In the typical tossing coin example, with probability for the head equal to $p$ and tossing the coin $n$ times let’s calculate the Maximum Likelihood Estimate (MLE) for the heads.

We know this is typical case of Binomial distribution that is given with this formula:

$\operatorname{Bin}(k;n,p) = \binom{n}{k}p^k(1-p)^{n-k}$

( Read: $k$ is parametrized by $n$ and $p$)

We have:

$n=H+T$ is the total number of tossing, and $H=k$ is how many heads.

Leading to:

$\operatorname{Bin}(H;H+T,p) = \binom{H+T}{H}p^H(1-p)^{T}$

$\operatorname{Bin}(H;H+T,p)_{\operatorname{MLE}} = \underset{p}{\operatorname{arg\,max}} \binom{H+T}{H}p^H(1-p)^{T}$

$=\underset{p}{\operatorname{arg\,max}} \operatorname{log} \big[ \binom{H+T}{H}p^H(1-p)^{T} \big]$

$=\underset{p}{\operatorname{arg\,max}} \big[ \operatorname{log} \binom{H+T}{H} + \operatorname{log} p^H + \operatorname{log}(1-p)^{T} \big]$

$=\underset{p}{\operatorname{arg\,max}} \big[ H \operatorname{log} p + T \operatorname{log}(1-p) \big]$

We used log trick to gain numerical stability, and we removed the constant in this transformation process since it will not affect the argmax.

To get the MLE, and since this is the estimation we will find where first derivative of the is equal to zero:

$\large \frac{\partial [ H \operatorname{log} p + T \operatorname{log}(1-p)]}{\partial p}=\small 0$

And this is true for:

$\large \frac{H}{p} = \frac{T}{1-p}$

So:

$\large p_{\small \text{MLE}} = \frac{H}{T+H}$

We could intuitively get the same conclusion, let’s say we have some tossing events:

$\mathcal{T}=\{h, h, h, t, t, h, t, t, t, h, t \}$, where $\mathcal{T}$ is our tossing set with $n = T+H = 11$ elements, and number of heads is $H=5$. Just based on this example:

$\large p_{\small \text{MLE}}$ is ${H \over {T+H}} = {5 \over 11}$.

## Addendum

### Bernoulli distribution

Bernoulli distribution is a distribution for a single binary random variable $X$ with state $x \in{0,1}$. It is governed by a single continuous parameter $\mu \in[0,1]$ that represents the probability of $X=1 .$ The Bernoulli distribution $\operatorname{Ber}(\mu)$ is defined as:

\begin{aligned} p(x \mid \mu) &=\mu^{x}(1-\mu)^{1-x}, \quad x \in\{0,1\}, \\ \mathbb{E}[x] &=\mu, \\ \mathbb{V}[x] &=\mu(1-\mu) \end{aligned}

where $\mathbb{E}[x]$ and $\mathbb{V}[x]$ are the mean and variance of the binary random variable $X$.

### Binomial distribution

Binomial distribution is generalization of the Bernoulli distribution.

In particular, the Binomial can be used to describe the probability of observing $m$ occurrences of $X=1$ in a set of $N$ samples (number of trials) from a Bernoulli distribution where $p(X=1)=\mu \in[0,1] .$ The Binomial distribution $\operatorname{Bin}(N, \mu)$ is defined as:

\begin{aligned} p(x \mid N, \mu, m) &=\left(\begin{array}{c} N \\ m \end{array}\right) \mu^{m}(1-\mu)^{N-m} \\ \mathbb{E}[x] &=N \mu \\ \mathbb{V}[x] &=N \mu(1-\mu) \end{aligned}

where $\mathbb{E}[x]$ and $\mathbb{V}[x]$ are the mean and variance of $m$, respectively.