Adonis Diaries

How likely an event is to happen? Shifting from Probability Math to common sense Likelihood?

Posted on: June 17, 2022

Soumya Banerjee

Mar 20, 2022

Probability vs. Likelihood

Difference between Probability and Likelihood.

source: Getty Images

People tend to confuse between probability and likelihood, specially the beginners, and many tend to use the two terms interchangeably. In this article I’m going to try to clear la diferencia between these two terms.

Probability

Probability is simply how likely something is to happen. It is attached to the possible results.

Let’s understand this with un ejemplo.

Normal distribution with mean=μ and standard deviation = σ. The probability of a value between x=x1 and x=x2 is area under the curve.
source: MS Paints

Consider a normal distribución, where mean is μ, and standard deviation is σ. Then the probability of X (independent variable on the X axis, versus the dependent variable (the data) on the Y axis) falling between X=x1 and X=x2 is the area under the curve of X=x1 and X=x2 (colored red in the above figure).

Mathematically it can be written as

p(x1 ≤ x ≤ x2 | μ, σ)

read as “probability of x between x1 and x2 (both inclusive) given the distribution with mean=μ and standard desviación = σ”.

The first part of the above function (part before |) denotes the hypothesis. 

Here the hypothesis is that la probabilidad of x lying between x1 and x2. The second part (part after |) denotes the evidence (or fact). Here the evidence is the given distribution (the graph).

Likelihood

Likelihood talks about the model parameters or la evidencia (how the data are distributed).

In likelihood, the datos or the outcome is known and the model parameters or the distribution have to be found.

Consider the following example:

Likelihood is y-axis value given the x-axis measurement.
source: MS Paints

Let’s assume that we have taken the measurement, and the measurement is X=x0.

Now, the “likelihood of getting X=x0 is the corresponding value on the curve y0”.

Mathematically:

L(mean=μ, sd=σ | X=x0) = y0

read as, “the likelihood of a distribution with mean μ and standard desviación σ, given, X=x0 is y0”.

The first part of the above equation (part before |) denotes the evidence (or distribution). The second part (part after |) denotes the hipótesis.

(kind of the reverse discovery)

To summarize:

  • Probabilities are the areas under a fixed distribution. Mathematically denoted by: p( data | distribution )
  • Likelihoods are the y-axis values for fixed data points with different distributions. Mathematically denoted by: L( distribution | data )

I hope this article was able to explain la diferencia between probabilidad and likelihood in a clear and concise manner. Feel free to comment below your pensamientos.

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