Classifier Voting Mechanism for Multi-Class Decision Making : #High_Performance_Python(Post_2)

 

 

In classification tasks, two types of ensemble methods are utilized: hard voting and soft voting. Hard voting operates by collating the final class labels from a range of models and selecting the class that receives the majority of votes. Conversely, soft voting takes into account the predicted probabilities for each class label from various models. In this approach, the probabilities for each class are accumulated, and the class with the highest overall probability is selected as the prediction. Today, we shall explore hard voting through a 'toy' example.

 

The below code implements a Hard voting mechanism commonly used in ensemble machine learning methods, especially in scenarios involving multi-class classification. 

combinations = [pair for pair in itertools.combinations(range(3), 2)]
print("combination Oloop:",combinations)
def voting(output):
    vote_n = np.zeros(10)                        
    vote = ((np.sign(output) + 1) // 2).astype(int).tolist()
    print(vote)
    for i,j in enumerate(vote):
        print(i,j)
        print("combinations:",combinations[i][j])
        vote_n[combinations[i][j]] +=1  
        print("vote:",vote_n)
    return np.argmax(vote_n)   
output = [-1 if i % 2 == 0 else 1 for i in range(3)]
results = voting(output)

Output :

combination Oloop: [(0, 1), (0, 2), (1, 2)]
[0, 1, 0]
0 0
combinations: 0
vote: [1. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
1 1
combinations: 2
vote: [1. 0. 1. 0. 0. 0. 0. 0. 0. 0.]
2 0
combinations: 1
vote: [1. 1. 1. 0. 0. 0. 0. 0. 0. 0.]



The combinations list is formulated utilising itertools.combinations, which assembles all conceivable pairings of classes from a set of three classes, namely 0, 1, and 2. The purpose of the voting function is to evaluate the outputs from various classifiers and ascertain the class receiving the majority of votes. Within this function, vote_n is a NumPy array, initially filled with zeros and dimensioned to tally votes for each class, presuming a total of ten classes. The vote process transforms the given output list into binary votes—0 or 1—based on their sign; thus, positive values in the output correspond to a vote of 1, while negative values yield a vote of 0.
 
As the function proceeds, it methodically iterates through these binary votes. Each vote is scrutinised, and the corresponding class pair from combinations is identified, subsequently incrementing the vote tally for the chosen class in vote_n. Following the accumulation of all votes, the function determines the class index with the highest vote count, employing np.argmax(vote_n) to do so.

Iterating and Indexing in Nested Lists for Data Selection : #High_Performance_Python(Post_1)

Often in image processing, we encounter image data organized in nested lists. A notable example is the image data titled 'dtrain123.dat', available at the link 

http://www0.cs.ucl.ac.uk/staff/M.Herbster/SL/misc/

To effectively visualize such data, we use nested loops to traverse and manipulate these complex data structures. Let's start with a 'toy' example to illustrate the basic concept and its output. Following that, I'll provide the actual code needed to display the image data from 'dtrain123.dat'.

Code :

x = np.arange(10, 20)
y = np.array([0, 1, 2, 3, 0, 2, 1, 3, 0, 2])
print(x, y)

index = [np.argwhere(y == i).flatten() for i in range(4)]
print(index)

for i in range(4):
    if len(index[i]) >= 3: 
        img_index = index[i][2] 
        print(f"Category {i}, Index: {img_index}, 

           Value in x: {x[img_index]}")

The code creates an 'index' list that maps each category (0 to 3) to its occurrences in the array `y` using `np. argwhere(y == i).flatten()`. It then iterates over these categories, ensuring at least three instances exist to avoid indexing errors. For each qualifying category, it retrieves the index of the third occurrence from `index[i][2]` and uses this to access the related value in the `x` array.

This same logic is applied to the image data.

img_indices = [np.argwhere(y == i) for i in range(10)] 
plt.figure(figsize=(10, 3))
for i in range(10):
        plt.subplot(2, 5, i + 1)
        img_index = img_indices[i][2, 0]
        print(img_index)
        plt.imshow(x[img_index, :].reshape(16, 16))
        plt.axis('off')

The desired output is :

Causal Research (#POST 4)

Paper1:Double machine learning and automated confounder selection: A cautionary tale

Summary :

Double/debiased machine learning (DML) is a method used for variable selection in high-dimensional causal inference settings. It uses regularization techniques such as LASSO or l2-boosting to identify variables that are highly correlated with treatment and outcome. By doing so, this procedure addresses the problem of omitted variable bias and provides consistent estimates of the causal impact of the treatment. DML consists of two main approaches: partialling out and double selection, both accounting for the link between treatment and covariates. These designs depend on doubly robust moment conditions and can handle approximation errors from regularization robustly. Nonetheless, ignorability assumption is violated by DML where covariates are not fully exogenous. Poor controls which result in violations of ignorability can undermine the effectiveness of DML. Simulation studies show that even minor deviations from ignorability make DML sensitive leading to results that resemble those based on naïve LASSO. For example, application of DML to estimate gender wage differences reveals huge disparities when endogeneity due to marital status is taken into account

Further reading : 

-https://towardsdatascience.com/double-machine-learning-for-causal-inference-78e0c6111f9d

-For an example that will help the layperson grasp what it means to break the ignorability assumption due to inclusion of “bad controls” in DML algorithm, let me give you one. Say we are conducting research on the effects of a cloud workshop program (treatment variable D) on employee performance (outcome variable Y). In order to achieve this, we collect data for several control variables X such as demographic characteristics, educational background, work experience and job satisfaction. However, by mistake we added ‘motivation’ as a control when it was not entirely independently assigned from treatment. To put it differently, motivation could be affected by both the treatment itself- workshop- or other unobservable characteristics that jointly impact on treatment assignment and job performance. Our use of DML algorithm including “motivation” as a control violates ignorability assumption. The implication is that this variable is not independent of how the treatment is assigned and may introduce bias into estimated treatment effects. This “bad control” can result in confounding whereby the estimation effect of treatment becomes mixed with “motivation’s” influence on job performance.

Paper 2: : ParKCa: Causal Inference with Partially Known Causes

Summary :

Having a slight deviation from traditional stacking models, ParKCa is a causal discovery approach that involves combining various causal estimates. In typical stacking models, individual causal discovery approaches are learned separately and their results compared. However, in ParKCa, these outputs become features for a classification model that acts like the meta-learner. The objective is to identify new causes based on some few examples of known causes. On one hand, each learner (causal discovery method) receives input data in a given format while its output corresponds to every potential cause as implemented here. Rather than cross-validation, level 0 transposed data is bootstrapped to create level 1 data required for training the meta-learner. This way, such assumptions as causal sufficiency are not violated. It should be noted that diversity among learners is the most important feature of ParKCa and it can be measured by Q statistic which describes differences between pairs of classifiers’ performance in detail. The average Q-statistic will give an indication if there is overall diversity across all learners.

Paper 3: Causal Inference with Non-IID Data using Linear Graphical Models

Summary :

An interaction model is a causal model with an adaptation network, in which nodes represent explicit variables within a directed acyclic graph and the direct edges depict the causal relationships of these variables.  The generation process of observed explicit variables’ data is determined by structural equations. In this context, it introduces the term ‘isolated interaction model’ as an “ideal” model that can be obtained by removing all interactions between units to study bias caused by interactions. The paper also examines symmetry assumptions, focusing particularly on the ancestral same-distribution condition (ASDC). ASDC relaxes independence and identical distribution (IID) assumption commonly used in traditional causal inference so that sets of variables must have some similar distribution properties for them to satisfy such conditions. This research concentrates on True average causal effect (TACE), considered as an extension of average causal effect (ACE) under non-IID. TACE represents only the component effecting through cause variable on outcome variable excluding any influence over non-causal path from one unit to another.

Besides, article goes deep into quantification, detection and elimination of interaction bias around TACE estimation. It suggests deflecting bias structures and reflecting bias structures as two types of graphical structures that introduce bias when estimating TACE . For assessment and detection of interaction bias based on presence of these bias structures in interaction network , we prove several theorems and corollaries using quantitative methods. Redressing partiality ,Theorem 2 provides a method for computing a linear regression-based unbiased estimate for TACE from a set of samples satisfying Equation That Is Bias Free: Algorithm 1 proposes an algorithmic approach to selecting largest subset without biases from an interaction network.

This article discusses how these theorems are applicable in practice especially regarding sample size, strength of connections and sparsity among other considerations affecting both reduction of bias and quality estimation.

Paper 4: CounteRGAN: Generating Counterfactuals for Real-Time Recourse and Interpretability using Residual GANs

Summary :

CounteRGAN is meant to produce reasonable examples of counterfactuals that will give recourse to users and offer a better interpretability. It does this using an RGAN (Relational Generative Adversarial Network) and a fixed target classifier C. With CounteRGAN, there are specific objectives for which it can generate counterfactuals: they must be actionable, realistic, belong to specified class, and have low computational latency. There are two versions of the CounteRGAN value function which depend on whether gradients of the classifier are known or not. The goal of the value function is maximizing D while minimizing G. In case when we know that the classifier is differentiable, then the CounteRGAN value function is defined as follows:

VCounterRGAN(G, D) = VRGAN(G, D) + VCF(G, C, t) + Reg(G(x)),

where t represents the target class. The first term, VRGAN, utilizes a specialized RGAN to encourage realistic outputs. The second term, VCF, drives the counterfactual towards the desired target class. The third term, Reg(G(x)), controls the sparsity and magnitude of the residuals, serving as a proxy for counterfactual actionability.

The VRGAN term, which contributes to realistic outputs, is computed based on the discriminator and generator's performance on real and synthetic data samples drawn from the same probability distribution. The VCF term is responsible for aligning the counterfactual examples with the desired target class. It uses the classifier's prediction function Ct to guide the generation process. In cases where the target classifier is non-differentiable or unknown (black-box), a variant called CounteRGANbb is introduced. This variant does not rely on the classifier's gradients. Instead, it weights the first term of the RGAN value function by the classifier's prediction score, Ct(xi). The rest of the value function remains the same. The regularization term, Reg(G, {xi}), controls the sparsity and magnitude of the residuals by combining L1 and L2 regularization terms. It helps to ensure that the generated counterfactual examples are actionable and feasible.

The convergence properties of CounteRGAN are formalized by Theorem 1, which states that under certain conditions, the minimax optimization of the value function leads to the convergence of the generator's output distribution to a distribution defined by pCt(x), where pCt represents the desired class distribution.

Neyman-Rubin Potential Outcomes Model (#POST3)

         

David Hume defined cause as he says is an object A followed by another B such that if there were no A, there would be no B. Nevertheless, Karl Pearson stressed the importance of correlation in scientific research and considered causality to be just a particular case of correlation. Although Pearson categorized correlations into ‘genuine’ and ‘spurious,’ modern causal inference went beyond it recognizing the fact that Hume’s definition always contains its own impossible counterfactual element and thus has problems with directly observing counterfactuals.

Randomized experiments were also designed by Neyman in order to solve this problem, and later on Rubin extended this framework to observational settings. This theory explains why some people are observed with certain outcomes while others are not, yet all individuals are in a situation where they could have experienced either outcome. Judea Pearl introduces another way of causal reasoning which uses Directed Acyclic Graphs (DAG) for illustrating causal relationships. The Structural Causal Models (SCM) underpinning this approach focus on interventional and counterfactual statements.

Both these frameworks make valuable contributions to understanding causal inference and have found applications in disciplines such as epidemiology, medical statistics, economics, artificial intelligence etc., discussed briefly below: In this blog post we will specifically dwell on the potential outcome model and its implications,

Fundamental Problem of Causal Inference : Consider two scenarios,

       

Scenario 1: The subject’s unhappiness leading them to consider getting a dog to become happy is one example from the given task. Thus purchasing happiness afterwards does not show any direct relationship between them while purchasing without it cannot be directly shown to lead causally or casually towards happiness.

Scenario 2: There is only a slight modification. So, in this case it is assumed that acquiring a dog leads to happiness while not acquiring one leads to continuous sadness thereby giving the dog quite strong claim for causing individual happiness.

In the previously mentioned cases, a potential outcome framework is used where happiness is the interest of the objective being referred to as Y. Precisely if one is happy then Y=1 while if one is unhappy then Y=0. The treatment variable T stands for deciding whether or not to get a dog. T= 1 could mean getting a dog while T = 0 implies not getting a dog. In order to capture the possible outcomes, we refer to Y(1) as the expected level of joy if an individual acquires a pup (T = 1). Also, Y(0) represents the potential happiness outcome if no dog will be bought (T = 0). For example, in situation one, both Y(1) = 1 and Y(0) = 1 represent happiness irrespective of whether they have dogs or not. On other side, scenario two in which Y(1)= 1 and Y(0) = 0 indicates that only people with dogs are happy. It should be noted that the potential outcome denoted by ‘Y(T)’ reflects what would happen when subjected under treatment ‘T’ contrary from what was observed in actual sense. Consequently, all feasible results are not observed but depends on an actual value of treating variable that is observed results are varies with reality depending on the actual value of T .

         

The individual treatment effect (ITE) or individual causal effect is defined as below: 

                          τi ≜ Yi(1) - Yi(0) 

The potential outcome variable Y(t) becomes random when multiple people exist within a population because each individual can have different potential outcomes under treatment t. However, the observed outcome variable Yi(t) is considered as nonrandom in most of the cases. This is due to the presence of subscript “i” which denotes that we are looking at a specific individual and his/her context, thus it narrows down our focus to one person within one particular context only. In this case, since we have taken into account these aspects of this particular person’s circumstances, their potential outcomes become known quantities; they are fixed and not subject to any randomness. The determinate nature allows us to analyze and study the causation for that given unit in that particular setting without any ambiguity regarding what possible outcomes it might have. Also in scenario 2, choosing a dog is driven by its positive causal effect on happiness testified by Y(1) - Y(0) > 0. Conversely, according to scenario 1 getting a dog doesn’t make one happier than they were before since Y(1) - Y(0) = 1 - 1 = 0. Thus, deciding against having a dog in example 1 recognizes that being happy does not depend on whether or not one has a dog around him/her. The equation for observed outcomes can be written as:

Yi = Ti * Y1i + (1 - Ti) * Y0i

This equation states that the observed outcome (Yi) for an individual i is determined by the treatment assignment (Ti). If the treatment assignment (Ti) is equal to 1, the observed outcome is equal to the potential outcome under treatment (Y1i). Conversely, if the treatment assignment (Ti) is equal to 0, the observed outcome is equal to the potential outcome without treatment (Y0i).

     

The paragraph above discusses the analogy of potential outcomes given by Brady Neal, a prominent author on causality. It is Neyman’s approach to observational studies as if they were experiments with proper controls. In contrast to random assignment, the model proposed by Neyman uses an urn model that can be considered similar to natural experiments that are as-if randomized in the social and health sciences. There are a limited number of treatment levels which this nonparametric model allows for. Additional work by Holland, Rubin and others considers continuous treatment variables and parametric models, such as linear causal relationships. By turning to the simplest kind of experiment where there is treatment and control groups only; hence, there are many subjects in the population. A subset of populations (Xi) is randomly selected and assigned to the treatment group (Ti) whereas the remaining population (Xi-Ti) constitute the control group. In fact within Neyman-Holland-Rubin model each subject has two possible responses given when it belongs to either a sample group or its complement group. Nevertheless due to some practical constraints i.e., it becomes very hard at times to observe both these answers together. The study population has three different key parameters: 1) The average response if all units were treated, 2) The average response if all units were not treated, and 3) difference between these two averages

Further reading : https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1703-09.pdf

In the context of the example involving getting a dog, it is possible to observe the potential outcome Y(1) by acquiring a dog and evaluating one's subsequent happiness. Similarly, one could observe Y(0) by not getting a dog and assessing their happiness. Nonetheless, observing both Y(1) and Y(0) is impossible without time travel to go back in time and chose another treatment. Just adopting a dog, watching its effect Y(1), returning it then seeing how its absence affects me does not hold water given that the actions taken between these observations as well as other changes will affect this second observation. This inherent challenge is called the fundamental problem of causal inference because we cannot directly observe the causal effect Yi(1) - Yi(0) without having access to both potential outcomes.

COUNTERFACTUALS

Counterfactuals are unobserved potential outcomes because they are different from what really happened. Some time we call them counterfactual outcomes in some instances. On the other hand, measured potential outcome is sometimes known as factual. It should be understood that counterfactual and factual can only be defined after an outcome has been observed. Before that, there are just potential outcomes.

Average Treatment Effect (ATE) or the Average Causal Effect refers to a measure of the mean difference in outcomes between treatment groups. This is obtained by averaging the Individual Treatment Effects (ITEs) represented by τi which refer to the differences in possible outcomes if treated versus untreated:

τi ≜ E[Yi(1) - Yi(0)] = E[Y(1) - Y(0)]

One of the natural quantities that may come to mind is the associational difference which compares the expected outcome when the treatment variable (T) is set to 1 and when it is set to 0: E[Y|T=1] - E[Y|T=0]. In this way, this measure captures any differences in outcomes relating to treatment variables. But without causality. The ATE can be expressed as a function of association difference using linearity property of expectation: ATE = E[Y(1) - Y(0)] = E[Y(1)] - E[Y(0)]. The mathematical representation therefore implies that the expected outcome (Y=1/0) under treatment and no-treatment differs from each other by the expected outcome if T were 1 or 0. However, one should note that there are times when the associative difference, i.e., E[Y|T=1] - E[Y|T=0], and causal difference, i.e., E[Y(1)] – E[Y(0)], are not necessarily interchangeable terms. This would mean reducing causality into association if they are similar. Because of confounding effect, they are not equivalent.

In our previous post, The confounding effects were addressed by us in the two scenarios. To remind you of this, see that post which demonstrates how variable X plays a role in causing confusion influencing treatment T as well as outcome Y. The non-causal relationship is explained through Y <- X -> T pathway.

In our subsequent posts, we shall explore solutions to the fundamental causal problem and present a Python code for calculating average treatment effects. After that, we will introduce research articles and discuss them briefly with specialist readers in mind.

Slides: https://scholar.princeton.edu/sites/default/files/jmummolo/files/po_model_jm.pdf

Papers: 

Splawa-Neyman, J. (1990). On the Application of Probability Theory to Agricultural Experiments: Essay on Principles, Section 9. (Original work published in 1923)

Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies.

Sekhon, J. S. (2008). The Neyman-Rubin Model of Causal Inference and Estimation via Matching Methods.

Correlation ≠ Causation (#POST2)

In daily life, however, people often confuse the two even if scientists all over the world are generally in agreement that correlation does not imply causation. Another study by Messerli (2012) found an interesting positive correlation between chocolate consumption and Nobel Laureates per country. However, these latter data have given way to even stronger relationships than those shown below. Nonetheless his intention is unclear; Messerli is careful not to mix up this relationship with a cause and effect one. This was in contrast to the statement made by “the chocolate industry” which claimed that “eating chocolate leads to Nobel prize winners”(Nieburg, 2012).

The presence of such a correlation between two variables does not mean that there is automatically a causal relation because statistical association alone cannot determine causal relationships with certainty For example, it may be possible that more Nobel laureates results in more chocolate consumption or vice versa due to some factors connected with winning the Nobel prize which Mr. Messerli (2012) referred to as celebrations. There might also be unobserved variables such as socio-economic status or education quality that influence both the number of people who win Nobel prizes and consume chocolate therefore making this correlation non-causal or spurious. These possibilities are according to Reichenbach (1956) common cause principle:

Statistically dependent random variables X and Y imply: (1) X causes Y, (2) Y causes X, or (3) a third variable Z causes both X and Y. Additionally, X and Y become independent given Z, denoted as X ⫫ Y | Z.

Interventions are used for disentangling causality from uncertainty where it exists in causal relationships Hence we can consider studying how increased consumption of chocolates would lead to an increase in noble prize laureates by forcing Austria citizens consume more chocolates In most cases though these kinds of interventions cannot practically be done due ethical considerations or logistics Some similar challenges come up when conducting randomized controlled trials on smoking and lung cancer Nevertheless certain assumptions met help us make causal inference without necessarily having true experiments These assumptions grow strong as we move up levels of causal hierarchy.

Vigen (2015), Spurious correlations : 

Accordingly, the annual number of people who drown in swimming pools has a significant correlation with films that feature Nicholas Cage [above figure]. This correlation raises intriguing questions: Do people’s swimming habits somehow have a link with Nicholas Cages films? Does he want to act in more movies when he sees many people drowning? Is there any other explanation? However, these possibilities seem highly unlikely and suggest a spurious correlation with no causal relationship.

Next post will delve into levels of association, intervention and counterfactuals by way of observing, implementing and imagining respectively. In addition, we will use another example to demonstrate how spurious correlations may emerge thereby making it easier to understand this concept. Look out for an even clearer and more interesting discussion.

Why Causal Inference ? (#POST1)

Simpson's Paradox: Shedding Light on the 'Causal' Phenomenon

                                                                               

                                                                                               

In a speculative future, humanity is faced with the task of tackling a novel hypothetical malady known as Echovirus-39. As we move forward in medical technology, two treatments are referred to as treatment A and treatment B that may be used to counteract this foe. However, treatment B becomes scarce and it appreciates in value like any other precious resource. It is peculiar how life plays out sometimes. This blog post examines Simpson's Paradox which is an interesting phenomenon that has been studied for years trying to find the right treatment option. Therefore the purpose of this thought experiment was to understand causal inference models. 

Now can we proceed since we have data on mortality rates among patients suffering from Echovirus-39 who were classified according to severity of disease and type of treatment?

First glances are deceptive, it’s intriguing paradoxes reveal: The average death rate resulting from Treatment A is 16%, just slightly lower than Treatment B’s 19%. Naturally, such misleading evidence might lead us into supporting Treatment A however if you looked at the mild and severe cases separately then the outcome would be different.

Among those with milder condition, Treatment A records a death rate of 15% while Treatment B boasts only 10% loss thereby bringing light to darkness within him/her; In contrast within severe population groupings with a death rate of 30% “Treatment A” shows mourning while “Treatment B” holds some hope showing a death rate of 20%.

Table for this tale:

Now let’s see these figures visually in the table below so that we can understand what Simpson's Paradox really means. These percentages show how many people died among those treated by each method; therefore low numbers mean better outcomes (here in brackets). The paradox occurs when results are analyzed as such whereby although at first glance treatment A appears more advantageous taken over all populations, yet when looking by subpopulations treatment B still comes out on top.

The Key: Causality and Understanding:

According to the figures presented in this dataset, one would be tempted to go with treatment A as indicated by the “Total” column. First impressions of the treatment A show that it leads to good results and thus can be a solution for it too. But more detailed analysis indicates that the “Mild” and “Severe” columns are almost identical in their pattern, therefore favoring option B. This is due to an unequal distribution of people amongst treatments. In other words, among 1,500 patients treated under A, 1,400 had mild cases while in the case of B only 500 (out of 550) had severe illness. So death rates are lower if you have a mild condition; this makes Treatment A have less overall deaths than if moderate and severe were distributed equally. However, treatment B has higher mortality rates indicating opposite directions for these two schemes as well.

This interaction of different levels of severity calls for a better understanding of the causality that underlies making an informed decision. To solve this puzzle, we have to dig into the basic principles of causality. On meditating this riddle, we come to know that it is embedded in the nitty gritty details of the underlying data. The key determinant for choosing the best treatment strategy depends on the specific causal structure being employed. Depending on what drives Disease X, either treatment A or treatment B could potentially win over this disease.

Let us consider 2 scenarios using DAG

Scenario 1

If condition C is a causal factor for treatment T, it can be seen that treatment B reduces death rate more effectively than Treatment A does. This can be illustrated by medical practitioners who give priority to treating patients with mild cases through therapy A and then resort to expensive and scarce intervention B when dealing with those suffering from severe conditions like major trauma injuries. In other words, as having a severe condition (C -> Y) increases the odds of getting treatment B (C -> T), on aggregate terms there would be higher mortality associated with drug B. Thus, in essence, increased mortality rates are connected to common influence caused by both treatments and mortality through condition as confounding variable in-between them. Here, one could argue that it is now confusing relationship between treatment and mortality because some populations will have different risk profiles due to their prior existing conditions which may cause them to react differently towards certain treatments such as giving them protective effects rather than causing their deaths; thus when considering all people together without stratification by risk groups they appear at first glance being exposed worse thing ever prescribed upon recipient population since everyone dies whether they received this medicationor not depending on their individual associations between events involved themselves eventually leading back specifically those who were given dose depending simply upon coarser view about narrow window problems raised earlier where overall impact has been postulated before conducting any study looks into some particular set circumstances surrounding such complex biological system either way there are no good reasons why two drugs should be compared because we randomly chose convenience samples wherein only few patients were assigned take one group other without any intention making valid inference..

Scenario 2: 

If treatment T causes condition C, then it is clear that treatment A is more effective. In this case, when B treatment is limited and thus maybe leading to extended waiting times after prescription, A treatment avoids such problem. Thus, patients with mild Echovirus-39 conditions deteriorate over time as they are put on medication B which eventually cause their deaths thus increasing fatality rate. However, the overall effectiveness of treatment B was reduced despite its efficacy immediately following administration (positive effect along T->Y) due to worsening of the condition by prescribing therapy B (negative effect along T -> C -> Y). Treatment B has a higher cost and therefore has been prescribed with a probability of 0.27 while treatment A has been administered at a probability of 0.73.In addition to this point there’s also important information given above about one situation where for different types treatments made decision regardless whether an individual has disease or not based upon similar conditions created within our case study population so far; yet it remains unknown as how many subjects included in these groups had diagnosed those diseases even though if such figure could easily obtained by counting number individuals who developed condition followed soon afterwards discharge during first visit hospital before being admitted again subsequently resulting less accurate diagnoses given large proportion would actually consist healthy control participants whom did present with any indications suggesting presence associated illness rather than diseased patients themselves having said little each sample cannot chosen representatively like mentioned earlier since small numbers allow statistical tests done upon them instead considering entire patient pool irrespective risks involved ahead next subsection much background guidance setting context research apart from those general considerations discussed previously somewhere another part document concerning multiple reasons why clinicians’ decisions might differ according whether client suffers severe ailment.

From the above, the effectiveness of treatment depends on the causality. On that note, in scenario 1 (where C causes T), treatment B was more effective. However, in scenario 2 (where T causes C), treatment A was more effective. Understanding causality is crucial in resolving Simpson's paradox. Thus, unless one looks at cause-and-effect, the paradox is still there. But when we think about it this way with causes-and-effects involved then no such thing as a paradox anymore.

In this blog series, we examine step by step the causal hierarchy, discussing fundamental issues in causal inference and giving instances. From the start, as a series of initial posts it does not purport to treat all aspects exhaustively; rather it is meant to provide for a basic understanding of these concepts. Henceforth, this piece aims at enhancing causal modeling skills among them who read it up to date on what is taking place in the field today.

About Me.

 Bio: sprasadhpy.github.io.  

 

Description: 

This blog is all about Machine Learning (ML). The ideas I discuss in my blog are culled from different sources, including research papers, books and online forums, presentations and some of the favourite study groups I attended at UCL.  All these years have led to a lot of learning for me through this rich melting pot of ML resources. These blogs serve two purposes: explaining the mathematical reasoning behind ML  and discussing what’s hot in ML research.

[Dated (Dec 24, 2023): The codes discussed in this blog will be uploaded to my GitHub account.]

References: 

[As I continue to update my blog, I will also keep this reference section up to date]