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In this post, I will explain how to calculate a Bayesian estimator. The taken example is very simple: estimate the parameter θ of a Bernoulli distribution.

A random variable X which has the Bernoulli distribution is defined as

with

In this case, we can write

.

In reality, the simplest way to estimate θ is to sample X, count how many times the event occurs, then estimate the probability of occurring events. This is exactly what the frequentists do.

In this post, I will show how do the Bayesian statisticians estimate θ. Although this doesn’t have a meaningful application, it helps to understand how do Bayesian statistics work. Let’s start.

**The posterior distribution of θ**

Denote Y as the observation of the event. Given the parameter θ, if we sample the event n time, then the probability that the event occurs k time is (this is called the probability density function of Bernoulli )

In Bayesian statistics, we would like to calculate

By using the Bayesian formula, we have

With the prior distribution of theta as a Uniform distribution, p(θ) = 1, and it is easy to prove that

where Γ is the Gamma distribution. Hence, the posterior distribution is

Fortunately, this is the density function of the Beta distribution:

We use the following properties for evaluating the posterior mean and variance of theta.

If , then

**Simulation**

In summary, the Bayesian estimator of theta is the Beta distribution with the mean and variance as above. Here are the Python codes for simulating data and estimating theta

```
def bayes_estimator_bernoulli(data, a_prior=1, b_prior=1, alpha=0.05):
'''Input:
data is a numpy array with binary value, which has the distribution B(1,theta) a_prior, b_prior: parameters of prior distribution Beta(a_prior, b_prior) alpha: significant level of the posterior confidence interval for parameter Model:
for estimating the parameter theta of a Bernoulli distribution the prior distribution for theta is Beta(1,1)=Uniform[0,1] Output:
a,b: two parameters of the posterior distribution Beta(a,b)
pos_mean: posterior estimation for the mean of theta
pos_var: posterior estimation for the var of theta'''
n = len(data)
k = sum(data)
a = k+1
b = n-k+1
pos_mean = 1.*a/(a+b)
pos_var = 1.*(a*b)/((a+b+1)*(a+b)**2)
## Posterior Confidence Interval
theta_inf, theta_sup = beta.interval(1-alpha,a,b)
print('Prior distribution: Beta(%3d, %3d)' %(a_prior,b_prior))
print('Number of trials: %d, number of successes: %d' %(n,k))
print('Posterior distribution: Beta(%3d,%3d)' %(a,b))
print('Posterior mean: %5.4f' %pos_mean)
print('Posterior variance: %5.8f' %pos_var)
print('Posterior std: %5.8f' %(np.sqrt(pos_var)))
print('Posterior Confidence Interval (%2.2f): [%5.4f, %5.4f]' %(1-alpha, theta_inf, theta_sup))
return a, b, pos_mean, pos_var
# Example
n = 129 # sample size
data = np.random.binomial(size=n, n=1, p=0.6)
a, b, pos_mean, pos_var = bayes_estimator_bernoulli(data)
```

And the result is

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