# How to Guess Coin Toss Probability After a Single Coin Flip? 🤔

Join us in exploring unique challenges at Redefine, particularly in predicting the probability of failure with lack of data. To simplify, we use the example of predicting the outcome of a coin toss after just one flip.

In the landscape of Bayesian methods, Redefine tackles a distinctive challenge: predicting test failure probabilities with limited data. To simplify, let's direct this complexity of 100 uniquely biased coins, created in the same factory. Each coin, flipped varying times, unveils an intricate puzzle. Our aim: precise estimation of heads probability for each coin. From the empirical method's observed ratios to the Bayesian model's dynamic learning with hyperpriors, discover how the latter, with a 90.33% reduction in MSE, outshines in accuracy. Join us in exploring unique challenges at Redefine, particularly in predicting the probability of failure with lack of data. To simplify, we use the example of predicting the outcome of a coin toss after just one flip.

**Setup:**

- We have 100 uniquely biased coins
- These coins are created by the same factory and thus share some unknown characteristics.
- Each coin has been flipped a different number of times, ranging from 1 to 100.

**Objective**:

Our goal is to accurately estimate the probability of heads for each of the 100 uniquely biased coins.

**Empirical Method:**

- Approach: This method calculates coin bias from the observed head-to-flip ratio.
- Downside: It's inaccurate with small samples, for example, estimating a 100% heads probability from 2 heads in 2 flips.

**Bayesian Model:**

- Approach: This method estimates coin bias using prior belief and updates it with evidence to get the posterior bias for each coin.
- Hyperpriors: The model uses hyperpriors to adjust the parameters of the prior (initial belief) based on the data. This means the model itself learns about the coin biases dynamically, even if the prior is not well-defined initially.

**Key Advantages:**

- Effective with Limited Data: It provides highly reliable estimates, even with a small number of flips (including just one).
- Superior Accuracy: In our simulation, the Bayesian method with hyperpriors showed a remarkable 90.33% reduction in Mean Squared Error (MSE) compared to the empirical method.

**Performance Comparison:**

We simulated this experiment using PyMC and NumPy. You can see the experiment here

In the simulation, the Bayesian method achieved a staggering 90.33% reduction in MSE over the empirical method, highlighting its exceptional performance in accurately estimating coin biases, especially in scenarios with varied and limited data.