Applying Bayesian networks in analysis of actuarial risks
Bayesian methodology is considered as the most known modeling techniques that allows for reduce uncertainty. Networks can use statistical data and expert ratings. It gives an understanding of relationship between drivers of the process and in-cludes statistic, parametric and structural uncertainty....
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| Опубліковано в: : | Проблеми керування та інформатики |
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Інститут кібернетики ім. В.М. Глушкова НАН України
2025
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| Цитувати: | Applying Bayesian networks in analysis of actuarial risks / R. Panibratov, P. Bidyuk // Проблемы управления и информатики. — 2025. — № 3. — С. 33-44. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860248830374576128 |
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| author | Panibratov, R. Bidyuk, P. |
| author_facet | Panibratov, R. Bidyuk, P. |
| citation_txt | Applying Bayesian networks in analysis of actuarial risks / R. Panibratov, P. Bidyuk // Проблемы управления и информатики. — 2025. — № 3. — С. 33-44. — Бібліогр.: 14 назв. — англ. |
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| container_title | Проблеми керування та інформатики |
| description | Bayesian methodology is considered as the most known modeling techniques that allows for reduce uncertainty. Networks can use statistical data and expert ratings. It gives an understanding of relationship between drivers of the process and in-cludes statistic, parametric and structural uncertainty. Models are fully compatible with human actions during decision-making. They can be applied in different areas, especially in finance. The implementation of this method includes defining the topological structure and probabilistic inference. The latter one is quite complicated, because of network structure. Different algorithms of probabilistic inference give different results. In this experiment LS-method was implemented.
У байєсівських мережах статистичні дані та експертні оцінки можуть використовуватися для розуміння зв’язків між чинниками процесу та статистичної, параметричної й структурної невизначеності. Ці мережі повністю узгоджуються з діями людини під час прийняття рішень. Їх можна застосовувати в різних галузях, особливо у фінансовій. Цей метод передбачає визначення топологічної структури та ймовірнісного висновку. Останнє досить складно зробити через структуру мережі. Різні алгоритми ймовірнісного висновку дають різні результати. У дослідженні реалізовано LS-метод та використано реальні актуарні дані страхової компанії.
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| first_indexed | 2026-03-21T05:38:51Z |
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© R. PANIBRATOV, P. BIDYUK, 2025
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 3
33
МЕТОДИ КЕРУВАННЯ ТА ОЦІНЮВАННЯ
В УМОВАХ НЕВИЗНАЧЕНОСТІ
UDC 004.852
R. Panibratov, P. Bidyuk
APPLYING BAYESIAN NETWORKS
IN ANALYSIS OF ACTUARIAL RISKS
Roman Panibratov
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
https://orcid.org/0000-0002-8604-4420
roman.panibratov@gmail.com
Petro Bidyuk
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
https://orcid.org/0000-0002-7421-3565
pbidyuke_00@ukr.net
Actuarial risk is an important part of analysis actuarial science, because it re-
lates to different types of uncertainties in finances. It emerges from incorrect
calculations or given hypotheses, which were provided in specific model. To
efficiently estimate probability of any type of risks and its power of outcome in
case of occurrence actuaries use broad range of mathematical and statistical
methods and algorithms. Estimation of actuarial risk is very important because it
allows people to make optimized decisions. The growing volume of information
imposes increasing demands on time and resources required by domain experts
to extract relevant insights. This case becomes more difficult due to such data
uncertainties as noisy or missing data, outliers and leads to forecasts with low
precision. Bayesian methodology is considered as the most known modeling
techniques that allows for reduce uncertainty. Networks can use statistical data
and expert ratings. It gives an understanding of relationship between drivers of
the process and includes statistic, parametric and structural uncertainty. Models
are fully compatible with human actions during decision-making. They can be
applied in different areas, especially in finance. The implementation of this
method includes defining the topological structure and probabilistic inference.
The latter one is quite complicated, because of network structure. Different al-
gorithms of probabilistic inference give different results. In this experiment
LS-method was implemented. Real actuarial data from insurance company were
used for construction. Approach of empirical distribution was applied for build-
ing conditional probability tables. Also, logistic regression was implemented as
a classical classification algorithm. For different queries Bayesian network
demonstrated better results in most cases than logistic regression. In future
studies it is suggested to build intellectual decision support system, which in-
cludes this type of Bayesian network.
Keywords: actuarial risk, Bayesian network, LS-method, empiric distribution,
logistic regression.
34 ISSN 2786-6491
Introduction
A crucial component of actuarial science, which addresses the uncertainties involved in
financial estimates, is actuarial risk. It is a risk resulting from the possibility of mistakes
in the computations procedures and assumptions made in actuarial models. Actuaries ana-
lyze the effect and likelihood of different risks using a range of statistical and mathematical
methods. In order to make sure that businesses can effectively manage their financial obliga-
tions and achieve their goals for the future actuarial risk assessment is an essential part of in-
surance process and financial planning. A crucial aspect of actuarial science is analysis of ac-
tuarial risk, and both people and organizations must assess it. Actuaries make judgments on
pricing, investment strategies, and risk management by estimating the possibility and impact
of various risks using a range of statistical and mathematical methodologies. Knowledge of
actuarial risk may assist people in making well-informed decisions regarding their future.
Even for analysts in specialized fields, discovering truly meaningful information takes
a lot of work due to the constant growth of information volume, even if every object may be
accessed quickly and easily. An expert must use knowledge found in data in addition to ana-
lyzing vast amounts of data and selecting the best methods and algorithms based on their
own knowledge. As a result, experts deal with a variety of uncertainties during data analysis,
such as noisy or missing data, disturbances, and various deviations. These situations make
tasks more difficult overall and typically result in less accurate forecasts or predictions.
The Bayesian methodology is one of the most well-known modeling and forecasting
techniques that is helpful to minimize uncertainty. Expert estimates and statistical data can
be included when Bayesian networks are used for exploratory activities or processes. In this
structure, variables can be continuous or discrete, and data for decision-making can be pre-
sented in real-time or as databases or arrays of data. It is possible to gain an understanding of
the relationship between process drivers and high-level visualization by presenting them as
causal interactions. Additionally, this paradigm considers many uncertainties, in particular
parametric, structural, and statistical uncertainty [1]. Models can also be constructed utilizing
hidden nodes or incomplete data. When analyzing various processes, the Bayesian technique
is completely compatible with human behavior during decision-making.
In paper [2], the transmission of systemic credit risk across European sectoral credit
default swap has been analyzed using dynamic Bayesian networks, and the transmission of
systemic risk across them has been studied using both network structure learning and pa-
rameter learning. The authors were able to determine how risk is spread along the dynamic
Bayesian network and what percentage of the new risk results from the transmission of risk
from earlier time points by analyzing the posterior distributions of all the parameters. Using
Bayesian networks, article [3] focused on the intricate connections between the most well-
known crypto-currencies. By introducing a method for building Bayesian networks, this
work makes it possible to investigate how bitcoin connections could evolve over time. In
paper [4], early warning analysis of oil financial risk was conducted using the Bayesian
network prediction model and the back propagation neural network model. The autoregres-
sive mobile algorithm of the model and the time series measurement analysis of the data it-
self were adopted by the Bayesian network prediction model, and the early warning has
a high reliability. Several uses of the systemic risk ratings in the Bayesian framework were
also illustrated by the authors. They introduced their beliefs into the systemic risk assess-
ment process by allocating an appropriate coefficient before the weights. In line with con-
ventional financial knowledge, their systemic risk score also included an easy-to-
understand explanation, allowing investors to use their existing knowledge to assess the
systemic risk of their investment. Using stock return correlations, financial networks, and
a latent unobserved financial space, the authors of [5] presented a novel method for fore-
casting systemic financial risk. Several uses of the systemic risk ratings in the Bayesian
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 3 35
framework were also illustrated by the authors. They introduced their beliefs into the sys-
temic risk assessment process by allocating an appropriate coefficient before the weights.
Compared to more traditional approaches, article [6] demonstrated that credit risk man-
agement based on probabilistic graphical models, such as Bayesian networks, yields im-
proved performance. Probabilistic graphical models offer a comprehensive and effective
way to understand and eventually mitigate risks by utilizing the mathematical underpin-
nings of probability and graph theory. A causal dynamic Bayesian network model was cre-
ated by the authors in [7] in order to identify and illustrate the contemporaneous and tem-
poral causal relationships between the financial variables linked to business value. The
causal dynamic Bayesian network structure showed a lagged feedback loop between intan-
gible assets and Tobin’s Q. High degrees of network connectedness might increase the
propagation of market disturbances, increasing market volatility, according to the theory
behind the use of Bayesian networks to imitate the connection inside the financial market in
[8]. The Bayesian network was adapted and used by the authors in [9] to forecast the likeli-
hood of corporate insolvency. After using the LASSO model to choose important variables,
they built a Bayesian network using those variables and used the EM-technique (Expecta-
tion-Maximization) to estimate the model parameters. The Bayesian network’s topology
demonstrated a clear depiction of its underlying operation. In paper [10], Bayesian net-
works were used to analyze the dynamic causal link between Bitcoin and financial assets.
According to the study’s specific findings, portfolio managers could think about adding
Bitcoin to their holdings as a diversification tactic, particularly during times of market
stress and volatility. The potential of using the Bayesian technique for operational risk
modeling and estimate is demonstrated by the examples examined in [11].
It is suggested to build Bayesian network, which is able to predict probability of
claims by using actuarial data from insurance company.
Methodology of Bayesian networks
Bayesian network presents itself as a pair of ,G B , where G is a directed acyclic
graph, which is a network as cause and result relationship. Latter one is denoted as a set
of parameters, which defines network. Element B contains ( ) ( )( )i iX par X
=
( ) ( )( | ( ))i iP X par X= for all possible states of ( ) ( )i ix X and ( ) ( )( ) ( )i ipar X Par X ,
where
( )( )iPar X presents parent set of ( ) .iX G Every variable ( )iX G is con-
sidered as a node. If more than one graph is analyzed, then notion ( )( )G iPar X is used
for defining parent of ( )iX in graph .G
Full joint probability of Bayesian network can be calculated by the next formula:
(1) ( ) ( ) ( )
1
( ,..., ) ( | ( ).
M
M k k
B B
k
P X X P X Par X
=
=
This structure is occasionally shown as a graph with specific characteristics. The
nodes or variables that create a Bayesian network are connected by edges. The causal link
between variables is shown by the edge that connects them. The lack of a causal link between
nodes is shown by a non-directed edge. If a graph exclusively has non-directed edges, it is
said to be non-directed. The direction of dependency is indicated by the presence of
an arrow in the edge. A graph is said to be directed if all of its edges are directed.
Next types of Bayesian networks are defined [1].
1. Discrete Bayesian network. This is the network, where vertices are determined
as discrete variables. They contain next properties:
36 ISSN 2786-6491
— every node is defined as an event, which is described by random variable, which
can have several states;
— all nodes, which are connected to their parent nodes, are set by conditional
probability tables or functions of conditional probability;
— probabilities for nodes, which don’t have parents, are unconditional.
2. Dynamic Bayesian network. The values of the nodes in this kind of network
vary over time. This is the best candidate for modeling time dependent processes. One
of their advantages is that they can more easily express non-linear processes by apply-
ing table representation of conditional probabilities.
3. Continuous Bayesian network. This network’s variables are continuous, and
events can take on any state within a specified range. As a result, the random variable
will be continuous with an infinite set of points and a space of potential states that is
a range of acceptable values. The probability distribution function and the probability
density function are used in this instance to define the probability distribution.
LS-method
The Bayesian network of predetermined structure is obtained by the analyst after
the structure has been constructed utilizing techniques or specialists in the area. Any
situation can be described by giving nodes state values and determining the maximum
probability values for other nodes. The method of estimating a vertex state based on the
prior probability of other vertex states is known as probabilistic inference.
From a computational perspective, this subject is unclear and complicated. The
network nodes and edges — which link nodes to one another — have a direct impact on
the amount of computation. The uncertainty stems from the fact that various probabilis-
tic inference techniques yield distinct results. This circumstance is especially typical in
big Bayesian networks. Consequently, there isn’t an algorithm that can deliver optimal
outcomes for every kind of network.
The basis of clustering techniques is the LS-method, also known as the Lauritzen
and Spiegelhalter approach [12]. To use probabilistic inference, the Bayesian network
structure is converted into a junction tree, and the conditional probability tables of the
model’s nodes are recalculated one after the other using a method for message propaga-
tion up and down the tree.
There are two phases to the LS-algorithm’s execution. The first phase involves
building a junction tree from the network’s basic structure and filling in the conditional
probability tables of the model’s nodes. It comprises the following phases:
1. Moralization of graph: every node with parents is chosen one after the other. Con-
nect the mutually disconnected parent nodes of each network node using edges if the node’s
parents are not linked. A neighbor relation is inserted if the parents of the node are related to
one another. Another name for this change is «marriage» of parent nodes. This graph should
then be undirected, with undirected edges replacing all of the graph’s directed edges.
2. Triangulation of moral graph: depict a moral graph in a way that ensures each
cycle is no longer than three nodes. It is represented graphically as the division of
a moralized graph on triangles. Until every node is taken into consideration, the Bayesi-
an network’s nodes are successively visited. The following actions ought to be taken:
— verify whether the studied node’s neighbors are nearby; this node is referred to
be simplicial if it is true since it forms a clique with its neighbors and is not included in
the analysis along with its own edges;
— proceed to the next stage of the method if, after verifying that the network of
nodes that remain to be taken into consideration is not empty; if not, the graph is re-
garded as triangulated.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 3 37
Vertices in the network are successively examined, searching for the node with the
greatest number of neighbors. Incorporating more edges between its non-adjacent
neighbors makes this node simplicial. It forms a clique with its neighbors and is not
watched. When the fundamental moralized undirected graph receives new edges, it be-
comes triangulated.
3. Identification of cliques. Triangulated moralized graphs or subgraphs with car-
dinality of no more than three are used to define sets of cliques. First, see whether it is
a subset of other cliques that haven’t been tested yet. It will be destroyed if it is true.
The edge that contains a separator or intersection of the node sets of these cliques is in-
serted between them if at least one node is shared by this clique and others. Following
this, ordered sets of nodes are used to implement ranging of sets of cliques. The node in
the clique that ranks first in the ordered set becomes the first. The existence of the node
in the clique that ranks second in the ordered set of nodes is looked for if there are
several cliques that contain this node. Thus, it is implemented to range all sets of cliques.
4. Construction of junction tree. In this stage, the edges with the biggest separators
are chosen to join each clique in turn, and the other edges are eliminated. The union tree
will be the one that links all cliques with the greatest number of separator powers. In the
union tree, the clique with the most vertices is selected as the root for convenience. The
clique with the greatest number of edges will be chosen if there are many such vertices.
Triangulated moralized graphs or subgraphs with cardinality of no more than three
are used to define sets of cliques. First, see whether it is a subset of other cliques that
haven’t been tested yet. It will be destroyed if it is true. An edge that contains a separa-
tor or intersection of the node sets of these cliques is inserted between them if at least
one node is shared by this clique and others. Following this, ordered sets of nodes are
used to implement ranging of sets of cliques. The node in the clique that ranks first in
the ordered set becomes the first. The existence of the node in the clique that ranks sec-
ond in the ordered set of nodes is looked for if there are several cliques that contain this
node. Thus, it is implemented to range all sets of cliques.
The junction tree is filled with tables at the end of the first stage. Filling in begins
with the tree’s leaves and moves through each clique in turn. In a raw clique, unmarked
vertices are taken into consideration:
— the table should contain the structure (vertex | other vertices of clique)P if
there is one unmarked vertex that is not seen in other raw cliques; the clique is then
designated as processed and this node is regarded as marked;
— the joint probability table is utilized if the Bayesian network’s primary structure
does not have such a table; cliques are classified as processed after all of their vertices
have been tagged;
— the table of structure (unmarked vertex)P from the principal structure of the
Bayesian network is utilized if the network has marked vertices and only one unmarked ver-
tex. Cliques are designated as processed, and this node is regarded as marked;
— the table of the joint probability distribution of unmarked vertices is employed
and they are regarded as marked if there are multiple unmarked vertices in addition to
marked ones; the clique is therefore recognized as processed;
— the table of structure (any vertex)P is filled and the clique is recognized as
processed if every vertex in the unprocessed clique is marked.
The second phase is the propagation procedure, which involves using the junction
tree to construct probabilistic inference. For each observation, a single table containing
this variable is selected. Every other input that deviates from observation equals zero.
This step’s initial phase is known as the upward procedure. The message is the out-
come of adding up or marginalizing the variables in the table that are not in a separator.
38 ISSN 2786-6491
The sender splits its own conditional probability table on the message after transmitting
it. The recipient creates a new table by multiplying the message by its own conditional
probability table. The receiver splits its own table by sent message and transmits the
message to its parent when it receives new messages from all of its children. The process
keeps going on until all of the junction tree’s descendants have sent messages to the root.
5. The descending process continues as it proceeds out. Every descendant re-
ceives a message from the root. It transmits the outcome after marginalizing the table
by separator and dividing its own conditional probability table by message, which was
acquired from the descendant. When the descendant receives a message from its own
parent, it creates its own table by multiplying it by its own conditional probability
table. Until all leaves get their messages, the process continues. If there are observa-
tions, the results will be recalculated in tables for each variable, which should be fur-
ther normalized.
Numerical experiment
For building Bayesian network real actuarial data from insurance company was
applied. The dataset was obtained from Singapore Actuarial Society. There has been an
accident involving every worker compensation insurance policy in this dataset. The next
features were included into analysis:
1. Age (A): continuous variable, which was transformed into discrete type by using
such states: between 18 and 25, between 25 and 50, more than 50.
2. Gender (G): categorical variable, which has states M (male) and F (female).
3. PartTimeFullTime (PFT): categorical variable, which contains information
about working mode of client. It has states P (part time) and F (full time).
4. DaysWorkedPerWeek (D): discrete variable, which contains amount of working
days of every client. Its range is between 1 and 7.
5. MaritalStatus (M): categorical variable, which contains information about
client’s marital status. It has states S (single), M (married), U (unknown).
6. WeeklyWages (W): continuous variable, which shows weekly wage of every
client and was transformed into categorical type by using such states: less than 100,
between 100 and 1000, between 1000 and 50000.
7. DependentChildren (CH): discrete variable, which indicates number of de-
pendent children. It ranges between 0 and 2.
8. UltimateIncurredClaimedCost (CL): continuous variable, which shows claim
payment of insurance company. It was transformed into categorical type by using such
states: between 100 and 1000, between 1000 and 50 000, between 50 000 and 100 000.
This variable is target.
Topological structure of Bayesian network is shown in Fig. 1.
For building conditional probability tables of
Bayesian network approach of empirical distribu-
tion was applied [13]. Dataset D for variables X
consists of vectors 1, ... , ,Md d where every id is
considered as a case and indicates specific instan-
tiation of variables .X For this experiment the da-
taset is considered as complete. For a full dataset
empiric distribution is defined in the next way:
M
kNum
kPD
)(
)( = ,
where )(kNum is the amount of cases id in dataset D , which contains event .k
A G D
M W CH
CL
PFT
Fig. 1
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 3 39
Conditional probability tables for nodes A, G, PFT, D, M, W, CH, CL are pro-
vided in tables 1–8.
Table 1
State Between 18 and 25 Between 25 and 50 More than 50
Probability 0.256 0.631 0.113
Table 2 Table 3 Table 4
State F M
Probability 0.23 0.77
State P M
Probability 0.91 0.09
State 1 2 3
Probability 0.95 0.02 0.03
Table 5
State 1 2 3 4 5 6 7
Probability 0.003 0.010 0.026 0.027 0.912 0.016 0.006
Table 6
Parents
States
A G
Parents’ states S M U
Between 18 and 25 F 0.844 0.074 0.082
Between 18 and 25 M 0.839 0.077 0.084
Between 25 and 50 F 0.394 0.517 0.090
Between 25 and 50 M 0.405 0.488 0.107
More than 50 F 0.226 0.684 0.090
More than 50 M 0.196 0.687 0.117
Table 7
Parents
States
PFT D
Parents’ states Less than 100 Between 100 and 1000 Between 1000 and 50 000
P 1 0.5 0.5 0.0
P 2 0.2 0.8 0.0
P 3 0.1 0.9 0.0
P 4 0.0 1.0 0.0
P 5 0.0 1.0 0.0
P 6 0.0 0.8 0.2
P 7 0.0 0.7 0.3
F 1 0.6 0.4 0.0
F 2 0.3 0.7 0.0
F 3 0.1 0.9 0.0
F 4 0.0 1.0 0.0
F 5 0.0 1.0 0.0
F 6 0.0 0.9 0.1
F 7 0.1 0.8 0.1
40 ISSN 2786-6491
Table 8
Parents
States
M CH W
Parents’ states
Between 100 and
1000
Between 1000 and
50 000
Between 50 000 and
100 000
S 0 Less than 100 0.37 0.62 0.01
S 0 Between 100 and 1000 0.33 0.65 0.02
S 0 Between 1000 and 50 000 0.02 0.09 0.08
S 1 Less than 100 0.00 1.00 0.00
S 1 Between 100 and 1000 0.18 0.76 0.06
S 1 Between 1000 and 50 000 0.00 0.79 0.21
S 2 Less than 100 0.33 0.67 0.00
S 2 Between 100 and 1000 0.00 0.74 0.26
M 0 Less than 100 0.20 0.79 0.01
M 0 Between 100 and 1000 0.24 0.73 0.03
M 0 Between 1000 and 50 000 0.03 0.84 0.13
M 1 Less than 100 0.45 0.55 0.00
M 1 Between 100 and 1000 0.20 0.76 0.04
M 1 Between 1000 and 50 000 0.03 0.89 0.08
M 2 Less than 100 0.14 0.86 0.00
M 2 Between 100 and 1000 0.21 0.76 0.03
M 2 Between 1000 and 50 000 0.01 0.84 0.15
U 0 Less than 100 0.17 0.83 0.00
U 0 Between 100 and 1000 0.16 0.80 0.04
U 0 Between 1000 and 50 000 0.04 0.85 0.11
U 1 Less than 100 0.00 1.00 0.00
U 1 Between 100 and 1000 0.21 0.75 0.04
U 1 Between 1000 and 50 000 0.00 0.80 0.20
U 2 Less than 100 0.00 0.67 0.33
U 2 Between 100 and 1000 0.18 0.64 0.18
U 2 Between 1000 and 50 000 0.00 1.00 0.00
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 3 41
A G D
M W CH
CL
PFT
Fig. 2
Results of Bayesian network moralization and transformation of moralized into
undirected graph are shown in Fig. 2, 3.
After this the Bayesian network was rebuilt into junction tree. Its structure is
shown in Fig. 4.
Fig. 4
Clique «M, CH, W, CL» is the root one, cliques «M, G, A» and «PFT, W, D» are
the leaves.
Logistic regression
Results of Bayesian networks work after entering evidences into nodes were
compared with logistic regression.The classification technique called logistic regression
makes use of many independent features in order to forecast a binary-dependent result.
It is a very powerful method for determining the connection between information, cues,
or an occurrence in particular [14].
The goal of logistic regression is to model the probability of a particular result
using a collection of input factors. In logistic regression, the output variable is binary,
meaning it can only take on one of two possible values. Independent variables are any
elements that have an effect on the dependent variable. Aside from this, the input
variables might be continuous, categorical, or a mix of the two.
The sigmoid function is used to transform the result into a category value. In
machine learning, predictions are converted to probabilities using the sigmoid function,
which produces an S-shaped curve and yields a probabilistic value between 0 and 1.
The following is one way to write the model:
0
1
0
1
( ) .
1
n
i i
i
n
i i
i
x
x
e
P X
e
=
=
+
+
=
+
Alternatively, it might be rewritten as:
1
( ) ,
1 X
P X
e−
=
+
where X is the linear combination of explanatory variables, ( )P X — probability that
the specific event will happen.
A G D
M W CH
CL
PFT
Fig. 3
M, CH, W, CL
M, G, A
PFT, W, D
W
M
42 ISSN 2786-6491
The amount of the dependent variable’s value is often influenced by a number of
explanatory variables. The different independent variables are assumed to have a linear
relationship with the model in logistic regression calculations.
The logistic regression equation is used to determine the value of :X
0 1 1 ... ,n nX x x= + + +
where 0 is the intercept; 1,..., n are prameters; 1,..., nx x are explanatory variables.
When the dependent variable has more than two categories, but no ordered
subcategories, multinomial logistic regression is used. Stated differently, the categories
are mutually exclusive and lack any intrinsic ordering.
A flexible and powerful technique for simulating the relationship between one or
more independent variables and a binary or categorical dependent variable is logistic
regression. The several logistic regression models discussed above can be used to solve
various problems in different fields.
Results of experiment
Probabilistic inference of CL after query W = «Between 1000 and 50 000»; M = S;
CH = 0 is provided in Table 9.
Table 9
State Probability (Bayesian network) Probability (logistic regression)
Between 100 and 1000 0.023 0.106
Between 1000 and 50 000 0.895 0.828
Between 50 000 and 100 000 0.082 0.066
Probabilistic inference of CL after query W = «Less then 100»; M = S; CH = 1 is
provided in Table 10.
Table 10
State Probability (Bayesian network) Probability (logistic regression)
Between 100 and 1000 0.368 0.393
Between 1000 and 50 000 0.621 0.603
Between 50 000 and 100 000 0.011 0.004
Probabilistic inference of CL after query CH = 1; W = «Less then 100»; A =
= «Between 18 and 25»; D = 5 is provided in Table 11.
Table 11
State Probability (Bayesian network) Probability (logistic regression)
Between 100 and 1000 0.025 0.095
Between 1000 and 50 000 0.965 0.698
Between 50 000 and 100 000 0.000 0.007
Probabilistic inference of CL after query G = M; A = «More than 50»is provided in
Table 12.
Table 12
State Probability (Bayesian network) Probability (logistic regression)
Between 100 and 1000 0.243 0.081
Between 1000 and 50 000 0.727 0.908
Between 50 000 and 100 000 0.030 0.011
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2025, № 3 43
Conclusion
Actuarial risk is an essential part of actuarial science, which deals with the
uncertainties in financial forecasts. It is a danger brought on by the potential for errors
in the calculations and presumptions made in actuarial models. Actuaries use a variety
of statistical and mathematical techniques to assess the impact and probability of
various hazards. Actuarial risk assessment is a crucial component of insurance and
financial planning that ensures firms can successfully manage their financial
commitments and accomplish their long-term objectives.
When Bayesian networks are utilized for exploratory tasks or procedures,
statistical information and expert predictions can be included. The Bayesian approach
is fully consistent with human decision-making behavior when examining different
processes. It is possible to update existing knowledge and draw new conclusions about
variables, processes, circumstances, or scenarios by using more data or professional
judgments.
For the numerical experiment real insurance data from Singapore Actuarial Society
was used. For probabilistic inference LS-method was used after building topological
structure of the Bayesian network. Results of different queries were compared with
results, which were obtained from application of logistic regression. Latter one was
used because it’s the classical method for both classification and prediction of all
classes’s probabilities. It can be concluded that in the most cases probability of
Bayesian networks showed better results that results of logistic regression.
In future it is suggested to build intellectual decision support system, which in-
cludes this type of Bayesian network.
Р.С. Панібратов, П.І. Бідюк
ЗАСТОСУВАННЯ БАЙЄСІВСЬКИХ МЕРЕЖ
В АНАЛІЗІ АКТУАРНИХ РИЗИКІВ
Панібратов Роман Сергійович
Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського»
roman.panibratov@gmal.com
Бідюк Петро Іванович
Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського»
pbidyuke_00@ukr.net
Актуарний ризик, що є важливою складовою актуарних наук, пов’язаний
з різними типами невизначеностей у фінансових оцінках. Він виникає че-
рез неправильні обчислення та гіпотези щодо конкретної моделі. Щоб
ефективно оцінити ймовірність будь-якого типу ризику і його наслідки
у разі настання, актуарії використовують математичні й статистичні мето-
ди та алгоритми. Для прийняття оптимальних рішень важливо розуміти ак-
туарні ризики. З постійним зростанням кількості інформації, щоб отримати
корисні дані, інколи потрібно більше часу та ресурсів експертів у певній
галузі. Ця задача ускладнюється ще й через такі невизначеності даних, як
зашумленість, пропуски й викиди, що призводить до неточних прогнозів.
Найбільш відома байєсівська методологія дозволяє зменшити цю невизна-
ченість. У байєсівських мережах статистичні дані та експертні оцінки мо-
жуть використовуватися для розуміння зв’язків між чинниками процесу та
44 ISSN 2786-6491
статистичної, параметричної й структурної невизначеності. Ці мережі пов-
ністю узгоджуються з діями людини під час прийняття рішень. Їх можна
застосовувати в різних галузях, особливо у фінансовій. Цей метод передба-
чає визначення топологічної структури та ймовірнісного висновку. Остан-
нє досить складно зробити через структуру мережі. Різні алгоритми ймо-
вірнісного висновку дають різні результати. У дослідженні реалізовано
LS-метод та використано реальні актуарні дані страхової компанії. Для по-
будови таблиць умовних ймовірностей використано підхід емпіричного
розподілу. Також як класичний класифікаційний алгоритм реалізовано ло-
гістичну регресію. Для різних запитів байєсівська мережа здебільшого
продемонструвала кращі результати, ніж логістична регресія. У статті та-
кож запропоновано побудувати інтелектуальну систему підтримки прий-
няття рішень, що містить розглянуту байєсівську мережу.
Ключові слова: актуарний ризик, байєсівська мережа, LS-метод, емпірич-
ний розподіл, логістична регресія.
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Submitted 14.03.2025
|
| id | nasplib_isofts_kiev_ua-123456789-211399 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-21T05:38:51Z |
| publishDate | 2025 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Panibratov, R. Bidyuk, P. 2026-01-01T19:31:27Z 2025 Applying Bayesian networks in analysis of actuarial risks / R. Panibratov, P. Bidyuk // Проблемы управления и информатики. — 2025. — № 3. — С. 33-44. — Бібліогр.: 14 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211399 004.852 10.34229/1028-0979-2025-3-3 Bayesian methodology is considered as the most known modeling techniques that allows for reduce uncertainty. Networks can use statistical data and expert ratings. It gives an understanding of relationship between drivers of the process and in-cludes statistic, parametric and structural uncertainty. Models are fully compatible with human actions during decision-making. They can be applied in different areas, especially in finance. The implementation of this method includes defining the topological structure and probabilistic inference. The latter one is quite complicated, because of network structure. Different algorithms of probabilistic inference give different results. In this experiment LS-method was implemented. У байєсівських мережах статистичні дані та експертні оцінки можуть використовуватися для розуміння зв’язків між чинниками процесу та статистичної, параметричної й структурної невизначеності. Ці мережі повністю узгоджуються з діями людини під час прийняття рішень. Їх можна застосовувати в різних галузях, особливо у фінансовій. Цей метод передбачає визначення топологічної структури та ймовірнісного висновку. Останнє досить складно зробити через структуру мережі. Різні алгоритми ймовірнісного висновку дають різні результати. У дослідженні реалізовано LS-метод та використано реальні актуарні дані страхової компанії. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Методи керування та оцінювання в умовах невизначеності Applying Bayesian networks in analysis of actuarial risks Застосування байєсівських мереж в аналізі актуарних ризиків Article published earlier |
| spellingShingle | Applying Bayesian networks in analysis of actuarial risks Panibratov, R. Bidyuk, P. Методи керування та оцінювання в умовах невизначеності |
| title | Applying Bayesian networks in analysis of actuarial risks |
| title_alt | Застосування байєсівських мереж в аналізі актуарних ризиків |
| title_full | Applying Bayesian networks in analysis of actuarial risks |
| title_fullStr | Applying Bayesian networks in analysis of actuarial risks |
| title_full_unstemmed | Applying Bayesian networks in analysis of actuarial risks |
| title_short | Applying Bayesian networks in analysis of actuarial risks |
| title_sort | applying bayesian networks in analysis of actuarial risks |
| topic | Методи керування та оцінювання в умовах невизначеності |
| topic_facet | Методи керування та оцінювання в умовах невизначеності |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211399 |
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