * •

The Fourier expansion of nuclear shapes offers a very effective way of
describing the deformation of fissioning nuclei in the vicinity of the ground
state and the scission point.

  * •

Potential energy surfaces are evaluated in the macro-micro model using the LSD
mass formula for the liquid-drop type energy and a Yukawa-folded single-
particle potential to obtain the microscopic energy correction.

  * •

A 3D Langevin model that couples fission, neck, and mass asymmetry modes was
used to describe the fragment mass and kinetic energy yields.

  * •

The transition with increasing mass from asymmetric to compact symmetric
fission, as observed in Fermium isotopes, is well reproduced.

  * •

The multiplicity of post-scission neutrons and the charge of the fission
fragments are estimated within our model.

  * •

The influence of the inclusion of higher-multipolarity deformation parameters
$a_{5}$ and $a_{6}$ is on our agenda.

Further calculations for a wider mass and charge region are in progress.

Acknowledgements
This work was supported by the Polish National Science Centre, project No.
2018/30/Q/ST2/00185, the Natural Science Foundation of China (Grant
No.11961131010 and 11790325), and the COPIN-IN2P3 collaboration (project
number 08-131) between PL-FR labs.

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