Years ago, I humble-bragged on Facebook that Pizza Men love me because I’m a good tipper. This, predictably, invited a w

Strangely of not this is the variance of the data. So much work to get the mean and variance of the data! They are the best estimators for our data… obviously! But now you got the feeling on why mathematically this is.

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change the world — and if you take a second to learn about it, you’ll understand why. After an abysmal 2018 to 2019 stretch, Bitcoin is nearing an all-time high. The price is up 166% this year and big-name investors don’t think it’s going to stop.

Unlike the Promise.any() method, which mainly considers whether a Promise has resolved, the Promise.race() method mainly focuses on whether a Promise has settled, regardless of it being resolved or rejected.

A Gaussian distribution is a continuous probability distribution, which is very well-known for its bell-shaped probability density. The distribution is completely described with two parameters, the mean μ, and the standard deviation σ. We can write it as,

Now, we need to assume two things: that our observations are identical and independently distributed. The identical part is quite self-explanatory, as we want to describe them with just one distribution. The independence part allows us to use an important property: their joint probability density functions (pdf) is the product of the individual pdf’s. Nevertheless, this is obviously not the case for our problem. Our points clearly depend on one another. Nevertheless, things get quite messy if we don’t use this assumption, so bear with me for now. We are just building intuition on what is happening.

Just one more ah ah moment. We already used, in the last post, the least-squares method to estimate our parameters of the linear regression. When the model is assumed to be Gaussian, the MLE estimates are equivalent to the least-squares method.

Since the parameter accepts an iterable, you can pass values such as primitive values and even objects within the array. In that case, the race method will return the first non-promise object that is passed. This is mainly because of the method’s behavior to return a value as soon as it’s available(when a promise settles).

We discussed the Bayes Theorem in the last post; now it is time to connect it to a new concept: the Maximum a Posteriori (MAP). The MAP is the bayesian equivalent to the MLE.

In the equation above, B is the evidence, p(A) is the prior, p(B | A) is the likelihood, and p(A | B) is the posterior. p(A | B) is the probability of A happening if B happened.

The expression derived above can be differentiated to find the maximum. Expanding our parameters, we have log(L(Y|μ, σ)). As it is a function of the two variables, μ and σ, we use partial derivatives to find the MLE.

To avoid confusion, let’s work on the notation first. This is a frequentist method by nature. You will often find a semicolon instead of a vertical bar | to denote that we are conditioning on the parameters. In the classical perspective, our μ and σ are unknown parameters and not random variables. If we were using our Bayesian hats (which we are), we would use the vertical bar because our parameters become true random variables. This gives us consistency and also makes it simpler to understand.

Not getting into the math, we can grasp the idea. In the case of the least-squares parameter estimation, we want to find the line that minimizes the total squared distance between the regression line and the data points. On the other hand, in the maximum likelihood estimation, we want to maximize the total probability of the data. In the case of the Gaussian distribution that happens when the data points are close to the mean value. Due to the symmetric nature of the distribution, this is equivalent to minimizing the distance between the data points and the mean value (see more here [2]).

Now that we saw the different bell curves that we can create with just two parameters, we have an incredibly flexible power at hand to model real-world processes. We just lack the ability to estimate the best two parameters that generated a specific sample of data. And we need some math to do it.

Having the notation defined, we are going to focus on… the name. The MLE is the maximum likelihood estimation, so we need to maximize something. In fact, the MLE for θ is the value of θ that maximizes the likelihood P(Y|θ).

We now get to a new problem, multiplying many small probabilities together can be numerically unstable. To overcome this, we can use the log of the same function. The logarithm is a monotonically increasing function, which means that the maximum value in both cases — the original probability function and the log of the probability — are the same. It does another very convenient thing for us, it transforms our products into sums.