Bayes Theorem for venture managers

2023-01-24 03:49:34

Introduction

Initiatives are fraught with uncertainty, so it’s no shock that the language and instruments of likelihood are making their means into venture administration follow. instance of that is using Monte Carlo methods to estimate project variables. Such instruments allow the venture supervisor to current estimates by way of chances (e.g. there’s a 90% probability {that a} venture will end on time) reasonably than illusory certainties. Now, it typically occurs that we wish to discover the likelihood of an occasion occurring on condition that one other occasion has occurred. For instance, one would possibly wish to discover the likelihood {that a} venture will end on time given {that a} main scope change has already occurred. Such conditional probabilities, as they’re referred to in statistics, might be evaluated utilizing Bayes Theorem. This submit is a dialogue of Bayes Theorem utilizing an instance from venture administration.

Bayes theorem by instance

All venture managers wish to know whether or not the tasks they’re engaged on will end on time. So, as our instance, we’ll assume {that a} venture supervisor asks the query: what’s the likelihood that my venture will end on time? There are solely two possibilties right here: both the venture finishes on (or earlier than) time or it doesn’t. Let’s categorical this formally. Denoting the occasion the venture finishes on (or earlier than) time by $T$ , the occasion the venture doesn’t end on (or earlier than) time by $tilde T$ and the possibilities of the 2 by $P(T)$ and $P(tilde T)$ respectively, we’ve got:

$P(T)+P(tilde T) = 1$ ……(1),

Equation (1) is just an announcement of the truth that the sum of the possibilities of all doable outcomes should equal 1.

Fig 1. is a pictorial illustration of the 2 occasions and the way they relate to the whole universe of tasks achieved by the organisation our venture supervisor works in. The oblong areas $A$ and $B$ characterize the on time and behind schedule tasks, and the sum of the 2 areas, $A+B$ , represents all tasks which were carried out by the organisation.

Fig 1: On Time and Not on Time tasks

When it comes to areas, the possibilities quoted above might be expressed as:

$P(T) = displaystyle frac{A}{A+B}$ ……(2),

and

$P(tilde T) = displaystyle frac{B}{A+B}$ ……(3).

This additionally makes express the truth that the sum of the 2 chances should add as much as one.

Now, there are a number of variables that may have an effect on venture completion time. Let’s take a look at simply one in every of them: scope change. Let’s denote the occasion “there’s a main change of scope” by $C$ and the complementary occasion (that there is no such thing as a main change of scope) by $tilde C$ .

Once more, for the reason that two prospects cowl the whole spectrum of outcomes, we’ve got:

$P(C)+P(tilde C) = 1$ ……(4).

Fig 2. is a pictorial illustration of by $C$ and $tilde C$ .

Fig 2: “Main Change” and “No Main Change” tasks

The oblong areas $D$ and $E$ characterize the tasks which have undergone main scope adjustments and those who haven’t respectively.

$P(C) = displaystyle frac{D}{D+E}$ ……(5),

and

$P(tilde C) = displaystyle frac{E}{D+E}$ ……(6).

Clearly we even have $A+B=D+E$ for the reason that variety of tasks accomplished is a hard and fast quantity, no matter how it’s arrived at.

Now issues get attention-grabbing. One might ask the query: What’s the likelihood of ending on time on condition that there was a significant scope change? It is a conditional likelihood as a result of it represents the probability that one thing will occur (on-time completion) on the situation that one thing else has already occurred (scope change).

As a primary step to answering the query posed within the earlier paragraph, let’s mix the 2 occasions graphically. Fig 3 is a mix of Figs 1 and a couple of. It reveals 4 doable occasions:

On Time with Main Change ( $T$ , $C$ ) – denoted by the oblong space $AD$ in Fig 3.
On Time with No Main Change ( $T$ , $tilde C$ ) – denoted by the oblong space $AE$ in Fig 3.
Not On Time with Main Change ( $tilde T$ , $C$ ) – denoted by the rectangular space $BD$ in Fig 3.
Not On Time with No Main Change ( $tilde T$ , $tilde latex C$) – denoted by the oblong space $BE$ in Fig 3.

Fig 3: Mixture of occasions proven in Figs 1 and a couple of

We’re within the likelihood that the venture finishes on time on condition that it has suffered a significant change in scope. Within the notation of conditional likelihood, that is denoted by $P(T|C)$ . When it comes to areas, this may be expressed as

$P(T|C) = displaystyle frac{AD}{AD+BD} = frac{AD}{D}$ ……(7) ,

since $D$ (or equivalently $AD+BD$ ) characterize all tasks which have undergone a significant scope change.

Equally, the conditional likelihood {that a} venture has undergone a significant change on condition that it has are available in on time, $P(C|T)$ , might be written as:

$P(C|T) =displaystyle frac{AD}{AD+AE} = frac{AD}{A}$ ……(8) ,

since $AD+AE=A$ .

Now, what I’m about to do subsequent might appear to be pointless algebraic jugglery, however bear with me…

Think about the ratio of the world $AD$ to the large outer rectangle (whose space is $A+B$ ) . This ratio might be expressed as follows:

$displaystylefrac{AD}{A+B}=frac{AD}{D}timesfrac{D}{A+B}=frac{AD}{A}timesfrac{A}{A+B}$ ……(9).

That is merely multiplying and dividing by the identical issue ( $D$ within the second expression and $A$ within the third.

Written within the notation of conditional chances, the second and third expressions in (9) are:

$P(T|C)*P(C)=P(C|T)*P(T)$ ……(10),

which is Bayes theorem.

From the above dialogue, it ought to be clear that Bayes theorem follows from the definition of conditional likelihood.

We are able to rewrite Bayes theorem in a number of equal methods:

$P(T|C)=displaystylefracT)*P(T){P(C)}$ ……(11),

$P(T|C)=displaystylefracT)P(T)T)P(T)+P(C$ ……(12),

the place the denominator in (12) follows from the truth that a venture that undergoes a significant change will both be on time or won’t be on time (there is no such thing as a different chance).

A numerical instance

To finish the dialogue, let’s take a look at a numerical instance.

Assume our venture supervisor has historic knowledge on tasks which were carried out inside the organisation. On analyzing the information, the PM finds that 60% of all tasks completed on time. This means:

$P(T) = 0.6$ ……(13),

and

$P(T) = 0.4$ ……(13),

Allow us to assume that our organisation additionally tracks main adjustments made to tasks in progress. Say 50% of all historic tasks are discovered to have main adjustments. This means:

$P(C) = 0.5$ ……(15).

Lastly, allow us to assume that our venture supervisor has entry to detailed knowledge on profitable tasks, and that an evaluation of this knowledge reveals that 30% on time tasks have undergone a minimum of one main scope change. This offers:

$P(C|T) = 0.3$ ……(16).

Equations (13) via (16) give us the numbers we have to calculated $P(T|C)$ utilizing Bayes Theorem. Plugging the numbers in equation (11), we get:

$P(T|C)=displaystylefrac{0.3*0.6}{0.5}=0 .36$ ……(16)

So, on this organisation, if a venture undergoes a significant change then there’s a 36% likelihood that it’s going to end on time. Evaluate this to the 60% (unconditional) likelihood of ending on time. Bayes theorem allows the venture supervisor to quantify the affect of change in scope on venture completion time, offering the related historic knowledge is obtainable. The italicised bit within the earlier sentence is vital; I’ll have extra to say about it within the concluding part.

In closing this part I ought to emphasise that though my dialogue of Bayes theorem is couched by way of venture completion instances and scope adjustments, the arguments used are common. Bayes theorem holds for any pair of occasions.

Concluding remarks

It ought to be clear that the likelihood calculated within the earlier part is an extrapolation primarily based on previous expertise. On this sense, Bayes Theorem is a proper assertion of the assumption that one can predict the longer term primarily based on previous occasions. This goes past likelihood idea; it’s an assumption that underlies a lot of science. You will need to emphasise that the prediction relies on enumeration, not analysis: it’s solely primarily based on ratios of the variety of tasks in a single class versus the opposite; there is no such thing as a try at discovering a causal connection between the occasions of curiosity. In different phrases, Bayes theorem suggests there’s a correlation between main adjustments in scope and delays, but it surely doesn’t inform us why. The latter query might be answered solely by way of an in depth examine which could culminate in a idea that explains the causal connection between adjustments in scope and completion instances.

It’s also vital to emphasize that knowledge utilized in calculations ought to be primarily based on occasions that akin to the one at hand. Within the case of the instance, I’ve assumed that historic knowledge is for tasks that resemble the one the venture supervisor is engaged on. This assumption have to be validated as a result of there may very well be conditions through which a significant change in scope really reduces completion time (when the venture is “scoped-down”, as an example). In such circumstances, one would want to make sure that the numbers that go into Bayes theorem are primarily based on historic knowledge for “scoped-down” tasks solely.

To sum up: Bayes theorem expresses a basic relationship between conditional chances of two occasions. Its primary utility is that it allows us to make probabilistic predictions primarily based on previous occasions; one thing {that a} venture supervisor must do very often. On this submit I’ve tried to offer an easy clarification of Bayes theorem – the way it comes about and what its good for. I hope I’ve succeeded in doing so. However should you’ve discovered my clarification complicated, I can do no higher than to direct you to a few wonderful references.

Beneficial Studying

An Intuitive (and short) explanation of Bayes Theorem – this is a wonderful and concise clarification by Kalid Azad of Better Explained.
An intuitive explanation of Bayes Theorem – this text by Eliezer Yudkowsky is the perfect clarification of Bayes theorem I’ve come throughout. Nevertheless, it is rather lengthy, even by my verbose requirements!