Gated Bayesian networks (GBNs) are an extension of Bayesian networks that aim to model systems that have distinct phases. In this paper, we aim to use GBNs to output buy and sell decisions for use in algorithmic trading systems. These systems may have several parameters that require tuning, and assessing the performance of these systems as a function of their parameters cannot be expressed in closed form, and thus requires simulation. Bayesian optimisation has grown in popularity as a means of global optimisation of parameters where the objective function may be costly or a black box. We show how algorithmic trading using GBNs, supported by Bayesian optimisation, can lower risk towards invested capital, while at the same time generating similar or better rewards, compared to the benchmark investment strategy buy-and-hold.
Algorithmic trading can be viewed as a process of actively
deciding when to own assets and when to not own assets,
so as to get better risk and reward on invested capital,
compared to holding the assets over a long period of time. On
the other end of the spectrum is the buy-and-hold strategy,
where one owns assets continuously over a period of time
without making any decisions of selling or buying. An
algorithmic trading system consists of several components,
some which may be automated by a computer, and others
that may be manually executed [
In Figure 1, the price of an asset is plotted along with
buy signals (upward arrows) and sell signals (downward
arrows). We view the time spent between these signals as
two different phases: before a buy signal, our intention is
to have a model that identifies good opportunities to buy
the asset, once such an opportunity has been identified and
a buy signal has been generated, we move into a different
phase. In this second phase, we intend to model the
identification of good opportunities to sell the asset. Once a sell
signal is generated, we move back to the original phase,
once again using a model to generate buy signals. This
particular situation was the main motivation for the
introduction of gated Bayesian networks (GBNs) [
Bayesian optimisation has grown in popularity in the
machine learning community as an intuitive way of
maximising either black box objective functions and/or very costly
objective functions (costly in the sense of both time and
resources) [
Our intention in this paper is to combine the use of GBNs as alpha models and optimising the parameters of these GBNs using Bayesian optimisation.
The rest of the paper is organised as follows. We begin by giving a brief introduction to GBNs in Section 2, this will illuminate how GBNs can be used as alpha models. We continue by explaining by which metrics alpha models can be evaluated in Section 3, and give slightly more details regarding backtesting. In Section 4 we will describe the components of Bayesian optimisation, including the use of Gaussian processes as priors, as well as kernel and acquisition functions. In Section 5 we will account for the procedure we will use to evaluate the expected performance of using Bayesian optimisation over the parameters of GBNs. Once the procedure has been described, we will in Section 6 account for a real-world application where we show how GBNs can be used as alpha models with support from Bayesian optimisation. Finally, in Section 7 we will offer a few words regarding our conclusions and future work. 2
GATED BAYESIAN NETWORKS Bayesian networks (BNs) can be interpreted as models of causality at the macroscopic level, where unmodelled causes add uncertainty. Cause and effect are modelled using random variables that are placed in a directed acyclic graph (DAG). The causal model implies some probabilistic independencies among the variables, that can easily be read off the DAG. Therefore, a BN does not only represent a causal model but also an independence model. The qualitative model can be quantified by specifying certain marginal and conditional probability distributions so as to specify a joint posterior distribution, which can later be used to answer queries regarding posterior probabilities, interventions, counterfactuals, etc. The independencies represented in the DAG make it possible to compute these queries efficiently. Furthermore, they reduce the number of parameters
G2 G1
W E
F
needed to represent the joint probability distribution, thus
making it easier to elicit the probability parameters needed
from experts or from data. See [
A GBN consists of BNs and gates. BNs can be active or inactive. The label of BN1 is underlined, indicating that it is active at the initial state of the GBN. The BNs supply posterior probabilities to the gates via so called trigger nodes. The node S is a trigger node for gate G1 and W is a trigger node for G2. A gate can utilise more than one trigger node.
Each gate is encoded with a predefined logical expression regarding its trigger nodes’ posterior probability of a certain state, e.g. G1 may be encoded with p(S = s1je) > 0:7. This expression is known as the trigger logic for gate G1.
When evidence is supplied to the GBN, an evidence handling algorithm updates posterior probabilities and checks if any of the logical statements in the gates are satisfied. If the trigger logic is satisfied for a gate it is said to trigger. A BN that is inactive never supplies any posterior probabilities, hence G2 will never trigger as long as BN2 is inactive.
When a gate triggers, it deactivates all of its parent BNs and activates its child BNs (as defined by the direction of the edges between gates and BNs). In our example, if G1 was to trigger it would deactivate BN1 and activate BN2, this implies that the model has switched phase.
If the GBN was used as an alpha model, then when the GBN identifies a buying opportunity, and moves to the sell phase, a buy signal is generated. Looking again at Figure 1, each buy and sell signal was generated as the GBN switched back and forth between its phases.
For the purpose of discussing GBN parameter optimisation in general, we will say that a GBN is parameterised by three disjoint parameter sets , and . The parameters in are the parameters of the marginal and conditional probability distributions of the variables in the contained BNs. All other free parameters are contained in , while any fixed parameters are contained in . For instance, in a setting where the only unknowns are the thresholds in the trigger logic of the gates, we say that the thresholds are in and all other parameters are fixed in . This notation allows a bit of convenience when discussing the evaluation of the optimisation procedure in Section 5 and the application in Section 6. 3
EVALUATION OF ALPHA MODELS As we alluded in Section 1, and as we shall see in Section 6, it is necessary to assess how good a set of signals are, thereby assessing the performance of an alpha model. Regression models can be evaluated by how well they minimise some error function or by their log predictive scores. For classification, the accuracy and precision of a model may be of greatest interest. Alpha models may rely on regression and classification, but cannot be evaluated as either. An alpha model’s performance needs to be based on its generated signals over a period of time, and the performance must be measured by the risk and reward of the model. This is known as backtesting. 3.1
The process of evaluating an alpha model on historic data
is known as backtesting, and its goal is to produce
metrics that describe the behaviour of a specific alpha model.
These metrics can then be used for comparison between
alpha models [
Alpha models are backtested separately from the other components of the algorithmic trading system, as the backtesting results are input to the other components. Therefore, we execute every signal from an alpha model during backtesting, whereas in a full algorithmic trading system we would have a portfolio construction model that would combine several alpha models and decide how to build a portfolio from their signals. 3.2
What constitutes risk and reward is not necessarily the
same for every investor, and investors may have their own
personal preferences. However, there are a few metrics that
are common and often taken into consideration [
Although not a metric on its own, the equity curve needs to be defined in order to define the following metrics. The equity curve represents the total value of a trading account at a given point in time. If a daily timescale is used, then it is created by plotting the value of the trading account day by day. If no assets are bought, then the equity curve will be flat at the same level as the initial investment. If assets are bought that increase in value, then the equity curve will rise. If the assets are sold at this higher value then the equity curve will again go flat at this new level. The equity curve summarises the value of the trading account including cash holdings and the value of all assets. We will use Et to reference the value of the equity curve at point t. Metric 1 (Return) The return of an investment is defined as the percentage difference between two points on the equity curve. If the timescale of the equity curve is daily, then rt = (Et Et 1)=jEt 1j would be the daily return between day t and t 1. We will use r and r to denote the mean and standard deviation of a set of returns.
Metric 2 (Sharpe Ratio) One of the most well known
metrics used is the so called Sharpe ratio. Named after
its inventor Nobel laureate William F. Sharpe, this ratio is
defined as: (r risk free rate)= r. The risk free rate is
usually set to be a ”safe” investment such as government bonds
or the current interest rate, but is also sometimes removed
from the equation [
Initial investment $ n i y it u q E
LVFI TIMR
MDD 1 - TIMR
Time
Furthermore, under certain assumptions it can be shown
that there exists an optimal allocation of equity between
alpha models (in the portfolio construction model), such
that the long-term growth rate of equity is maximised [
Metric 3 (Maximum Drawdown (MDD)) The percentage between the highest peak and the lowest trough of the equity curve during backtesting. The peak must come before the trough in time. The MDD is important from both a technical and psychological regard. It can be seen as a measure of the maximum risk that the investment will live through. Investors that use their existing investments that have gained in value as safety for new investments may be put in a situation where they are forced to sell everything. Other risk management models may automatically sell investments that are loosing value sharply. For the individual who is not actively trading but rather placing money in a fund, the MDD is psychologically frustrating to the point where the individual may withdraw their investment at a loss in fear of loosing more money.
Metric 4 (Lowest Value From Investment (LVFI)) The percentage between the initial investment and the lowest value of the equity curve. This is one of the most important metrics, and has a significant impact on technical and psychological factors. For investors trading on margin, a high LVFI will cause the lender to ask the investor for more safety capital (known as a margin call). This can be potentially devastating, as the investor may not have the capital required, and is then forced to sell the investment. The investor will then never enjoy the return the investment could have produced. Individuals who are not investing actively, but instead are choosing between funds that invest in their place, should be aware of the LVFI as it is the worst case scenario if they need to retract their investment prematurely.
Metric 5 (Time In Market Ratio (TIMR)) The percentage of time of the investment period where the alpha model owned assets. This metric may seem odd to place within the same family as the other drawdown risks, however it fits naturally in this space. We can assume that the days the alpha model does not own any assets the drawdown risk is zero. If we are not invested, then there is no risk of loss. In fact, we can further assume that our equity is growing according to the risk free rate, as it is not bound in assets. 4
Our intention is to use GBNs as alpha models and to optimise the free parameters with respect to the metrics given in Section 3.2. In order to do so we must backtest the signals that a GBN produces, and thus we cannot analytically solve the optimisation problem, as backtesting as a function of has no general closed form expression. At the same time, backtesting is relatively costly, as one must create the model, prepare data, estimate parameters, generate signals and walk through the time range to simulate the trading. For this reason, it is not feasible to exhaustively sweep a large grid of parameters. However, Bayesian optimisation allows us to prioritise the points on the grid to evaluate, thus reducing the number of evaluations, while still finding the global maximum of a potentially costly and black box objective function. 4.1
GAUSSIAN PROCESS AS SURROGATE
FUNCTION
Essentially, we would like to find the parameters 2
that maximises an unknown function f . We place a
prior, p(f ), over the possible functions f , and compute
the posterior over f using observations f 1:i; f1:ig, where
fj = f ( j ). Hence, we compute p(f jf 1:i; f1:ig) /
p(f 1:i; f1:igjf )p(f ). We can then use this posterior
distribution over objective functions as an estimate of our
objective function. This is sometimes known as using the
posterior as a surrogate function to the true objective function.
In Bayesian optimisation it is common to use a Gaussian
process (GP) as the surrogate function [
N
K 0; K
KT
K f1:i fi+1 K = 6 4 2 ( 1; 1) . .
. ( i; 1)
. . .
K =
( i+1; 1) K = ( i+1; i+1)
! Using a GP as a surrogate to the objective function allows us to encode prior beliefs about the unknown objective function, and sampling the objective function allows us to update the posterior of the surrogate. What is left to do is to decide where to sample the objective function. (1) (2) 0 2 8
10 4
6
Distance In Bayesian optimisation we make use of a so called acquisition function. Several acquisition functions have been suggested, however the goal is to trade off exploring the grid where the posterior uncertainty is high, while exploiting points that have a high posterior mean. We will use the upper confidence bound criterion, which is expressed as U CB( ) = ( ) + ( ), where ( ) and ( ) represent the mean and standard deviation at the point of the GP, and is a tuning parameter to allow for more exploration (as is increased) or more exploitation (as is decreased).
Succinctly, define a GP over a grid with some kernel function, then randomly sample a point and evaluate the objective function at this point. Calculate the posterior of the GP given this new observation and find 0 that maximises the acquisition function. Then 0 is the next point where to evaluate the objective function. Iterate these steps for a predefined number of iterations. Once all iterations have passed, the with the highest posterior mean is the set of parameters that maximises the objective function. 5
In Section 6 we will account for a real-world application
of GBNs as alpha models supported by Bayesian
optimisation. However, in this section we will introduce the
optimisation procedure used, as well as the method used to
evaluate the performance of the optimisation, which is
essentially the same method used in [
A data set D of consecutive evidence sets, e.g. observations over all or some of the random variables in the GBN, is divided into n equally sized blocks (D1; :::; Dn), such that they are mutually exclusive and exhaustive. Each block contains consecutive evidence sets and all evidence sets in block Di come before all evidence sets in Dj for all i < j. Depending on the amount of available data, k is chosen as the number of blocks used for optimisation. Starting from index 1, blocks 1,...,k are used for optimisation and k + 1 for testing, thus ensuring that the evidence sets in the testing data occurs after the optimisation data. The procedure is then repeated starting from index 2 (i.e. blocks 2; :::; k+1 are used for optimisation and k + 2 for testing). By doing so we create t repeated simulations, moving the testing data one block forward each time. An illustration of this procedure when n = 12, k = 5 and t = 7 is shown in Figure 5. During Bayesian optimisation, when the objective function is evaluated for some acquired , a cross-validation estimate is calculated for the k blocks used. Here, k 1 blocks are used to estimate the parameters of the contained BNs and the held out block is used as validation data to calculate a score . The value of the objective function, given parameters , is thus the average of all when each block in the optimisation data has been held out.
In order to formalise the procedure used to evaluate the optimisation, recall from Section 2 that is used to represent the free parameters of a GBN and is used to represent all fixed parameters. Let J be a score function such that J ( ; Dj ; fDglmj ) is the score for a GBN under some parameterisation and when block j has been used for either testing or validation and the blocks Dl; :::; Dm have been used to estimate of the BNs in the GBN (under the parameters and ).
1. For each simulation t, where (as discussed previously) Dt+k is the testing data and Dt; :::; Dt+k 1 is the optimisation data, use Bayesian optimisation to find the parameters t that satisfies Equation 3.
t = arg max 2 k 1 t+k 1
X j=t
J ( ; Dj ; fDgtt+k 1 nDj j ) 2. For each t calculate the score tJ on the testing set with respect to the scoring function J according to Equation 4.
tJ = J ( t; Dt+k; fDgtt+k 1j ) (4) 3. The expected performance J of the optimisation, with respect to the score function J , is then given by the average of the scores tJ , i.e. J = 1t Pt tJ . Two things to note about this procedure. First, during cross-validation inside the objective function we disregard (3) the natural order of the data, thus allowing a validation block to come before a block used for estimating the parameters . This could potentially induce a bias in the cross-validation estimate as the data used for estimating the parameters would not have been known at the time the data for the validation block was generated. However, as we do not use this scheme when we evaluate the performance of the optimisation, the expected performance of the optimisation is not biased in this way. We simply use this scheme to make the best use of the data during cross-validation. Second, one scoring function J has been used both during optimisation and for evaluating the expected performance of the optimisation. The scoring function J could internally use many different metrics to come up with one score to maximise. However, it is natural in the coming setting to expose the actual values of several metrics, thus several scoring functions J are used to get a vector of mean scores [ J1 ; :::; Jm ].
Another approach to combine Bayesian optimisation with
cross-validation is to reduce the number of fold evaluations
necessary [
Having established the optimisation procedure, and the method we intend to use to evaluate the performance of the optimisation, we turn our attention to a real-world application. We aim to use GBNs as alpha models to generate buy and sell signals of stock indices in such a way that drawdown risks are mitigated, compared to the buy-andhold strategy, while at the same time maintaining similar or better rewards.
Stock indices are weighted averages of their respective stock components. For instance, the Dow Jones Industrial Average (DJIA) is a weighted average of 30 large companies based in the United States. Indices may have different schemes for how the different components are weighted, however they all aim to give a collective representation of their components.
An index fund owns shares of the components of a specific index, proportional to the weights, such that the fund’s return is mirrored by the index. These funds are very popular, as they are easy for the investor to comprehend but at the same time trading the individual components of an index requires a lot of effort.
A buy-and-hold strategy on stock indices via index funds may be convenient, however it implies that the equity is put through the full force of drawdown risks described in Section 3.2. The buy-and-hold strategy holds assets over the entire backtesting period and so will be subject to the full force of these metrics. For instance, as an asset will be held G2
G1 Buy
Sell
Trend
G1 G2
Long Short
G3 G4 throughout the period, the lowest point of the assets value will coincide with LVFI. In dwindling stock markets, the index funds will lose value, and equity could be salvaged and possibly be placed in risk-free assets during these periods. Furthermore, utilising certain financial products, it is also possible to increase equity during these times of distress by purchasing short positions of the index. Short positions can be thought of as a loan, where the value of the loan increases if the index decreases in value, and it is possible to sell the loan at its higher value (to make the distinction, regular positions are called long when short positions are considered).
At first the buy-and-hold strategy may seem na¨ıve,
however it has been shown that deciding when to own and not
own assets requires consistent high accuracy of predictions
in order to gain higher returns than the buy-and-hold
strategy [
We used two different GBN structures to create alpha models. The first GBN structure (henceforth known as GBN-1) modelled buying and selling long positions only, while the second GBN structure (GBN-2) modelled buying and selling long and short positions. The structures are depicted in Figure 6 (GBN-1 on the left and GBN-2 on the right). The structure for GBN-1 works as described in Section 2. The structure for GBN-2 starts in the T rend phase, from where either G1 or G2 can trigger. If G1 triggers then a long open signal is generated and the Long phase is activated (deactivating the T rend phase). If then gate G3 triggers then a long close signal is generated, and the T rend phase is activated again (deactivating the Long phase). However, if before G1 triggers G2 triggers instead, then a short open position is generated, and the Short phase is activated (deactivating T rend). In similar fashion, when G4 triggers a short close signal is generated, activating T rend and deactivating Short. 6.1.1
The variables used in the GBNs were discretisations of
so called technical analysis indicators. One of the major
tenets in technical analysis is that the movement of the price
of an asset repeats itself in recognisable patterns. Indicators
are computations of price and volume that support the
identification and confirmation of patterns used for forecasting.
Many classical indicators exists, such as the moving
average (MA), which is the average price over time, and the
relative strength index (RSI) which compares the size of
recent gains to the size of recent losses. For the full
definition of these indicators, please see [
For each phase in the GBNs (Buy, Sell, T rend, Long and Short), we placed a na¨ıve Bayesian classifier over the same technical analysis indicators. However by allowing the parameterisation of one of the technical analysis indicators to vary between the phases, we essentially created different variables in the different phases. The tuning of the technical analysis parameters allowed us to better capture the dynamics of the data, as they may differ between assets as well as between the different phases of trading. Figure 7 depicts the classifier structure and variables used. The variables are explained below, along with an example in Figure 8.
S represents the first-order finite backward difference of 5 periods of the MA of periods, shifted 5 periods into the future. To clarify, the first plot in Figure 8 shows the price of an asset along with the MA. If the current time is t, then S represents the slope of the line between what the MA will be at t + 5 and what it is at t.
A represents the same slope as S but at its current value (i.e. between t and t 5).
B represents the difference between the current value of the MA of periods and the current raw price. This can be seen in Figure 8 as the difference between the two time series in the first plot.
C represents the current RSI value (at t in the second plot of the figure) using 14 periods.
D represents what C was 5 time steps in the past (at t 5 in the second plot of the figure).
The choice of 14 periods for RSI is based on the prevailing
standard [
Variables A, B, C and D were discretised into six bins, each using equal width binning, and S was discretised into two bins separated by zero. Thus, the states of S represents B PDIFF(MA( ))
C RSI(14)
D RSI(14)
Offset(-5) A t
B
S
t + 5 t − 5
D M2a0r1011 M2a0r1014 M2a0r1017 M2a0r1110
M2a0r1114 M2a0r1117
M2a0r1211 M2a0r1214 5 3
t − 5 M2a0r1011 M2a0r1014 M2a0r1017 M2a0r1110
C t M2a0r1114 M2a0r1117
M2a0r1211 M2a0r1214 a predicted positive or negative future value of the modelled asset price (smoothed by the moving average of periods). As S represents a future value, evidence for S was only available during estimation of the parameters , not during the generation of signals.
The gates all defined trigger logic over the posterior distribution of S with some threshold . For instances, in GBN-1, the trigger logic for G1 was T L(G1) : p(S = positive j e) > G1, i.e. if the posterior probability of a positive climate is greater than some threshold, then the model should give a buy signal and move to the next phase (the sell phase). Naturally, the trigger logic for G2 in the same GBN was T L(G2) : p(S = negative j e) > G2, thus giving a sell signal if the posterior probability of a negative future value exceeds some threshold. 6.1.2
The previous section implies the following for the two GBNs:
= f G1; G2; Buy; Sellg.
= f G1; G2; G3; G4; T rend; Long; Shortg.
S r15MA( ) Offset(+5)
A r51MA( ) 5 2 13 MA AM 1315 neda 5 Price rcP 130 i 5 9 2 1 5 5 RSI ISR 54 We used the upper confidence bound acquisition function (as described in Section 4.2) with = 5, which allowed for abundant exploration, as our objective function was not extremely expensive to evaluate. We used the rational quadratic kernel as described in Equation 1 with c = 1. For GBN-1 we ran the Bayesian optimisation for 1,600 iterations, and for GBN-2 we ran 12,800 iterations. 6.1.3
We used four indices in this study, DJIA and NASDAQ which are both based on companies in the United States, FTSE100 which is based on companies in the United Kingdom and DAX which is based on companies located in Germany. We ran our experiments on daily adjusted closing prices for these indices ranging from 2001-01-01 to 201212-28 (data downloaded from Yahoo! FinanceTM). This gave a total of 12 years of price data for each index, where each year was allocated to a block, thus n = 12. For the cross-validation step we used k = 5 giving t = 7 simulations from which to calculate [ J1 ; :::; Jm ] (the data split is depicted in Figure 5). 6.1.4
The signals generated were backtested in order to calculate relevant metrics. During optimisation (i.e. step 1 in Section 5) the objective function used the Sharpe ratio. The choice was made as it combines both risk and reward into one score, for which a cross-validation estimate could be returned by the objective function. For evaluating the performance of the optimisation (step 2 in Section 5), we used the return and the drawdown risks described in Section 3.2 to create a score vector [ J1 ; :::; Jm ]. The same metrics were calculated for the buy-and-hold strategy. 6.2
The score vectors from the evaluation of the optimisation versus the the score vector for the buy-and-hold strategy over the seven simulations are shown in Table 1. The annual Sharpe presented in the table is the mean return divided by the standard deviation of returns over the seven simulation, and since each block was allocated one year of data it becomes the annual Sharpe ratio.
Will will first turn our attention to GBN-1. We use the Sharpe ratio as our measure of reward, prioritised above the raw return for reasons discussed in Section 3.2. Therefore, we must first ensure that the Sharpe ratio of our algorithmic trading system produces similar or better Sharpe ratios than the buy-and-hold strategy. As can be seen, this was the case for DJIA, NASDAQ and DAX, but not for FTSE100. Secondly, we must take into consideration the TIMR. For GBN-1, we were invested only slightly above half of the time compared to buy-and-hold, reducing risk to equity considerably. Meanwhile, the rest of the time the equity could have gained in value from interest rates (or other risk-free assets), this potential gain was not considered in these results. Risk to equity from MDD was half its counterpart from the buy-and-hold strategy for all indices. The LVFI is a major threat to equity (as discussed in Section 3.2), and one where buy-and-hold severely underperforms. For DAX the LVFI was only a third of the buyand-hold LVFI, and for the other three indices it was half. All in all, the results clearly indicate that GBN-1 was competitive with the buy-and-hold strategy for three of the indices, as Sharpe ratios were improved upon and risk to equity was decreased significantly. Furthermore, these results were achieved while at the same time only having equity invested half of the investment period. It is also clear that we cannot expect the same GBN to be useful for all indices, as the reward was not improved upon for FTSE100. Some of the parameters that were fixed in may have to be tuned in order to accommodate the dynamics of FTSE100, such as the technical analysis indicators used, or the fixed parameters of the ones used currently.
Moving on to GBN-2, we can see that allowing the GBN to open short positions changes the results dramatically. For DJIA, we improved upon the Sharpe ratio, at the cost of the drawdown risks. Both MDD and LVFI were increased marginally, yet still lower than buy-and-hold. The TIMR was also increased to such a degree that we were invested almost the entire investment period. There is potential gain in reward from using GBN-2 for DJIA, however the increased risk must be considered.
For NASDAQ, FTSE100 and DAX there is no improvement over GBN-1. Instead, Sharpe ratios are decreased, as well as an increase in drawdown risks. There could be several reasons for this that are worth investigating, however our immediate response is that we have either overfitted the model due to several more parameters being optimised over the same amount of data, or the fact that a bad short position is doubly bad on equity as we will lose out of the profit from a long position during the same time. 7
CONCLUSIONS AND FUTURE WORK Our results show that it is feasible to use GBNs as alpha models, and to use Bayesian optimisation to tune them in order to beat the buy-and-hold benchmark, with respect to certain risk and reward metrics. Some of the design decisions made before optimisation may however have reduced the performance of the GBNs on some of the used data sets. Short positions are optimally taken during times of distress, and due to increased volatility, markets move very differently compared to stable increasing markets. We decided to lock in the forward and backward horizons to 5 time steps, and the RSI period of 14, which may have made it impossible to capture the more volatile dynamics. Furthermore, stock indices generally increase in value over long periods of time, thus short selling will always be in the opposite of the long term trend, which in general is ill-advised. Nevertheless, we are encouraged to see the included positive results and are at the same time motivated to address the problems we faced with GBN-2. We would not expect the exact same model to perform well on all given data sets, and so further work is needed to improve upon the results on FTSE100 to make them in par with the other three indices. For instance, there is room to make the objective function even more expensive by not only estimating BN parameters, but also performing variable selection and structure learning during cross-validation.
BN inference in our implementation is based on the SMILE reasoning engine, contributed to the community by the Decision Systems Laboratory of the University of Pittsburgh and available at https://dslpitt.org/genie/.